Topology Seminar

University of Kentucky Topology Seminar

We meet in POT 745 (or sometimes on Zoom) at 3:30PM on Thursdays

There is also a google calendar for this seminar. It usually contains a more up to date schedule of speakers, but less complete information about the talks.
Double check seminar expectations.

January 11 Organizational Meeting
January 18 No meeting
January 25 Millie Rose Functoriality of the abstract marks homomorphism
The Burnside ring of a finite group is one of the most fundamental objects in equivariant homotopy theory, and the marks homomorphism is a powerful way of studying it. The Burnside ring can be generalized from groups to orbit categories, and the marks homomorphism can be generalized in the presence of a finiteness condition. A few examples of these abstract Burnside rings have appeared in the work of the bourbon seminar. In this talk, I will define and describe the abstract marks homomorphism. In particular, I will provide sufficient conditions for a map on the Burnside ring to extend to marks and vice versa.
February 1 Lakshay Modi Presheaves and Day Convolution
The Day convolution theorem from category theory was used to resolve the long-standing problem of constructing a smash product of spectra. In this talk, I will build up the necessary categorical machinery to define the Day convolution product and give a sketch of the construction of the smash product.
February 8 Sarah Petersen (CU Boulder) (Zoom talk) Brown-Gitler Spectra: An introduction & current directions
Brown-Gitler spectra have many uses in homotopy theory, largely due to the fact that they realize finite pieces of the dual Steenrod algebra, which a key input for computing stable homotopy. However, Brown-Gitler spectra were not originally constructed with homotopical applications in mind. Instead, Brown and Gitler first constructed these spectra to aid in studying the immersion conjecture for differentiable manifolds. This talk will provide an overview of homotopical and geometric motivations for constructing Brown-Gitler spectra. We will describe four different constructions due to a range of authors, and discuss how these Brown-Gitler spectra are used in homotopy computations. These descriptions will motivate a sketch of current research directions in equivariant and motivic homotopy theory, and culminate in the statement of new results from work in progress with Guchuan Li and Elizabeth Tatum.
February 15 David Mehrle Topological Hochschild homology and trace methods
Topological Hochschild homology and its relatives (eg. topological cyclic homology) play a huge role in modern homotopy theory. The key structure that makes these objects so useful is the trace map from algebraic K-theory to topological Hochschild homology. In this talk, I'll introduce topological Hochschild homology and trace methods and give some applications.
February 22 no meeting
February 29 Jamin Kochman Profunctors
In my qualifying talk, I was asked if there are any connections between the Möbius function and Kate's work with Traces. Profunctors seem like a promising avenue to pursue such a connection. In this talk I will justify this line of inquiry, define the profunctor bicategory, and, time permitting, show some examples I've been thinking about.
March 7 Nate Cornelius Bordism and the MO spectrum
Closed real manifolds are an important class of spaces, useful in many areas like geometric topology and homotopy theory. But they turn out to be quite difficult to classify by homeomorphism. So we instead rely on a weaker notion of equivalence called Bordism. Work by Pontryagin and Thom showed that this relation induces a natural algebraic structure which is related to the stable homotopy groups of a particular ring spectrum, MO. During this talk, I will define Bordism, construct MO, and showcase some interesting facts about these stable homotopy groups.
March 21 Jesse Keyes The Tate Method and C2-Equivariant Eilenberg-MacLane Spectra
Isotropy separation theory gives rise to a way to break a G-spectrum into computationally easier pieces. Using this to compute the homotopy of some G-spectrum is known as the Tate method. In this talk, I'll showcase the Tate method as a computational technique by using it to compute the RO(C2)-graded homotopy of any Eilenberg-MacLane C2-Spectrum. These types of equivariant spectra are the starting inputs for slice spectral sequences, another major computational tool of the field.
March 28 Evan Franchere Multicategories Encoding Algebraic Structure
Multicategories turn out to be very useful for encoding other algebraic structures. By giving a multicategorical description of permutative categories and Segal's K-theory, Elmendorf and Mandell are able to leverage the power of multicategories to obtain algebraic structures on symmetric spectra. During this talk, I will explain the results of Elmendorf and Mandell, starting with the definition of multicategories and ending with examples of where they obtain algebraic structures.
April 4 Travis Wheeler Bicategorical Character Theory
In 2007, Nora Ganter and Mikhail Kapranov defined the categorical trace, which they used to define a character theory for 2-representations. In 2008, Kate Ponto defined a shadow functor for bicategories. With the shadow functor, Dr. Ponto defined the bicategorical trace, which is a generalization of the symmetric monoidal trace for bicategories. How are these two notions of trace related to one another? We've used bicategorical traces to define a character theory for 2-representations, and the categorical character is an example. In this talk, we'll define our character theory and produce the categorical character as an example.
April 11 Maximilien Peroux (Michigan State) Delooping trace methods in algebraic K-theory
The Dennis trace on a ring R connects the algebraic K-theory of R with its topological Hochschild homology THH(R), and is an essential resource for accessing computations in algebraic K-theory. In this talk, we present a Koszul dual of the Dennis trace. Just as any grouplike monoids M in spaces can be `delooped' into a connected space BM, any ring spectrum R can be `delooped' to a coalgebra spectrum BR. Topological coHochschild homology (coTHH) is the analogue invariant of THH for coalgebra spectra introduced by Hess-Shipley. We introduce new algebraic K-theories for coalgebras and present a program on how to connect these K-theories with coTHH, extending the Dennis trace in the coalgebraic setting. I will present joint work with Sarah Klanderman where we build explicitly coalgebraic refinements of the Hattori-Stallings that defines the Dennis trace on level zero. We show that coHochschild homology is a bicategorical shadow on bicomodules, and the challenges one the coalgebraic settings. The delooping increases the connectivity on our conditions with the trade off to be working with coalgebras.
April 16 Joshua Peterson The Stripping Process in the Classical and Motivic Steenrod Algebra
The Steenrod algebra A has a presentation in which the relations are the Adem relations among products of Steenrod squares. Using these relations, we may write the product of elements in A in terms of a certain basis. Rather than using these unwieldy relations directly, a process known as stripping gives us an easier way to carry out an equivalent computation. We will introduce the stripping process in the classical setting first, then generalize to the motivic setting.
April 18 Lewis Dominguez Adams operations on the Burnside ring from power operations
Topology furnishes us with many commutative rings associated to finite groups. These include the complex representation ring, the Burnside ring, and the G-equivariant K- theory of a space. Often, these admit additional structure in the form of natural operations on the ring, such as power operations, symmetric powers, and Adams operations. We will discuss two ways of constructing Adams operations. The goal of the talk is to understand these in the case of the Burnside ring.
April 25 Kiran Luecke (UIUC) Thom classes, twists, and transchromatic maps in equivariant elliptic cohomology
I will discuss some constructions of global equivariant elliptic cohomology at the Tate curve, as well as the (twisted) Thom classes associated to vector bundles. The homotopy classes of those twists relate to familiar characteristic classes as well as the representation theory of loop groups. Finally I will discuss an interesting redshift phenomenon displayed by this theory that will lead us toward finding its location within the chromatic framework of stable homotopy theory.
August 24 Organizational Meeting
August 31 Bert Guillou Dualizing A-modules
The cohomology of any space has an action of the Steenrod algebra A. The A-module structure on the Spanier-Whitehead dual can then be recovered using the antipode on A. The C_2-equivariant Steenrod algebra is not known to have an antipode, but I will discuss how to describe the equivariant A-module structure on a C_2-equivariant Spanier-Whitehead dual. This is joint work with Prasit Bhattacharya and Ang Li.
September 7 David Mehrle Partition Functors
Mackey functors are a ubiquitous structure in equivariant homotopy theory, and we have a lot of tools for working with them. There are also many situations where the data almost, but not quite, fit the pattern of a Mackey functor. In this talk I'll introduce partition functors, a variant on Mackey functors designed to capture combinatorial data with symmetric group action. Time permitting, I'll discuss the close relation between partition functors and bialgebras. This work with Millie Deaton and Nat Stapleton is an offshoot of last year's Bourbon Seminar.
September 14 Millie Rose Lower Semiadditivity
Higher semiadditivity is a connection between certain addition-related properties and structures on higher categories with particular relevance in chromatic homotopy theory. Fortunately, the lower levels of semiadditivity are completely visible at the 1-categorical level and do not require topology to understand. I will spend the majority of the time describing n-semiadditivity in 1-categories for n close to -2. In the vanishingly unlikely situation that I have extra time, I will explain a definition which is valid for all n.
September 21 Nate Cornelius Representations, Fancy Rings, and Divided Power Algebras
Recent work in the Kentucky Bourbon Seminar has led to some interesting exponential formulas for global Green functors. Much of this work was first developed by studying various ring constructions whose underlying pieces are representation rings of symmetric groups. In this talk, we will review some necessary facts about representation theory and then introduce one of these constructions, R[Σ], which has a rather unorthodox multiplication. Afterwards we will relate this ring to another ring, Cl(Σ,Z), constructed from integer-valued class functions on symmetric groups. This class function ring also has an unusual multiplication which is entirely combinatorial. Lastly if time permits, I'll also introduce the concept of a Divided Power Algebra and sketch a proof that Cl(Σ,Z) has this property.
September 28 Bridget Schreiner (Notre Dame) Cross effects and stability
A sequence of groups (or spaces) G_1 -> G_2 -> G_3... satisfies homological stability if H_i(G_n)≅ H_i(G_{n+1}) for n>>i. Regarding the sequence G_n as the image of a functor from the category (N,+,0) and noting that stability occurs when H_i(G_{n+1},G_n) ≅ 0 turns this into a statement about "differences" in the image of a functor. This talk will introduce a generalization of the cross effects of Eilenberg and MacLane that is suited to answer questions about homological stability in this context.
October 5 Jamin Kochman Euler Characteristics and Möbius Functions
Euler characteristic is a construction that exists in multiple contexts. It is usually a number which carries some defining information of an object. In graph theory, this can be used to determine whether a graph is planar. And in topology, it has connections to the lefschetz fixed point theorem, for example. Here we consider a generalization to a certain class of categories. In a closely related construction, we also generalize the well-known möbius function. The latter being a helpful tool in calculating the former and presenting exciting properties in its own right. We end with some results justifying generalization as the proper name for this construction.
October 12 Tanner Carawan (Virginia) 2-Segality and the S_\bullet-construction
Waldhausen's S_\bullet-construction gives a way to define the algebraic K-theory space of a category with cofibrations. Specifically, the K-theory space of a category with cofibrations C can be defined as the loop space of the realization of the simplicial topological space |iS_\bullet C |. Dyckerhoff and Kapranov observed that if C is chosen to be a proto-exact category, then this simplicial topological space is 2-Segal. A natural question is then what variants of this S_\bullet-construction give 2-Segal spaces. In this talk, we give the necessary background in this area and discuss work in progress that aims to address the preceding question.
October 19 Jesse Keyes The Bigraded Homotopy of C_2-Equivariant HA
One G-equivariant analogue of graded homotopy rings of ring spectra is multi-graded homotopy Green functors for "genuine" G-ring spectra. In this talk, I'll introduce (co)fiber sequences in both the stable and unstable setting and give an idea of how to compute the RO(C_2)-graded homotopy Green functor of HA, the Eilenberg-MacLane C_2-spectrum for the Burnside Mackey functor, A, for C_2. Time permitting, we will see a method for studying the multiplicative structure of this Green functor.
October 26 Joshua Peterson An Overview of the Peterson Hit Problem in the Classical and Motivic Settings
The ingredients necessary to understand the original formulation of the Peterson hit problem will be established. Hence, Wood's theorem, a powerful yet elementary result in the study of the hit problem, will be introduced. Once this groundwork is laid, we shall consider how the hit problem is affected by moving to the more general motivic setting.
November 2 Jordan Sawdy Chern Characters and Categorified Geometry
The Grothendieck-Riemann-Roch theorem is a major result in algebraic geometry that connects Euler characteristics of algebraic vector bundles with certain characteristic classes defined on them. We'll first look at this theorem and try to understand it through an analogy with smooth vector bundles on manifolds. Then, we'll briefly discuss a recent result of Hoyois, Safronov, Scherotzke, and Sibilla that establishes a categorified version of the GRR theorem in an infinity-categorical framework. Finally, we'll see some work I've done on a version of this categorified GRR theorem in the setting of symmetric monoidal bicategories, which manages to abstract away from some of the more geometric details of the other two.
November 9 Lewis Dominguez Adams Operations on A(G) from power operations
Topology furnishes us with many commutative rings associated to finite groups. These include the complex representation ring, the Burnside ring, and the G-equivariant K-theory of a space. Often, these admit additional structure in the form of natural operations on the ring, such as power operations, symmetric powers, and Adams operations. We will discuss two ways of constructing Adams operations. The goal of the talk is to understand these in the case of the Burnside ring.
November 16 Danika Van Niel (Michigan State) Computational approaches to twisted topological Hochschild homology
Topological Hochschild homology (THH) is an invariant of ring spectra and is a key component of the trace method approach to algebraic K-theory. One of the main computational tools for THH is the Bökstedt spectral sequence. The study of the algebraic structure of THH and the Bökstedt spectral sequence have advanced our computational ability. In recent years, a generalization of THH for equivariant ring spectra called twisted THH has been developed along with an equivariant version of the Bökstedt spectral sequence. In this talk I'll introduce THH, twisted THH, and discuss computations of twisted THH using the equivariant Bökstedt spectral sequence as well as work on the algebraic structure of twisted THH. I will also talk about Mackey functor fields, which are an equivariant analog of classical fields, and how they can be used in computations of twisted THH.
November 30 Travis Wheeler Kan Extensions
In mathematics, we often find ourselves wanting information about a difficult map. One way to do this is by factoring that map as a composition of maps that we have more information about. However, this isn't always possible. In category theory, given two functors $F: C\to E$ and $G:C\to D$, there is a way to approximate a functor (if it exists) from $D\to E$ satisfying a universal property. This is known as a Kan extension. In this talk, I will define Kan extensions and give a few examples and relevant theorems. Time permitting, I'll discuss how this fits in with my research.
December 7 Evan Franchere Infinite Loop Spaces, Omega Spectra, and E-Spaces
Building on the idea of loop spaces, we will use the Brown Representability Theorem to motivate the definition of infinite loop spaces and Ω-spectra. We will then discuss E-spaces and how they can help recognize infinite loops spaces. Finally, time permitting we will present Segal's method of defining E-spaces, as well as a method for constructing them from small permutative categories.
January 12 No Seminar
January 19 No Seminar
January 26 Jordan Sawdy Traces in Symmetric Monoidal Bicategories
One of the most useful facts in the trace theory of symmetric monoidal categories is that taking traces commutes with symmetric monoidal functors. We'll start by discussing traces in general as well as a slight reframing of this result that will be useful for our intended generalization. We'll then take a look at a recent theorem of Hoyois and coauthors that appears to be analogous in certain ways. However, 1-categories will unfortunately turn out to be insufficient to capture this similarity, so we'll have to categorify our original discussion of traces and consider symmetric monoidal bicategories. In this context, I'll present some of my recent work that I hope will eventually pin down this analogy precisely.
February 2 Millie Rose Equivariant homotopy theory by example
Equivariant K-theory is one of the simplest equivariant cohomology theories other than Bredon cohomology. I will develop the restriction and transfer maps for equivariant K-theory and use them to motivate a compact definition of Mackey functors. Time permitting, I will extend to the global equivariant case and obtain a definition of global Mackey functors.
February 9 Lewis Dominguez Factoring the Burnside Adams Operation through the Power Operation
Within representation theory, it's well known that the Adams operation can be factored through the power operation. Similar factorizations exist for some other cohomology theories. It is not known for the Burnside ring. Our primary goal is to describe such a factorization for the Burnside ring in certain cases. We begin by recalling the Burnside ring and these operations, then observing a theorem by Gay, Morris, and Morris which describes the Adams operation, and finally, describing the three major steps of the proof.
February 16 Lucy Yang (Harvard) Categorical dynamics on stable module categories
Topological entropy is a measure of the complexity of a continuous self-map of a compact topological space. Categorical entropy generalizes this measure to exact endofunctors of triangulated categories. In this work we ask: How can categorical entropy serve to quantify growth in stable homotopy theory, and how can the methods of stable homotopy theory aid in understanding and computing categorical entropy? We show that the categorical polynomial entropy of a twist functor on the stable module category of a finite connected graded Hopf algebra A over a field k recovers one less than the Krull dimension of H*(A;k), generalizing a computation of Fan, Fu, and Ouchi. Along the way, we will see how a stable homotopy theoretic-perspective permits us to make this refinement.
February 23 Shahzad Kalloo Chern Classes and Formal Group Laws
Starting from a first calculus course, one learns that one of the most useful ideas in mathematics is the concept of linearization. The concept of a vector bundle extends this concept to topology and allows us the use of linear algebra to produce homotopy invariants of spaces. In particular, when studying the cohomology of vector bundles we find that the cohomology rings are structured with respect to particular cohomology classes, called characteristic classes.

In this talk we will detail two parallel stories of particular characteristic classes: the classical theory of Chern classes in singular cohomology and the theory of generalized Chern classes arising from sufficiently nice cohomology theories. It turns out the theory of generalized Chern classes gives us a way to interpret our cohomology rings as algebro-geometric objects, called formal group laws. We then use this perspective to give an algebro-geometric interpretation of the cohomology ring of BU(n).
March 2 Jack Carlisle (Notre Dame) Complex cobordism and formal group laws in the equivariant setting
Complex cobordism is among the most important generalized cohomology theories that one encounters in homotopy theory. This is because of the deep connection between complex cobordism, complex orientations, and formal group laws. In this talk, we will investigate the G-equivariant analogue of this story, where G is some group. In particular, we will calculate the complex cobordism ring of Z/2-manifolds, and explain how this calculation leads to a new theory of "filtered" Z/2-equivariant formal group laws.

This seminar will be online

March 9 David Mehrle Universal Exponential Formulas
In finite group representation theory, the symmetric or exterior powers and Adams operations are related by means of an exponential formula - a formula reminiscent of the relation between ordinary generating functions and exponential generating functions. This can be viewed as an integral version of the height 1 case of a theorem of Ganter, which asserts an exponential relationship between certain cohomology operations in Morava E-theories at any height. We ask: for which cohomology theories do such exponential formulas exist? In this talk, I will describe a construction whose universal property yields exponential formulas for any global Green functor R that additionally admits sections of the transfer homomorphisms. We apply our construction to recover Ganter's theorem and produce a new exponential formula for E-theory at height 2 and prime 2.
This is joint work with the Kentucky Bourbon Seminar.
March 23 Van Winter Lecture
March 30 Hannah Housden (Vanderbilt) Equivariant Stable Homotopy Theory for Diagrams
A group can be viewed as a category with one object where every morphism is an isomorphism. In this context, a G-space is just a functor from G to Top, the category of topological spaces. This immediately generalizes: for any category D, a D-space is a functor from D to Top. We will explore this generalization, which relies heavily on the theory of "orbits," which generalize G-sets of the form G/H. The latter half of the talk will focus on the stable case and what it would mean to invert a sphere with D-action. Our particular model for D-spectra is spectral Mackey Functors, which admit many constructions one might hope for, such as Eilenberg-MacLane spectra and geometric fixed points.
April 6 Josh Peterson Introduction to the Peterson Hit Problem
The mod 2 Steenrod algebra is a powerful tool in the realm of cohomology. To be precise, it is the algebra of stable cohomology operations for F_2 cohomology. In this talk, I will be discussing the Peterson hit problem, which concerns the action of the Steenrod algebra on the F_2 cohomology ring of a certain collection of spaces. To demonstrate the intricasies of the hit problem, both computational methods and concrete examples will be included throughout.
April 13 Jesse Keyes The Picard Group of the C_n-Equivariant Stable Homotopy Category
G-equivariant cohomology theories are often graded over the real representation ring RO(G), but a more natural choice is to grade over the Picard group of the equivariant stable homotopy category. With this in mind, we will see a computation of the Picard group of the C_n-equivariant stable homotopy category. In particular, we will see that this Picard group splits as a sum of the Picard group of the Burnside ring for C_n and a factor related to dimensions of virtual representations. Further, explicit representatives for these invertible C_n-spectra will be given.
April 20 Jamin Kochman Lost in Trace
Sets, rings, topological spaces, analytic functions... These are all objects a mathematician might study. The goal of category theory is to take a step back, generalizing notions of object and morphism to build results which hold in many areas at once. The categorical trace is a prime example of this goal in action. Originating as the sum of the diagonal entries in a square matrix, the trace has been generalized to a notion which carries fixed point information in several categories. In this talk we go through several definitions and examples to develop the necessary background for understanding the trace.
April 27 Travis Wheeler Bicategorical Character Theory
In 2007, Nora Ganter and Mikhail Kapranov wrote a paper defining a character theory for 2-representations. In this talk, I'll use GK to motivate defining a character theory for bicategories. I'll end with the bicategorical character and how GK fits into the story.
August 25 Organizational Meeting
September 1 David Chan (Vanderbilt) Compatible multiplicative and additive structures in equivariant algebra
Equivariant cohomology theories are robust invariants for studying spaces with an action by a group G. In contrast to classical cohomology theories, these invariants do not take values in abelian groups. Instead, there are several possible algebraic structures, known collectively as incomplete Mackey functors, which provide a home for these additive theories. Similarly, multiplicative equivariant cohomology theories can take values in one of several different generalizations of rings. In this talk we will study examples of these different algebraic structures, and discuss recent work on the interplay between the various multiplicative and additive structures that arise in equivariant topology.
September 8 David Mehrle When free algebras are not flat: the weird world of equivariant algebra
In equivariant stable homotopy theory, we replace homotopy groups with homotopy Mackey functors. In many ways, the category of Mackey functors behaves like the category of abelian groups, but the analogy is not perfect. These differences can be a real headache for computations, but they also reflect how much richer the equivariant world can be. In this talk, we will highlight one surprising difference between algebra with Mackey functors and algebra as you know it - the free algebras are not flat as modules, in stark contrast with the non-equivariant case. This is joint work with Mike Hill and JD Quigley.
September 15 Lewis Dominguez Introductions to λ-rings and a 'concrete' formula for the Burnside Ring
There are a variety of special commutative rings which come equipped with operations mapping to themselves, notably λ-rings, β-rings, ψ-rings, and more. These notions are related and turn out to define one another. We'll motivate the construction of such rings and following this, present a concrete formula for the λ-structure on the Burnside Ring as discovered by Bouc and Rökaeus.
September 22 Millie Rose λ-rings and the ring Λ
I will provide a motivation for λ-rings coming from the K-theory of symmetric monoidal linear categories. This motivates the construction of a universal ring Λ. Then, we will discuss how this ring controls the category of λ-rings and reduces many results to calculations. Time permitting, I will use this calculational control to present an algorithm for computing the Bott cannibalistic class.
September 29 Emily Riehl (Johns Hopkins) Path induction and the indiscernibility of identicals
Mathematics students learn a powerful technique for proving theorems about an arbitrary natural number: the principle of mathematical induction. This talk introduces a closely related proof technique called "path induction," which can be thought of as an expression of Leibniz's "indiscernibility of identicals": if x and y are identified, then they must have the same properties, and conversely. What makes this interesting is that the notion of identification referenced here is given by Per Martin-Löf's intensional identity types, which encode a more flexible notion of sameness than the traditional equality predicate in that an identification can carry data, for instance of an explicit isomorphism or equivalence. The nickname "path induction" for the elimination rule for identity types derives from a new homotopical interpretation of type theory, in which the terms of a type define the points of a space and identifications correspond to paths. In this homotopical context, indiscernibility of identicals is a consequence of the path lifting property of fibrations. Path induction is then justified by the fact that based path spaces are contractible.
October 6 Jordan Sawdy Propositional and Contractible Groupoids
We'll discuss a logical system inspired by homotopy type theory but modeled in 1-groupoids that is useful for proving certain facts about equivalences of 1-categorical structures. As with homotopy type theory, there are analogues with classical logic, but there are also some major differences. This divergence is necessary, as this groupoidal logic enables us to talk about equivalence of not only properties, but also structures. However, when we restrict to certain types of groupoids (those that are propositional or (-1)-truncated), the logic becomes much more classical, but with the advantage that it will enable us to prove results about equivalences of certain structures provided they satisfy a sufficient "uniqueness" condition.
October 13 Jesse Keyes An Introduction to Bredon Homology Valued in Mackey Functors
A notion of Bredon homology for G-CW complexes with coefficients in Mack(G) will be introduced and demonstrated through its calculation for various representation spheres. Further, it will be demonstrated that Bredon homology can be used to determine (non-equivariant) integral homology of a G-CW complex, its associated orbit spaces, and its subspace of G-fixed points.
October 20 Hari Rau-Murthy (Notre Dame) The HKR theorem and factorization homology
The HKR theorem and factorization homology Classically, the Hochschild homology of the ring of smooth functions on a manifold is its differential forms. This result is due to Hochschild, Kostant and Rosenberg(HKR) and one may say the Hochschild homology of a ring was defined to make this true.

We prove the HKR theorem for smooth $E_\infty$ rings. This generalization is a brainchild of McCarthy-Minasian and my contribution is in correcting their proof of the theorem. We further generalize the result to factorization homology, which includes iterated and higher applications of Hochschild homology. We will also discuss implications for the dimensional reduction map of Euclidean field theories and the Hopkins-Kuhn-Ravenel-Stapleton character map.
October 27 Shahzad Kalloo The Homotopy Groups of the K(1)-Local Sphere

This week seminar will be 3:00-3:50!

Computing the homotopy groups of the sphere has shown to be extremely complicated, though throughout the past few decades Homotopy theorist have developed many methods and ideas to tackle the problem. The goal of this talk is to display some of the modern methods of homotopy theory at work by computing the homotopy groups of the K(1)-local sphere. We will introduce the category of spectra, discuss the notion of localization, and then compute the homotopy groups at odd primes. Provided there is time we will also discuss the result for p=2.
November 3 David Mehrle A Hochschild-Kostant-Rosenberg Theorem for Tambara Functors
The Hochschild-Kostant-Rosenberg (HKR) Theorem says that when A is a smooth k-algebra, the Hochschild homology of A is isomorphic to the Kahler differentials of A. This theorem is an extremely useful tool in calculations of not only Hochschild homology, but also cyclic homology, topological Hochschild homology, and topological cyclic homology. Inspired by recent progress in equivariant topological Hochschild homology, we ask if some version of a Hochschild-Kostant-Rosenberg theorem holds. For certain Tambara functors satisfying a flatness condition, we establish a relation between their Hochschild homology and their modules of genuine Kahler differentials (as defined by Hill and Leeman). This is joint work with J.D. Quigley and Michael Stahlhauer.
November 10 Kelly Pohland (Vanderbilt) The RO(C3)-graded Bredon cohomology of C3-surfaces
For spaces with an action of a group G, there is a cohomology theory graded on real G-representations called Bredon cohomology which serves as an equivariant analogue of singular cohomology. This talk will explore a recent family of computations in RO(C3)-graded Bredon cohomology where C3 is the cyclic group of order 3. We give an overview of the main results as well as demonstrate some of the techniques used through small examples.
November 17 Josh Peterson An Introduction to the Steenrod Algebra and Modules over Its Subalgebras
The mod-2 Steenrod algebra will be defined in terms of homomorphisms on the cohomology of spaces with coefficients in F_2. Hence, modules over subalgebras of the Steenrod algebra will be introduced with the intent of describing a method to easily determine whether a given module is free.
December 1 Justin Barhite Detecting Free Modules over Subalgebras of the Steenrod Algebra
Two weeks ago Josh introduced the Steenrod algebra and a tool for determining when modules are free over its subalgebras. I will present an application of this result (or a variation of this result due to John Palmieri) to a project with William Balderrama, Nick Kuhn, Don Larson, and Andres Mejia. I will introduce the Brown-Gitler modules, which are the central characters of this application, and illustrate how Q_1-homology can tell us about Dyer-Lashof operations.
December 8 Travis Wheeler Pseudo Limits
During my last talk, I computed an equalizer that implied a notion of induced representation for 2-categories was in fact co-induction. This computation also implied that I need a weaker notion of equalizer, which has led me to pseudo limits. During this talk, I’ll define pseudo limits and how they relate to my research.
January 13
No seminar - send an email with your speaking preferences!
January 20 Millie Rose Character Theory and Euler Characteristics of Categories
I develop a sort of character theory for familially representable presheaves on finite EI-categories and show that this generalizes the Burnside character for finite G-sets. Then, I show that the technology used in this character theory leads to a notion of an Euler characteristic of a category, which agrees with the Euler characteristic of its classifying space in many cases.
January 27 Lewis Dominguez Commutative Rings Associated to Groups and their relations
In this talk, we discuss relationships between the Burnside Ring and the Representation Ring. The primary goal is to show that the Adams Operation on these two rings are compatible, and we will make use of the Ring Of Class Functions and Table of Marks to do so.
February 3 Travis Wheeler Characters for 2-Representations as Bicategorical Traces
One of the main questions behind my research: Can the character map for induced 2-representations be recognized as a bicategorical trace? We don't know yet, but we do know some important properties of the classical character we'd like to hold for the generalization. During this talk, I'll discuss how the classical character map is a bicategorical trace and the familiar formulas for the induced and restricted character are examples of basic properties of the trace.
February 10 Carissa Slone Slices of \Sigma^n HZ over Q_8
The slice filtration focuses on producing certain spectra, called slices, from a genuine G-spectrum X over a finite group G. There are several constructions which inflate a G/N-spectrum X to a G-spectrum, which generally do not coincide. We will examine three such constructions, when they coincide, and how this relates to the slices of \Sigma^n HZ over Q_8.
February 17 Rok Gregoric (UT-Austin) Chromatic homotopy theory via spectral algebraic geometry
Chromatic homotopy theory studies the surprising relationship between the structural properties of the stable homotopy category and the algebro-geometric properties of the moduli stack of formal groups. In this talk, we will review this connection, and discuss how to make it precise in the world of non-connective spectral algebraic geometry.
February 24 Liz Tatum (UIUC) Splitting BP<2> ^ BP<2> at primes p > 5
S In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ^ bo and l ^ l. These splittings helped make it feasible to do computations using the bo- and l-based Adams spectral sequences. In this talk, we will construct an analogous splitting for BP<2> ^ BP<2> at primes p> 5.
March 3 Justin Barhite Traces and Fixed Points
Traces provide information about fixed points. We will look at some examples of this, then make this connection more precise by exploring the equivalence between trace operators and fixed point operators on a cartesian monoidal category. Time permitting, we will speculate about how this might help make sense of the bicategorical cotrace.
March 10 Jordan Sawdy Trace Theorems for Double Categories and More
Inspired by recent work of Hoyois, Safronov, Scherotzke, and Sibilla, we'll present a theorem about shadows and traces in the context of fibrant double categories. We'll discuss ways in which this theorem is a sort of decategorified version of a theorem of HSSS. Finally, we'll look at some in-progress work on extending this to tricategories and possibly other three-dimensional category-like structures in hopes of generalizing the HSSS theorem.
March 24 Jonathan Campbell (CCR) Toroidal Traces
Traces are shockingly useful in mathematics as a whole, and one of the only invariants of mathematical objects that we can actually compute. They are also the simplest invariants that get attached to manifolds in a topological quantum field theory (a trace corresponds to a circle in this case). I'll describe two results due to myself and Kate Ponto about iterated (or toroidal) traces, the sorts of invariants that get attached to tori in TQFTs. 2-categorical arguments guided by geometric intuition allow us to prove meta-theorems that recover many theorems that get called "Lefschetz" or "Hirzebruch-Riemann-Roch" theorems.
March 31 Shahzad Kalloo Formal Group Laws and Cohomology Theories
For the past several decades a large part of homotopy theory has been guided by the idea that the geometry of a certain algebraic object controls much of the stable homotopy category. This idea is the backbone of the chromatic approach to stable homotopy theory. In this talk we introduce the main ideas in understanding this correspondence between certain "nice" cohomology theories and formal group laws. We first discuss our nice class of cohomology theories and how to produce a formal group law from a cohomology theory. In order to understand how to go the other direction, we discuss some of the basic theory of formal group laws and the Landweber Exact Functor theorem.
April 7 Christy Hazel (UCLA) The cohomology of C_2-surfaces with constant integral coefficients
Let C_2 denote the cyclic group of order 2. In this talk, we’ll explore some recent computations done in RO(C_2)-graded cohomology with constant integral coefficients for C_2-surfaces. We’ll also explore some interesting patterns in these computations, and discuss how these might generalize to C_2-manifolds of higher dimension.
April 14 Ang Li A real motivic v_1 selfmap of the spectrum Y and its cofiber realizing real motivic A(1)
We consider a nontrivial action of C_2 on the type 1 spectrum Y:=S/2 smash S/\eta. This can also be viewed as the complex points of a finite real-motivic spectrum. One of the v_1 self-maps of Y can be lifted to a C_2 equivariant self-map as well as a real-motivic self-map. Further, the cofiber of the self-map of the R-motivic lift of Y is a realization of the real-motivic Steenrod subalgebra A(1). We also give a description of all the 128 A-module structures of the real-motivic Steenrod subalgebra A(1). This is joint work with Prasit Bhattacharya and Bertrand Guillou.
April 21 No seminar
April 28 Math Lab end of year celebration!
September 2 Lewis Dominguez Adams Operations & Power Operations on the Burnside Ring
The Burnside Ring for a group G allows for the construction of two important operations for each natural number, the Adams Operation and Power Operation, respectively. Using the embedding of the Burnside Ring into the Table of Marks, we will prove that the Adams Operation factors through the power operation in the case of abelian G and n=p^2, then use this to motivate a general formula.
September 9 Nathaniel Stapleton The multitude of functors from finite groups to commutative rings
Stable homotopy theory provides a rather large number of functors from finite groups to commutative rings. I'll discuss many of the most important functors, their special properties, and how they interact. The goal is to give a big picture of how these things fit together.
September 16 Bert Guillou Eilenberg-Mac Lane spaces and equivariant cohomology
Spaces with a single nonzero homotopy group play a crucial role in homotopy theory. Known as Eilenberg-Mac Lane spaces, they turn out to completely govern (ordinary) cohomology, in a way that I will make precise. And if you like that, then you better put a G on it. We will discuss equivariant Eilenberg-Mac Lane spaces and their relation to equivariant cohomology. Finally, we will discuss some example computations of equivariant cohomology.
September 23 Ang Li A comparison between C_2-equivariant and classical squaring operations
For any C_2-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from the RO(C_2)-graded cohomology to the classical cohomology. In this talk, I will compare the RO(C_2)-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.
September 30 Millie Rose Power operations in equivariant K-theory
We introduce Adams operations for equivariant complex K-theory which generalize the Adams operations in representation theory and non-equivariant K-theory. Using the Atiyah-Segal character map, which generalizes the classical character for representation theory and the Chern character for K-theory, we produce a formula for the Adams operations applied to the Bott element of the regular representation sphere for an abelian group.
October 7 Nima Rasekh (EPFL) Shadows are bicategorical Traces
Topological Hochschild homology (THH), first defined for ring spectra and then later dg-categories and spectrally enriched categories, is an important invariant with connections to algebraic K-theory and fixed point methods. The existence of THH in such diverse contexts motivated Ponto to introduce a notion that can encompass the various perspectives: a shadow of bicategories. On the other side, many versions of THH have been generalized to the homotopy coherent setting providing us with motivation to develop an analogous (∞,2)-categorical generalization of shadows. The goal of this talk is to use an appropriate bicategorical notion of THH to prove that a shadow on a bicategory is equivalent to a functor out of THH of that bicategory. We then use this result to give an alternative conceptual understanding of shadows and, in particular, an alternative proof for the Morita invariance. Moreover, we use this alternative characterization to give an appropriate definition of a homotopy coherent shadow of (∞,2)-categories. This is joint work with Kathryn Hess.
October 14 Sarah Petersen (Notre Dame) The RO(C_2) - Homology of C_2-Equivariant Eilenberg-Maclane Spaces
This talk describes work in progress computing the HF_2 homology of the C_2-equivariant Eilenberg-Maclane spaces associated to the constant Mackey functor F_2. We expand a Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the RO(C_2) setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which is quite complicated, lacking an explicit E_2 page and having arbitrarily long equivariant degree shifting differentials. We avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson along with norm and restriction maps to complete the computation.
October 21 Carissa Slone Characterizing 2-slices over C_2 and K_4
The slice filtration focuses on producing certain irreducible spectra, called slices, from a genuine G-spectrum X over a finite group G. We have a complete characterization of all 1-, 0-, and (-1)-slices for any such G. We will characterize 2-slices over C_2 and expand this characterization to K_4 = C_2 \times C_2.
October 28 Travis Wheeler Induced 2-Representations
Classical representation theory can be generalized to 2-representations: representations on 2-categories. This opens the door to talk about the character of a 2-representation, as well as the restricted and induced 2-representations. In this talk, I will define the induced 2-representation, and we'll see how the induced 2-representation is related to induced representations of groupoids.
November 4 Matt Feller (UVA) Generalizing quasi-categories via model structures on simplicial sets
Quasi-categories are particular simplicial sets which behave like categories up to homotopy. Their theory has been massively developed in the past two decades, thanks largely due to Joyal and Lurie, and they have become vital tools in many areas of algebraic topology, algebraic geometry, and beyond. Due to the success of quasi-categories, it would be nice to extend the theory to up-to-homotopy versions of objects more general than categories, such as the 2-Segal sets of Dyckerhoff-Kapranov and G\'alvez-Kock-Tonks. Such a generalization would ideally come with an associated model structure on the category of simplicial sets, but finding a model structure with a more general class of fibrant objects than a given model structure is a nontrivial and open-ended task. In this talk, I will explain how to use Cisinski's machinery to construct model structures on the category of simplicial sets whose fibrant objects generalize quasi-categories. In particular, one of these model structures has fibrant objects precisely the simplicial sets that satisfy a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects.
November 11 Jordan Sawdy The Generalized Nerve-Realization Adjunction
Do you like fundamental ∞-groupoids, fibrant replacement functors for simplicial sets, equivalences of models of (∞,1)-categories, or sheaves on topological spaces? Well then, do I have the construction for you! We'll spend some time building up an intuition for both colimits and presheaves on small categories that hopefully makes the needed technology easily understandable and maybe even (dare I say it?) obvious. Then, we'll give a quick proof that the eponymous adjunction is, in fact, an adjunction. Finally, we'll work through a number of examples, including but ideally not limited to those in the first sentence of this abstract.
November 16 Janet Huffman Using burnside rings to determine group solvability
In this talk we use the Burnside ring as a tool to study whether a group is solvable. For a group G, we will introduce the Burnside ring A(G). We will see that there is an injection from A(G) into a product of integers, which we can use to study certain properties of the Burnside ring. Our goal is to understand the relationship between the units in A(G) and the solvability of the group G.
December 2 Shahzad Kalloo Stacks as 2-Sheaves
Stacks were first introduced into algebraic geometry by Grothendieck in the 50's in order to formalize descent and moduli problems and have since found applications to many areas where the same types of problems would like to be solved. We first review the concept that the functions on a space form a sheaf of sets, and then use this example to motivate the definition of a stack. We then consider our main example of the stack of vector bundles over a space, and provided there is time we will discuss the Amitsur complex and faithfully flat descent of modules.
December 9 Justin Barhite Bicategorical Traces and Cotraces
Abstract: The familiar trace of a matrix generalizes to a notion of trace of an endomorphism of a dualizable object in a symmetric monoidal category. To extend these ideas to other settings, such as modules over non-commutative rings, we must further generalize the trace to a bicategory equipped with additional structure called a shadow functor. We propose a notion of bicategorical cotrace of certain maps involving dualizable objects in a closed bicategory equipped with a coshadow functor, and we use this framework to draw connections between work of Joseph Lipman (1987) and Ganter and Kapranov (2006).
February 4 Ozgur Bayindir The K-theory of THH(F_p)
In this work, we study THH(F_p) from various perspectives. We start with a new identification of THH(F_p) as an E_2-algebra. Following this, we compute the K-theory of THH(F_p). The first part of my talk is going to consist of an introduction to ring spectra, algebraic $K$-theory and the Nikolaus Scholze approach to trace methods. In the second part, I will introduce our results and the tools we develop to study the topological Hochschild homology of graded ring spectra. This is a joint work with Tasos Moulinos.
February 11 Morgan Opie Complex rank 3 vector bundles on CP^5
Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on CP^5. In particular, I will describe a classification of such bundles which involves a surprising connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on CP^5 from "simple" ones; and future directions related to this project.
February 18 Angelica Osorno Transfer systems and weak factorization systems
N infinity operads over a group G encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of G. In this talk, we will show that when G is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of G. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. We will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg-Hill saturation conjecture. This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.
February 25 Justin Barhite String Diagrams for Closed Bicategories
The familiar trace of a finite-dimensional vector space endomorphism generalizes to a notion of the trace of an endomorphism of a dualizable object in any symmetric monoidal category. To extend these ideas to other settings, such as modules over non-commutative rings, we must further generalize the trace to a bicategory equipped with additional structure called a shadow. In a 1987 paper, Joseph Lipman defines families of "trace" and "cotrace" maps via Hochschild homology and cohomology. We have identified his trace maps as bicategorical traces with Hochschild homology as the shadow, and we have proposed a "bicategorical cotrace" which exists in a closed bicategory and generalizes Lipman's cotraces (with Hochschild cohomology serving as a "coshadow"). Understanding the interaction of these traces and cotraces is made easier by string diagrams; we will describe how to draw and manipulate these string diagrams and demonstrate their utility by proving a bicategorical theorem that generalizes one of Lipman's results.
March 4 Nat Stapleton On the KU-local sphere
I hope to explain how to calculate the zeroth homotopy group of the localization of the sphere spectrum with respect to complex K-theory. A number of ingredients go into this: spectra, Bousfield localization, fiber sequences, p-completion/rationalization, the arithmetic fracture square, Adams operations, and real/complex K-theory. I'll try to give a bit of an overview of this stuff and perform the calculation. If there is time, I'll explain a recent result with Peter and Bert calculating the zeroth homotopy group of the localization of the equivariant sphere spectrum with respect to equivariant complex K-theory for odd p-groups.
March 11 Mona Merling Scissors congruence for manifolds via K-theory
The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German "schneiden und kleben," cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic. This is joint work with Hoekzema, Semikina, Rovi, and Wells.
August 27 Peter Bonventre Rigidification of Infinity Operads
Quasicategories are a model for "categories up to coherent homotopy", where composition is only defined up to a contractible choice. There are two ways to rigidify this structure: the homotopy category, which outputs a plain old category, and the W-construction, which outputs a simplicially enriched category. These two functors are the left adjoints to the more familiar "nerve" and "homotopy coherent nerve" functors, the latter of which is in fact an equivalence of homotopy theories, but in general are much more opaque. In this talk, I will discuss an explicit description of both of these adjoints, starting with work of Dugger-Spivak in the categorical setting, and building to a reformulation and operadic generalization in joint work with Luis Pereira.
September 3 Justin Barhite Bicategorical Traces and Cotraces
The familiar trace of a finite-dimensional vector space endomorphism generalizes to a notion of the trace of an endomorphism of a dualizable object in any symmetric monoidal category. In order to extend these ideas to other settings, such as modules over non-commutative rings, we must further generalize the trace to a bicategory equipped with additional structure called a shadow. In a 1987 paper, Joseph Lipman defines families of "trace" and "cotrace" maps via Hochschild homology and cohomology. We have identified his trace maps as bicategorical traces with Hochschild homology as the shadow, and we propose a "bicategorical cotrace" which specializes to Lipman's cotraces (with Hochschild cohomology serving as a "coshadow"); furthermore, these bicategorical traces and cotraces are related in a way that recovers Lipman's results.
September 10 Travis Wheeler Categorical Trace in a 2-Category
In a 2006 paper, Nora Ganter and Mikhail Kapranov defined the categorical trace for an endomorphism A in a 2-category. The definition for the categorical trace defines a functor from the category of endomorphisms of an object, x, to Set. This functor seems to act similarly to the bicategorical shadow functor defined in Kate Ponto and Michael Shuman's 2009 paper, Shadows and Traces in Bicategories. My current project is understanding these similarities and differences.
September 17 William Balderrama From power operations to E-infinity maps
A general heuristic in homotopy theory tells us that by understanding the operations which act naturally on the homotopy groups of a class of objects, one can build obstruction theories and so forth for working with these objects. In the setting of highly structured ring spectra, this leads one to obstruction theories based on theories of power operations. I'll talk about this, focusing on the particular case of Morava E-theory. Power operations in this context were first studied by Matt Ando to understand H-infinity complex orientations, and I'll explain how at low heights, they can be used to produce E-infinity complex orientations.
September 24 Carissa Slone Characterizing 2-slices over C_2 and K_4
The slice filtration focuses on producing certain irreducible spectra, called slices, from a genuine G-spectrum X over a finite group G. We have a complete characterization of all 1-, 0-, and (-1)-slices for any such G. We will characterize 2-slices for C_2 and expand this characterization to K_4 = C_2 x C_2.
October 1 Ang Li A real motivic v1 selfmap
We consider a nontrivial action of C_2 on the type 1 spectrum Y:=S/2 smash S/\eta. This can also be viewed as the complex points of a finite real-motivic spectrum. One of the v_1−self-maps of Y can be lifted to a C_2 equivariant self-map as well as as well as a real-motivic self-map. Further, the cofiber of the self-map of the R-motivic lift of Y is a realization of the real-motivic Steenrod subalgebra A(1). This is joint work with Prasit Bhattacharya and Bertrand Guillou.
October 15 Emily Rudman The cyclic homology of $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$
The Hochschild homology of the ring $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$ has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic homology of the ring. The calculations are relatively straightforward for $k=\mathbb{Q}$, but we see interesting torsion phenomena over $k=\mathbb{Z}$.
October 22 Joel Stapleton Twisted Weibel's conjecture for algebraic K-theory.
Weibel's conjecture for algebraic K-theory states that the negative K-groups of a d-dimensional quasi-compact quasi-separated scheme vanish below -d. Kerz--Strunk--Tamme have proven Weibel's conjecture in generality by establishing pro cdh-descent for algebraic K-theory. Using recent work of Land--Tamme, I will prove the same statement for twisted algebraic K-theory. To define twisted algebraic K-theory, we need a scheme and a cohomology class in H^2(X, G_m). We can then build an oo-category of perfect complexes of twisted modules and take its algebraic K-theory.
October 29 Jocelyne Ishak Rigidity of the K(1)-local stable homotopy category
In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the underlying model structure can we recover?This question has been investigated during the last decade, and some examples have been studied, but there are still a lot of open questions regarding this subject. Starting with the stable homotopy category Ho(Sp), that is the homotopy category of spectra, it has been proved to be rigid by S. Schwede. Moreover, the E(1)-local stable homotopy category, for p=2, has been shown to be rigid by C. Roitzheim. In this talk, we investigate a new case of rigidity, which is the localisation of spectra with respect to the Morava K-theory K(1), at p=2.
November 5 Lewis Dominguez A power operation on the table of marks
The table of marks is closely related to the Burnside Ring, Representation Ring, and Ring of Class Functions on a group. Each of these have a notion of power operation, with the exception of the Table of Marks. In this talk, we aim to construct and understand such an algebraic power operation on Marks(G), compatible with its relation to these other rings.
November 12 Hana Jia Kong The motivic Chow t-structure and the computational applications
In this talk, I will introduce the Chow t-structure on the motivic stable homotopy category over a general base field. This t-structure is a generalization of the Chow-Novikov t-structure defined on a p-completed cellular motivic module category in work of Gheorghe--Wang--Xu. Moreover, we identify the heart of this t-structure with a purely algebraic category, and expand the results of Gheorghe-Wang-Xu to integral results on the entire motivic category over general base fields. This leads to computational applications on determining the Adams spectral sequences in the classical stable homotopy category, as well as that in the motivic stable homotopy category over C, R, and F_p. This is joint work with Tom Bachmann, Guozhen Wang and Zhouli Xu.
November 19 Sarah Klanderman Computational Structure of Topological CoHochschild Homology
In recent work, Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology (coTHH). In 2018, Bohmann-Gerhardt-Hogenhaven-Shipley-Ziegenhagen developed a coBokstedt spectral sequence to compute the homology of coTHH for coalgebras over the sphere spectrum. However, examples of coalgebras over the sphere spectrum are limited, and we would like to have computational tools to study coalgebras over other ring spectra. In this talk I’ll describe a relative coBokstedt spectral sequence that I developed in order to study the topological coHochschild homology of more general coalgebra spectra. We’ll look at a few examples of coalgebra spectra and use the additional algebraic structure of the spectral sequence to complete coTHH calculations.
December 3 David Chan Comultiplication in the Serre Spectral Sequence
Coalgebraic structures can be a powerful tool for analyzing differentials in spectral sequences. Unfortunately, many examples of these spectral sequences can be difficult to construct and understand. In this talk we will discuss the comultiplicative properties of the Serre spectral sequence as an avenue toward generating interesting and accessible examples.
January 30 Nat Stapleton From unstable to stable: free loops and evaluation maps
Making use of covering space techniques from MA651, I will describe a recent theorem, which is joint work with Reeh and Schlank, that shows that there is an extension of a certain evaluation map for spaces to "transfer maps". No knowledge beyond MA551 and a bit of MA651 should be required.
February 6 Peter Bonventre Additive power operations in equivariant cohomology
Equivariant cohomology theories give rise to Mackey functors. Bert, Nat, and I have studied power operations in this setting and this led to the discovery of a canonical Mackey ideal in certain naturally occurring Green functors in this context.
February 13 Carissa Slone Slice towers of trivial suspensions of HF_2 over C_2
The slice filtration, developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire invariant one problem, is an equivariant generalization of the Postnikov filtration. The associated slice tower provides a method of building genuine G-spectra from certain ''irreducible" spectra. We will set up the slice filtration and characterize the towers of trivial suspensions of HF_2 over C_2.
February 27 Bert Guillou Additive Power Operations in Equivariant Cohomology
An H_infty ring spectrum comes with an mth power operation for any positive integer m, and this becomes additive only after collapsing a certain transfer ideal. I will discuss the analogous situation equivariantly, both in the case of G-spectra and in the global setting. In each case, I will identity precisely the minimal ideal that must be collapsed in order to make the mth power operation a map of Mackey functors. I will provide examples, such as the sphere spectrum and complex K-theory. This is joint work with Peter Bonventre and Nat Stapleton.
September 12 Shane Clark
The Reidemeister trace of an endomorphism of a CW complex is a lower bound for the number of fixed points (up to homotopy) of that endomorphism. For an endomorphism $f$, the Reidemeister trace of $f^n$ is a lower bound for the number of fixed points of $f^n$, however it can be a far from an optimal lower bound. There are many related invariants that refine the Reidemeister trace and have different strengths and weaknesses in regards to computability and realizability. In this talk we will describe a classes of spaces where these invariants can be computed and realized.
September 12 Peter Bonventre Equivariant trees and equivariant higher algebra
Trees provide a useful framework for understanding monoids, operads, and algebras over operads. However, in the equivariant setting, they are no longer sufficient, as they cannot detect which "norm multiplication maps" an equivariant operad encodes. In this talk, I will discuss a solution in the category of G-trees. I will introduce norm maps and show how they relate to G-trees. Finally, I will show how G-trees help to construct a combinatorial model for equivariant operads with norms up to homotopy.
September 19 Nat Stapleton An invitation to Morava's extraordinary E-theories
I will introduce the Morava E-theories, which are cohomology theories that have close connections to number theory, representation theory, and geometry. I hope to say something about these connections!
September 26 Prasit Bhattacharya Revisiting higher associativity of Moore spectra
The mod $p^i$ Moore spectrum $M_p(i)$, is the cofiber of multiplication by $p^i$ map on the sphere spectrum. $M_p(i)$ is nothing but the spectrum analogue of $\mathbb{Z}/p^i$ in abelian groups. However, it is well-known that $M_p(i)$ do not inherit nice multiplicative properties of the sphere spectrum. It is not $E_{\infty}$, in fact it is conjectural that $M_p(i)$ is not $E_1$ or $A_{infty}$. What are the obstructions? In this talk I will describe a method that suggests that the obstructions to associativity of $M_p(i)$ are related to certain elements in the image of J.
October 3 Jonathan Campbell Hilbert's Generalized Third Problem and Homotopy Theory
In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory. Two n-dimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble the pieces into $Q$. The scissors congruence problem, aka Hilbert Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent iff they have the same area. In three dimensions, there is volume AND another invariant, the Dehn Invariant. In higher dimensions, very little is known. I'll give an introduction to this very classical problem, and explain how homotopy theory can be used to get purchase on it. Prerequisites: The discussion of Hilbert's Third Problem and Dehn's invariant will be widely accessible. No knowledge of algebraic K-theory will be assumed. This is all joint work with Inna Zakharevich.
October 10 Ang Li In completing the Incomplete Guillou May Theorem
The Guillou May theorem gives a Quillen equivalence between the category of genuine G-spectra and the category of spectrally enriched functors from the spectral Burnside category to nonequivariant spectra, with respect to a complete universe U and a finite group G. In this talk we are retrieving the punchline of the proof, and bring up the discussion of what would be different under an incomplete universe. This is joint work with Hana Jia Kong.
October 17 Nat Stapleton More on symmetric monoidal categories
We will describe how to think about symmetric monoidal categories as op-fibrations over the category of pointed finite sets.
October 24 Tim Campion Duality in homotopy theory
We explore some implications of one basic observation. Namely, the $n$-sphere has the remarkable property that the “swap” map $\sigma: S^n \otimes S^n \to S^n \otimes S^n$ can be “untwisted”: it is none other than $(-1)^n \otimes 1$. This simple fact remains true in equivariant and motivic contexts. One consequence is a structural fact about for symmetric monoidal $\infty$-categories with finite colimits and duals for objects: it turns out that any such category splits as the product of three canonical subcategories (for instance, one of these subcategories is characterized by being stable). In stable equivariant homotopy theory, one inverts representation spheres. In stable motivic homotopy theory, one inverts $\mathbb{P}^1$. In both cases, these peculiar choices are sometimes motivated by their duality-related consequences. We sharpen these motivations, showing that if a representation sphere or $\mathbb{P}^1$ is forced to become dualizable, then it is almost forced to become invertible, so that genuine $G$-spectra are almost obtained from genuine $G$-spaces by freely adjoining duals.
October 30 Shane Clark Periodic Points: Vanishing and Realizability
The Reidemeister trace of an endomorphism of a CW complex is a lower bound for the number of fixed points (up to homotopy) of that endomorphism. For an endomorphism $f$, the Reidemeister trace of $f^n$ is a lower bound for the number of fixed points of $f^n$, however it can be a far from an optimal lower bound. One method of addressing this discrepancy constructs an equivariant map, the n^{th} Fuller trace $f$, which carries information about the periodic points of a map $f$. However, we must ask how much information is retained by this equivariant construction? In this talk we show that the n^{th} Fuller trace of $f$ is a complete invariant for describing a minimum set of periodic points for maps of tori.
November 7 Bert Guillou Navigating the world of G-spectra
Cohomology is a powerful tool for studying topological spaces. If we want to study cohomology theories, it is useful to recast them as spectra. I will discuss the equivariant version of this. Equivariantly, there are several possible interpretations of ``equivariant cohomology’’, leading to the different notions of equivariant spectra: Borel, naive, or genuine.
November 14 Justin Barhite Traces and Cotraces via Hochschild Homology
The familiar trace of a finite-dimensional vector space endomorphism generalizes to a notion of the trace of an endomorphism of a dualizable object in any symmetric monoidal category. In order to extend these ideas to other settings, such as modules over non-commutative rings, we must further generalize the trace to a bicategory equipped with additional structure called a shadow. In a 1987 paper, Joseph Lipman defines families of "trace" and "cotrace" maps via Hochschild (co)homology; we would like to understand these maps, and the theorems Lipman proves about them, in the framework of the bicategorical trace.
November 21 Foling Zou Nonabelian Poincare duality theorem in equivariant factorization homology
The factorization homology are invariants of $n$-dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following Miller-Kupers. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra.
December 5 Carissa Slone The Slice Filtration
The slice filtration, developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire invariant one problem, is an equivariant generalization of the Postnikov filtration. The associated slice tower provides a method of building genuine G-spectra from certain ''irreducible" spectra and yields a slice spectral sequence, which contains information about the homotopy of the original spectrum. We will set up the slice filtration and then examine the slice towers and corresponding spectral sequences for two genuine G-spectra.
January 31 Peter Bonventre Power operations in equivariant homotopy theory
A commutative ring spectrum gives rise to certain algebraic operations called power operations in the homotopy category of spectra. These operations are multiplicative, but not additive. However, they are additive after taking the quotient by a certain naturally occurring ideal. Bert, Nat, and I have been investigating the analogous algebraic structures associated to a commutative equivariant ring spectrum. I will explain what all of these words mean and talk about some results that we have found.
February 7 Nat Stapleton Chromatic musings around the Segal conjecture
The goal of this expository talk is to discuss the relationship between the category of finite group and unstable and stable homotopy theory. We will explain the statement of the Segal conjecture, the p-complete analogue and its relation to fusion systems, and why all of this raises more questions than it answers for a chromatic homotopy theorist.
February 14 Anna Marie Bohmann A multiplicative comparison of Segal and Waldhausen K-theory
In influential work of the 70s and 80s, Segal and Waldhausen each construct a version of K-theory that produces spectra from certain types of categories. These constructions agree, in the sense that appropriately equivalent categories yield weakly equivalent spectra. In the 2000s, work of Elmendorf--Mandell and Blumberg--Mandell produced more structured versions of Segal and Waldhausen K-theory, respectively. These versions are "multiplicative," in the sense that appropriate notions of pairings of categories yield multiplication-type structure on their resulting spectra. In this talk, I will discuss joint work with Osorno in which we show that these constructions agree as multiplicative versions of K-theory. Consequently, we get comparisons of rings spectra built from these two constructions. Furthermore, the same result also allows for comparisons of related constructions of spectrally-enriched categories.
February 21 Shane Clark Reaching Rock Bottom
The Reidemeister trace/ Lefschetz number/Neilsen number of an endomorphism of a CW complex gives a lower bound for the number of fixed points (up to homotopy) of that endomorphism. For an endomorphism $f$, the Reidemeister trace/ Lefschetz number/Neilsen number of $f^n$ is a lower bound for the number of fixed points of $f^n$, however it can be a far from an optimal lower bound. In previous talks we have discussed stronger invariants which give lower bounds for periodic points. This time we will discuss the second major problem of the fixed point theory, how to achieve this minimum number of periodic points. This question has been the focus of my research while attending UK and this talk will cover some of the main techniques we have used.
February 28 Niles Johnson A 2-categorical group-completion
We introduce a group-completion for symmetric monoidal 2-categories, generalizing Quillen's S-inverse-S construction. No background on 2-categories will be assumed. The talk will begin with some categorical algebra, recall Quillen's construction, and then state what we know about a 2-dimensional version. We will describe an application to algebraic K-theory in dimension 3, and close with some discussion of future plans. This talk is based on joint projects with Ernie Fontes, Nick Gurski, and Angélica Osorno.
March 7 Dan Berwick-Evans Modular forms, physics, and topology
Modular forms appear in a wide variety of contexts in physics and mathematics. For example, they arise in two dimensional quantum field theories as certain observables. In algebraic topology, they emerge in the study of invariants called elliptic cohomology theories. A long-standing conjecture suggests that these two appearances of modular forms are intimately related. After explaining the ingredients, I’ll describe some recent progress.
March 21 Ang Li The lower and higher Seifert-van Kampen theorems
The Seifert-van Kampen theorem is the first thing we encountered to compute homotopy groups, telling us how we could determine a space’s homotopy type by its open subsets’. In this talk, we are going to see the 1-dim usual and also higher version of van Kampen theorem, and pick up the notions we need on the way. Of course, no serious proof will be given.
March 28 Christina Osborne Decomposing the classifying diagram in terms of classifying spaces of groups
The classifying diagram was defined by Rezk and is a generalization of the nerve of a category; in contrast to the nerve, the classifying diagram of two categories is equivalent if and only if the categories are equivalent. In this talk, we will show that the classifying diagram of any category is characterized in terms of classifying spaces of stabilizers of groups. We will also prove explicit decompositions of the classifying diagrams for the categories of finite ordered sets, finite dimensional vector spaces, and finite sets in terms of classifying spaces of groups.
April 4 Rafael Rojas Concrete Nonsense: Compactness, Invertibility, and Dualizability in pleasant symmetric monoidal categories
Notions of compactness, invertibility, and dualizability show up in many places in various areas of mathematics. In this talk we will explore how these ideas can be extended into a more general categorical setting, however we shall restrict ourselves to particularly friendly categories. Our main purpose will be to explore the interaction of these notions with one another.
April 11 Carissa Slone Smith Theory and Bredon Cohomology
One of the main theorems in Smith theory is as follows: If a p-group G acts on a CW complex X, where the cohomology of X matches that of a sphere, then either the action is free or the subspace of fixed points is another cohomology sphere of no greater dimension. We will introduce tools including Bredon cohomology and then use these tools to prove the theorem.
April 18 Justin Barhite Dualizability and Trace in Symmetric Monoidal Categories
The familiar trace of a linear map generalizes to a notion of trace for any endomorphism of a dualizable object in a symmetric monoidal category. These traces often provide information about fixed points. Moreover, by recognizing traces in different categories as instances of this same abstract formulation, we can compare them functorially. We will introduce the notions of dualizability and trace, look at some motivating examples, and obtain the Lefschetz-Hopf theorem as an application of the functoriality of traces.
August 30 Nathaniel Stapleton Organizational meeting
September 6 Faculty Retreat
There will be no seminar.
September 11 Mike Hill Equivariant algebra and computations
The role of abelian groups in equivariant stable homotopy theory is played by certain diagrams of abelian groups called Mackey functors. These form an abelian category with enough projectives, so the ordinary tools of homological algebra can be applied, but they also have surprising additional multiplicative structures. I'll discuss the algebra surrounding Mackey functors and also show how it can be used to compute equivariant homology.
September 20 Peter Bonventre Symmetric Monoidal Mackey Functors
Symmetric monoidal categories and Mackey functors are two flavors of categorical algebra which often show up when working in algebraic topology. Recent work in equivariant homotopy theory suggests the need for a notion which combines them. In this talk, I will define Mackey functors, introduce a "coordinate free" version of monoidal categories, and use this added flexibility to define an equivariant generalization which recovers Mackey functors as a special case. Time permitting, I will describe "commutative monoids" in these categories.
September 27 Shane Clark Lets Talk Tori
Topological fixed points theory looks to assign algebraic invariants to an endomorphism in an attempt to detect/minimize the number of fixed points (up to homotopy) of that map. Classical fixed point invariants are given by the Lefschetz number, Nielsen number, and the Reidemeister trace give a ``complete" story to the fixed point theory, however they do not generalize to periodic fixed points. For this talk we introduce the periodic fixed point invariants and try to tackle the periodic point problem for endomorphism's of Tori. Time permitting, we will also cover some questions of about a 1 parameter family of endomorphisms instead of a single map.
October 4 Bert Guillou The Tate construction in homotopy theory and Lin's theorem.
If you splice together the homology and cohomology of a group G, you get what is known as the Tate (co)homology of G. I will discuss an analogous construction in stable homotopy theory. This construction is related to the Segal conjecture (which is a theorem). I will discuss the case for the group C_2, which was proved by Lin.
October 11 J.D. Quigley The parametrized Tate construction
The Tate construction is a powerful tool in classical homotopy theory, with applications to the Segal Conjecture, Mahowald invariants, blueshift for complex oriented spectra, and trace methods for algebraic K-theory. In this talk, I will describe an enhancement of the Tate construction to equivariant homotopy theory called the "parametrized Tate construction." I will describe the category of objects where this construction is defined and discuss some classical examples. I will then discuss equivariant analogs of the above applications, including C_2-equivariant versions of Lin's Theorem and the Mahowald invariant, blueshift for Real Johnson-Wilson spectra (joint work with Guchuan Li and Vitaly Lorman), and trace methods for Real algebraic K-theory (work-in-progress with Jay Shah).
October 18 Ang Li Homotopy pullback: theory and formula
Homotopy diagrams are good because if we replace some nodes in the diagram by something weak equivalent, the diagram still commute up to homotopy. In this talk we will talk about (total) derived functors between model categories and see a way to compute (small finite) homotopy (co)limits. We will use homotopy pullback as a throughout example, get formulas for homotopy pullback in Top (\infty,0)-cats) and Gpd ((1,0)-cats), and possibly see some connections.
October 25 Ben Riley Cut it Out: Cut-Paste Operations and Relations to Bordism
A Cut-paste operation on a smooth oriented closed manifold allows us to cut it along a codimension 1 submanifold, and then paste back using an orientation-preserving diffeomorphism of the resulting boundaries. This operation allows us to define an equivalence relation on n-dimensional closed manifolds, and form a group denoted SK_n. These groups, and a generalization called Cut-Paste-Controllable(SKK_n) groups, have surprising connections to bordism, through a series of three short exact sequences. I will present a fourth exact sequence characterizing the kernel of the map from SKK_n to SK_n, which is absent from the literature. Additionally, equivariant versions of these constructions will also be discussed. This work is joint with Carmen Rovi, and is the result of a summer REU at IU-Bloomington.
November 1 Jeremy Hahn Even spaces old and new
Call a space "even" if it has a cell structure with only even-dimensional cells and also its only homotopy groups are in even degrees. Many of the most important spaces in homotopy theory are even, such as CP^infinity and BU. Other important spaces naturally sit inside even spaces, such as in the chain of inclusions BO(n) inside BU(n) inside BU. I will discuss a classification result for even spaces due to Steve Wilson, and then describe as many extensions, applications, and conjectures, due to a variety of authors, as time allows.
November 8 Rafael Rojas Orthogonal Sequences, Spectra and Symmetric Monoidal Categories
Orthogonal sequences are a type of functor from the category of finite dimensional real inner product spaces to based topological spaces. We will discuss the idea of a symmetric monoidal category by building one from orthogonal sequences. We will then discuss the category of modules over the sphere sequence, S-mod, and its connection with orthogonal spectra. Lastly we will explore why the category S-mod is symmetric monoidal.
November 15 Nat Stapleton The Adams spectral sequence: an overview
I will discuss the Adams spectral sequence in some generality. We will learn about cosimplicial spectra, Hopf algebroids, and the dual Steenrod algebra. Finally we will discuss an element called b_10 in the cohomology of the dual Steenrod algebra at the prime 3.
November 22 Thanksgiving
November 29 Eva Belmont Localizing the E_2 page of the Adams spectral sequence
The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page for the sphere at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.
December 6 Ben Riley Nilpotence in the Steenrod Algebra 2
The Steenrod algebra is a powerful tool in algebraic topology, generated by elements called squares, which define stable cohomology operations. Due to their topological origins, the algebraic properties of these squares carry topological implications. Last semester, I gave an overview of the Steenrod algebra and described how Dr. Guillou and I established a lower bound on the nilpotence height of a certain family of Steenrod Squares. This talk will give another overview on the Steenrod Algebra, with emphasis given to those items which have potential application to establish an upper bound on the squares of interest. This work is joint with Bert Guillou and provides the basis for my Honors Thesis. Be there or Be Square.
January 11 Organizational meeting
January 18 Peter Bonventre (Equivariant) (Dendroidal) Segal Spaces and Categorical Homotopy Theory
Segal spaces offer a convenient model for "categories up to homotopy". In this talk, we will first introduce Segal spaces and their connection to generic homotopy theories. Then, using the technology of G-trees, we will build modern generalizations which present equivariant multicategories with norm maps.
January 25 Nat Stapleton A whirlwind tour of complex cobordism in classical chromatic homotopy theory
Complex cobordism is a surprisingly powerful cohomology theory with deep connections to classical questions in stable homotopy theory and the theory of formal groups in algebraic geometry. In this talk we will attempt to introduce these connections and some of the most important theorems regarding complex cobordism without getting bogged down in the technicalities (ie. expect few proofs or even precise definitions).
February 1 Bert Guillou An introduction to equivariant cohomology
Cohomology is a very useful and powerful tool that is used in the study of spaces. In the equivariant world, when the spaces come equipped with group actions, there are several distinct constructions that go by the name of equivariant cohomology. I will discuss these theories and the relationships between them.
February 9 Clover May (University of Oregon) A structure theorem for RO(C2)-graded cohomology
Computations of singular cohomology groups are very familiar. An equivariant analogue is RO(G)-graded Bredon cohomology with coefficients in a constant Mackey functor. Computations in this setting are often more challenging and are not well understood, even for the cyclic group of order two C2. In this talk I will present a structure theorem for RO(C2)-graded cohomology with Z/2 coefficients that substantially simplifies computations. The structure theorem says the cohomology of any finite C2-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will give some examples and sketch the proof, which depends on a Toda bracket calculation.
February 15 Ang Li An introduction to model categories, or, a `to do' list before getting to the homotopy category
Want to understand a Homotopy Category? You need a model structure first! Don't know what that is? Not a problem! Come to my talk and we will start from the definitions, doing some "clean hand" category theory and hopefully get to some examples without making our hands dirty,
March 1 Ben Riley Nilpotence Height in The Steenrod Algebra
The Steenrod algebra is a powerful tool in algebraic topology, generated by elements called squares, which define stable cohomology operations. Due to their topological origins, the algebraic properties of these squares carry topological implications. I will be giving a brief overview of the Steenrod Algebra and its properties, with an emphasis on the nilpotence of certain families of squares. I will end with some recent progress on the height of the family Sq(2^n-2). This is joint work with Bert Guillou. Be there Square.
March 8 Paul VanKoughnett (Northwestern University) Localizations of E-theory
Chromatic homotopy theory uses the theory of formal groups from algebraic geometry to construct new topological invariants. The tightest link between the two worlds is Morava E-theory, a homotopical avatar of the space of deformations of a formal group of fixed height. We study what happens when E-theory undergoes chromatic localization, forcing the height of this formal group to decrease. We give modular descriptions of the resulting objects, and applications to the study of power operations in homotopy theory.
March 22 Julie Vega Does not commute: A journey through persistent homology
In topological data analysis (TDA) there are two main questions: (1) How can we understand the structure of large amounts of data and (2) How can we learn about global structure from discrete points? To aid our understanding we use persistent homolgy. In this talk, we will explore the basics of TDA and persistent homology. Our focus will be on the nerve lemma which arises in relation to the complexes that are produced from the data and extending the nerve lemma to play nicely with persistent homolgy.
March 29 Shane Clark Equivariant Fixed Point theory and Periodic Points
The Reidemeister trace of an endomorphism of a CW complex gives a lower bound for the number of fixed points (up to homotopy) of that map. Therefore, for an endomorphism f, the Reidemeister trace of f^n is a lower bound for the number of fixed points of f^n. However, it can be far from an optimal lower bound. In an attempt to remedy this we change the question into an equivariant one and use an equivariant analog of the Reidemeister trace. In this talk we will discuss the transition to the equivariant Reidemeister trace and apply it to a very specific map to give us a "correct" bound for the number of periodic points of f.
April 5 Kaelin Cook-Powell Line bundles!
Have you ever asked yourself: "What is a line bundle?" (Hint: It's a bundle of lines) or "Where can I find a line bundle?" (Hint: Out in the wild). If you said, "Yes," to either of these questions, then this talk might be for you. This week we introduce the concept of a line bundle, look at a number of supposedly motivating examples, and then ask ourselves, "Why should we care?" Hopefully, at least one of these questions will be answered (offer void where prohibited by law).
April 10 Jonathan Campbell (Vanderbilt University) An Introduction To, and Extension Of, Algebraic K-Theory
In this talk I'll introduce algebraic K-theory, and then explain how it can be extended to many non-algebraic situations. In fact, K-theory should be thought of as a machine for "breaking things into pieces" and I'll provide evidence for this by constructing K-theory for both polytopes and algebraic varieties, objects that in no way fit into abelian categories, but nevertheless have a notion of decomposition. I will sketch applications for this extension -- for example, the rank filtration in algebraic K-theory due to Quillen seems closely related with the classical scissors congruence group. This is joint work with Inna Zakharevich.
April 19 Kalila Sawyer The Truth About Lie Groups
What is a Lie group? What is a Lie algebra? In this talk we'll give a general answer to these questions, with a goal of understanding why people care about these objects and what they can be used for.
April 26 Inbar Klang (Stanford University) Factorization homology
This talk will be an introduction to factorization homology via labeled configuration spaces. It will mostly be about configuration space models for certain mapping spaces, but time permitting, I might say a bit about factorization homology in the category of spectra.
August 24 Organizational meeting
August 31 Bert Guillou An introduction to equivariance
If G is a group, then an action of G on a set (or space) X means that each element g of G acts as a symmetry of X. We will explore the interaction of group actions with homotopy theory.
September 7 Peter Bonventre Operads and Exotic Multiplications
One standard goal of algebraic topology is to determine "geometric" information using algebraic methods. A classic example is that loop spaces are characterized by a "multiplication" which is only associative if we allow ourselves certain topological flexibility. Encoding all of this data can be quite tricky, and inspired (or compelled) the definition of an operad. In this talk, we will give some examples of "exotic" multiplications, introduce operads, and discuss the connections between these two concepts.
September 14 Jonathan Rubin (University of Chicago) Categorical models of equivariant spectra
Numerous spaces, such as universal principal bundles, Eilenberg-Mac Lane spaces, and algebraic K-theory spaces can be built using combinatorial methods. I will survey some classical results on how spaces and spectra can be constructed from small categories, and then I will turn to more recent developments. In particular, I will discuss some work in progress on the presentation of equivariant symmetric monoidal categories, and some potential applications. All terms will be defined during this talk, and no prior knowledge of equivariant homotopy theory will be assumed.
September 21 Ang Li The Hopf Invariant 1 theorem
Hopf invariant 1 is a celebrated theorem of F. Adams, proved in 1960 via secondary cohomology operations. Later, a less painful approach arose using a number-theory-argument in complex K-theory. We are going to have a glance at the second approach and pick up a bunch of things needed on the way.
September 28 Luis Pereira (Notre Dame University) Genuine equivariant operads
In this talk I will talk about one piece of a current joint project with Peter Bonventre which aims at providing a more diagrammatic understanding of Blumberg and Hill's work on G-operads. Our work uses a notion of $G$-trees, which are a somewhat subtle generalization of the trees of Cisinski-Moerdijk-Weiss. More specifically, I will describe a new algebraic structure, which we dub a "genuine equivariant operad", which naturally arises from the study of $G$-trees and which we conjecture to be the analogue of coefficient systems in the "correct" analogue of Elmendorf's theorem for $G$-operads. This is joint work with Peter Bonventre.
October 5 Kalila Sawyer The Grassmannian is Always Greener...
In this talk, we'll focus mainly on a general introduction to all sorts of bundles. Via fiber bundles and vector bundles, our goal will be to see how the Grassmannian is used to categorize bundles, and we'll end with a brief application to algebraic geometry.
October 12 Shane Clark Vector Bundles and Cohomology, how can that be?
Last week we talked about a few types of bundles, how to classify them, and how they relate to questions in algebraic geometry. This week we will explore the glory of vector bundles and show how they helped motivate a generation of mathematics. We will explore operations on vector bundles, Grothendieck completion and a classical result of Bott which can be used to show ordinary cohomology actually has something to do with vector bundles.
October 19 Eric Kaper An Extension of the Notion of Derivative to the Context of More Than Two Variables
We will discuss the de Rham complex. In particular, this complex gives rise to a cohomology theory on differentiable manifolds (which happens to coincide with singular cohomology!). Additionally, many of the notions that appear in a standard third semester calculus class have nice explanations from this point of view. The notion of differentiable manifold will be introduced, so no differential topology/geometry background will be assumed. However, a basic understanding of the differentiable structure on Rn will be taken for granted.
October 26 Kaelin Cook-Powell Continuing The French Agricultural Metaphor: A Sheafy Business
This week we introduce the notion of pre-sheaves and sheaves, objects that help us study topological spaces/geometric objects via the functions defined on them, rather than by just studying their points. We'll be looking at a number of examples to try and get a better feel for what they are, touching on their categorical flavor, and looking at a process called "sheafification." If you aren't already familiar with these objects, hopefully this talk helps you see how they just appear in the wild.
November 2 Bert Guillou The slice filtration for certain RO(K_4)-graded suspensions of HF_2
A space X can be described by its Postnikov tower, whose stages have only the homotopy groups of X in a range. Equivariantly, there is an analogue of the Postnikov filtration called the slice filtration. After reviewing some previously known examples, I will describe the slice filtration for twisted Eilenberg-Mac Lane spectra when the group of equivariance is K_4, the Klein four group. This is joint work with C. Yarnall.
November 16 Gabe Angelini-Knoll (Michigan State University) The Segal conjecture for topological Hochschild homology of Ravenel spectra
In algebraic topology, we use a tool called a spectral sequence to reduce a geometric problem to an algebraic problem. For example, suppose we want to show that a map between nice spaces is a homotopy equivalence. If we can show the map induces an isomorphism on the inputs of the spectral sequence then by Boardman's theorem that map of spectral sequences induces an isomorphism of homotopy groups. Then, by the Whitehead theorem, the map is, in fact, a homotopy equivalence. This technique was used by Lin to prove a conjecture of Segal's for the cyclic group of order 2. Segal's conjecture can be phrased as the statement that categorical G-fixed points of the sphere are weakly equivalent to the homotopy G-fixed points of the sphere for a finite group G. One may also ask whether the categorical G fixed points of topological Hochschild homology of a ring spectrum are weakly equivalent to the homotopy G-fixed points of topological Hochschild homology of a ring spectrum for G a finite subgroup of the circle. In my talk, I will discuss how to apply techniques of Lunoe-Nielson and Rognes to answer this question for the Ravenel spectra X(n) and T(n). This project is joint work with J.D. Quigley.
November 30 Rafa Rojas Eilenberg-Maclane Spaces, Principal Bundles, and Classifying Spaces
In this talk we will discuss Eilenberg-Maclane spaces, specifically K(G,1) spaces. We will look at some common examples and discuss how to get a K(G,1) space for any topological group G. We will then review some properties of principal G-bundles with the intent of introducing the universal G-bundle, leading to the notion of a classifying space for a topological group. Time and resources permitting, we may also discuss some characteristic classes associated with classifying spaces.
December 7 Ang Li The baldness problem for (even) spheres
We knew from the Hairy Ball Theorem that there is no nonvanishing tangent vector field on even-dimensional n-spheres. As to odd spheres, the condition gets better, yet they still can't have as many hairs as possible. Of course only linearly independent "hairs" count. We will see why this is true via some results in real topological K-theory, based on computations using the Atiyah Hirzebruch Spectral Sequence. (And don't worry too much about spheres, they are still pretty with or without hairs!)
February 9 A discussion about spectra and localization.
February 16 Gabriel Valenzuela (Ohio State University) The Chromatic Splitting Conjecture for Noetherian Commutative Ring Spectra
The work I will discuss in this talk is joint with Tobias Barthel and Drew Heard. The goal is to formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum R, and present a positive answer to this conjecture in the case when \pi_*R is Noetherian. Our approach relies on a novel decomposition of the local cohomology functors constructed previously by Benson, Iyengar, and Krause as well as a generalization of Brown-Comenetz duality. If time permits, I will discuss how these results provide a new local-to-global principle in the modular representation theory of finite groups.
March 2 Dominic Culver (Notre Dame) Towards a calculation of the cooperations algebra for truncated Brown-Peterson spectra
Following ideas of Mahowald, Mark Behrens has initiated a program to investigate the tmf-based Adams spectral sequence. In order to pursue this, one needs to know the cooperations algebra for tmf. This has been attempted before, most notably in BOSS, but a complete description has been elusive. In this talk, I will describe a method for computing the cooperations algebra of a closely related spectrum, called $\mathrm{BP}\langle 2\rangle$.
March 9 Carolyn Yarnall The Equivariant Slice Filtration
The slice filtration is an equivariant analogue of the Postnikov tower. After a brief introduction to equivariant stable homotopy theory and recalling the Postinikov tower, we will define the slice tower and provide some examples that demonstrate the similarities and differences between it and the Postnikov tower. More specifically, we will investigate the role of suspensions and connectivity in the slice tower.
March 23 Niles Johnson (Ohio State University) Categorical models in homotopy theory
This will be a survey-style talk describing ways that categorical algebra helps us understand calculations in homotopy theory. Our main result is the 2-dimensional Stable Homotopy Hypothesis; we will explain what this means and how it relates to basic questions in topology and basic tools in algebra. We will close with some indications of the crucial technical results and what they have taught us about higher-dimensional categories. Much of the talk is based on joint work with Nick Gurski, Angelica Osorno, and Marc Stephan.
March 30 Shane Clark The Intersection Problem and Applications to Fixed Points
Determining whether an endomorphism is homotopic to a map without fixed points can be reinterpreted as an intersection problem up to homotopy. This transition leads us to discovering obstructions to the existence of a fixed point free map with algebraic data. Oddly enough, these obstructions live in (stable) homotopy groups of a recognizable space. The purpose of this talk is to illuminate the transition from a question about fixed points to the world of stable homotopy. I will follow the work of paper by Klein and Williams' paper ''Homotopy Intersection I'' and Kate Ponto's exposition found in her thesis. All are welcome and questions encouraged!
April 6 Ang Li The Yoneda Lemma
The Yoneda Lemma is a result in category theory which appears in many areas of math. It describes how a construction (functor) can be understood in terms of its relation to other basic ones. I will give a brief proof and present some examples and applications.
April 13 Rob Denomme Spiders, webs and quantum knot invariants
This talk is an introduction to the graphical calculus of spiders and webs. These are a diagrammatic tool for understanding the generators and relations in a certain category with a tensor product. A knot diagram can easily be interpreted as a morphism in this category, a concept that lead to the cutting-edge knot invariants of Khovanov Homology. The generators and relations for many similar categories are yet to be discovered, and this is an active area of research.
April 20 Bert Guillou Strictification, or: How I Learned to Stop Worrying and Love Bicategories
If a category has ''higher morphisms'' between morphisms, this gives a 2-category. When the composition of morphisms is only associative up to these higher morphisms, we get a ''weak'' 2-cateogry, or bicategory. We will see examples and discuss in what sense bicategories may be replaced by equivalent strict 2-categories. (No previous knowledge of 2-category theory will be assumed).
August 25 Organizational Meeting
September 1 Bert Guillou An introduction to Topological Hochschild Homology
Algebraic K-theory tells us about manifolds. Or it would, if we could calculate it. In the 1980's, Bokstedt, Goodwillie, Waldhausen, and others proposed other means of understanding K-theory. Importing the algebraic notion of Hochschild homology into the world of topology gives one approximation to K-theory. I will describe a construction of this theory and discuss the calculation of THH of a loop space. This material will appear again in the talk of John Lind.
September 8 Kate Ponto Transfers!
One really important fact in algebraic topology is that maps of spaces induce homomorphisms in the same direction on homology and homotopy groups and the opposite direction on cohomology. For some very nice maps, there are associated maps on homology and cohomology that reverse these familiar directions (transfer maps!). As preparation for next week's talk I will discuss one (very general) approach to these transfers.
September 15 John Lind (Reed College) The Transfer Map of Free Loop Spaces
Associated to a fibration E --> B with homotopy finite fiber is a stable wrong way map LB --> LE of free loop spaces coming from the transfer map in THH. This transfer is defined under the same hypotheses as the Becker-Gottlieb transfer, but on different objects. I will use duality in bicategories to explain why the THH transfer contains the Becker-Gottlieb transfer as a direct summand. The corresponding result for the A-theory transfer may then be deduced as a corollary. When the fibration is a smooth fiber bundle, the same methods give a three step description of the THH transfer in terms of explicit geometry over the free loop space. (Joint work with C. Malkiewich)
September 22 Dustin HedmarkA beginner's guide to Schubert Calculus
In my research I've encountered a certain polynomial that sums over all partitions lambda that fit in an m by n box. It turns out these are exactly the partitions that form a basis for the cohomology of the Grassmanian G(m,n+m). In this talk we will work through some examples of Schubert calculus.
September 29 Nathan Druivenga Knots and Invariants
I will give a brief introduction to knot theory and then discuss knot invariants. Specifically, I will defined the Jones polynomial and its quantum counterpart, the colored Jones polynomial using skein theory and representations of quantum groups.
October 6 Wesley Hough Discrete Morse Theory
Determining the homotopy type of a given CW complex is often difficult to do in practice, but we can usually gain some traction by reducing the complex under consideration to something that is homotopy-equivalent and (hopefully) much smaller or easier to study. One such way to reduce a given CW complex is by using techniques from discrete Morse theory to define useful cellular collapses that will produce a "nicer" quotient complex to consider. In this talk, we will discuss the main theorems of discrete Morse theory for simplicial complexes, generalize to the cellular case, and then discuss how to calculate the degree of the attaching maps in the quotient complex.
October 13 Carolyn YarnallTowers and Filtrations
Often in mathematics, we try to better understand an object by studying how it is built out of pieces via a filtration. In homotopy theory, the Postnikov tower gives us a way of constructing a space or spectrum from its homotopy groups. In this talk, I will describe the construction and basic properties of the Postnikov tower and then end by provided a few additional examples of similar towers in algebraic topology.
October 20 Shane Clark Fixed Point Invariants
How can we tell if a particular map has fixed points? Topological fixed point theory uses invariants from algebraic topology to give information about fixed points. I will primarily discus two of these invariants. The first is the fixed point index which gives us information about the maps behavior around a given fixed point. The second is the Lefschetz number which is a global invariant defined using maps between homology groups induced from the original endomorphism. The final invariant is the Reidemeister trace which takes themes from both the fixed point index and the Lefshcetz number. Albrecht Dold gave a particularly nice proof that the fixed point index and the Lefschetz number coincide for "nice" spaces. I will give Dold's proof of this equality and some further developments.
October 27 Bert Guillou Introduction to computational stable homotopy theory
In the first half of the twentieth century, it was noticed that the homotopy groups of spheres fit into certain `stable' families. For example, maps S^4 --> S^3 correspond bijectively to maps S^5-->S^4. In preparation for an upcoming visiting speaker, I will give an introduction to the Adams spectral sequence, which is a tool for computing these stable homotopy groups.
November 10 Amelia Tebbe (UIUC) Computing Polynomial Approximations of Atomic Functors
A functor from finite sets to chain complexes is called atomic if it is completely determined by its value on a particular set. In this talk, we present a new resolution for these atomic functors, which allows us to easily compute their Goodwillie polynomial approximations. By a rank filtration, any functor from finite sets to chain complexes is built from atomic functors. Computing the linear approximation of an atomic functor is a classic result involving partition complexes. Robinson constructed a bicomplex, which can be used to compute the linear approximation of any functor. We hope to use our new resolution to similarly construct bicomplexes that allow us to compute polynomial approximations for any functor from finite sets to chain complexes.
November 17 Julie Vega Topological Techniques in Combinatorics
In '55 Martin Kneser conjectured for any partition of n-subsets of a (2n+k)-element set into k+ 2 classes, no class will contain disjoint n-subsets. While showing that k+2 is an upper bound is more straight forward, it took about 20 years, some innovation, and the magic we call topology before Lovász was able to show k+2 is a lower bound as well. In this talk, we will consider the techniques employed by Lovász and prove k + 2 is the lower bound.
December 1 Eric Kaper Group Cohomology and Homology
We will discuss constructions of group cohomology and homology with the goal of understanding what the first few of each of these groups represent. Some homological algebra will be assumed and tensor products will be used to change coefficient rings---though the focus of the talk will be to demonstrate the novel connections that this framework makes between representation theory, topology, homological algebra and group theoretic constructions. As such, constructions that do not rely on derived functors (i.e ext and tor) will be emphasized.
January 28 Kristen Mazur (Elon University) An Introduction to Mackey Functors and Tambara Functors
Mackey functors make frequent appearances in algebra and topology. For example, the stable homotopy groups of G-spectra are Mackey functors. Tambara functors are Mackey functors with a lot of extra structure. The theory of Tambara functors is not nearly as well developed, but these objects are proving to play a key role in equivariant stable homotopy theory. This talk will focus on understanding the basic properties of Mackey functors and Tambara functors. We will discuss how to think about them and work through some examples. We will end by developing a new structure on the category of Mackey functors that provides a nice characterization of Tambara functors.
February 4 Carolyn Yarnall Slices and Suspensions
The equivariant slice filtration is an analogue of the Postnikov tower for G-spectra. However, unlike the Postnikov tower, the slice tower does not commute with taking ordinary suspensions and, in fact, what results when suspending slice towers is not understood in general. In this talk, after recalling the construction of the slice tower, we will look at the slice towers for integer-graded suspensions of HZ and compare them to complementary results of Hill, Hopkins, and Ravenel concerning λ-suspensions. We will conclude with a brief look at future directions regarding the interplay between suspension and the slice filtration for general G-spectra.
February 11 Shane Clark Multiplication on Cohomology Theories
Topologists use various functors and algebraic invariants to classify spaces. Some quintessential examples are cohomology, H^* (X ; R), and K-Theory, K^*(X), which turn topological spaces into (graded) rings. This talk will cover the construction and ring structure of the two rings mentioned above as well as compute some basic examples. (This talk assumes knowledge of topology I, II, & III.)
February 25 Prasit Bhattacharya (Notre Dame) On the spectrum that admits a 1-periodic v_2-self-map at the prime 2
At the prime 2, Behrens-Hill-Hopkins-Mahowald showed M(1,4) admits 32-periodic v_2-self-map and more recently B-Egger-Mahowald showed A_1 also admits 32-periodic v_2-self-map. This leads to the question, whether there exists a finite 2-local complex with periodicity less than 32. This talk will answer the question by producing a finite 2-local complex Z which admits 1-periodic v_2-self-map. Apart from admitting 1-periodic v_2-self-map, Z has other remarkable properties such as tmf_* Z =k(2)_* and E \wedge Z = K(2) \wedge (G_{48})_+, where E is the height 2 Morava E-theory and G_{48} is the maximal profinite subgroup of Morava stabilizer group.
March 3 Dustin Hedmark 2 Categories and Bicategories
In this talk we will explore the notion of a 2-category, a new type of category where the hom sets act like categories themselves. Weakening some conditions of the 2-category, we obtain a bi-category. We will look at lots of examples of both types. No topological background will be needed, except for a working understanding of categories and functors.
March 10 Eric Kaper Notions of Orientability of Smooth Manifolds
We will discuss definitions of orientability in terms of relative homology, determinants of differentials, and vector bundles---focusing on relating the various definitions. We will then attempt to contextualize orientability as it relates to the first Stiefel-Whitney class.
March 24 erica Whitaker Constructing noncongruence subgroups using graphs on surfaces
There is a known correspondence between certain bipartite graphs on surfaces and certain subgroups of PSL2(Z). We will define these graphs and groups and discuss the correspondence. We will also define congruence and noncongruence subgroups, and learn how to construct examples of noncongruence subgroups by drawing their graphs. (This talk will include some elementary ideas from topology, graph theory and number theory.)
March 31 Thomas Barron Monads and Adjunctions
In category theory, a monad is a construction involving an endofunctor and two natural transformations. In this expository talk, we learn about monads, their algebras, and the relationship between monads and adjunctions. Examples abound.
April 14 Robert Cass An Introduction to Sheaves and Their Cohomology
In this talk, we will introduce sheaves on a topological space and look at some basic examples. Then we will study the exponential sheaf sequence in complex geometry to motivate the definition of sheaf cohomology. Finally, we will discuss some applications in complex and algebraic geometry.
April 21 Serge Ochanine Generalizations of the Borsuk-Ulam theorem and Stiefel-Whitney numbers
Let X be an m-dimensional closed smooth manifold and let t: X —> X be a smooth free involution. The triple (X, t, n) has the Borsuk-Ulam property if for every map f: X—> R^n, there is a point x in X for which f(t(x))=f(x). Given (X,t), an interesting (and difficult) problem is to find the smallest n for which the triple (X, t, n) has the Borsuk-Ulam property. A recent paper by Crabb-Goncalves-Libardi-Pergher gives a partial solution to this problem within the Z/2-cobordism class of (X,t).
April 28 Bert Guillou Spectra and (co)homology theories
Homology and cohomology were introduced as algebraic invariants that you can actually calculate and therefore use to distinguish spaces. It was later realized that these were specific examples of ``generalized cohomology theories''. These, in turn, correspond precisely to "spectra" in the stable homotopy theory sense. I will discuss these connections and provide examples. No prior knowledge of cohomology or stable homotopy theory will be assumed.
September 10 Jonathan Campbell (UT Austin) The Algebraic K-Theory of Varieties
The Grothendieck ring of varieties is a fundamental object of study for algebraic geometers. As with all Grothendieck rings, one may hope that it arises as π0 of a K-theory spectrum, K(Vark). Using her formalism of assemblers, Zahkarevich showed that this is in fact the case. I'll present an alternate construction of the spectrum that allows us to quickly see the E-structure on K(Vark) and produce various character maps out of K(Vark). I'll end with a conjecture about K(Vark) and iterated K-theory.
September 17 Carolyn Yarnall An introduction to Mackey functors
In equivariant stable homotopy theory, Mackey functors play the role abelian groups play in the nonequivariant setting. In this talk, I will provide the definition of a Mackey functor and useful diagrams for depicting such objects. After investigating a collection of examples, we will briefly discuss some applications and results concerning Mackey functors.
September 24 Anna Marie Bohmann (Vanderbilt) Constructing equivariant spectra
Equivariant spectra determine cohomology theories that incorporate a group action on spaces. Such spectra are increasingly important in algebraic topology but can be difficult to understand or construct. In recent work, Angelica Osorno and I have created a machine for building such spectra out of purely algebraic data based on symmetric monoidal categories. Our method is philosophically similar to classical work of Segal on building nonequivariant spectra. In this talk I will discuss an extension of our work to the more general world of Waldhausen categories. Our new construction is more flexible and is designed to be suitable for equivariant algebraic K-theory constructions.
October 1 Chris Hays Exotic 4-manifolds
One of the primary goals when studying manifolds is to determine when two smooth manifolds are homeomorphic, but not diffeomorphic. This question is particularly interesting in dimension 4, as this is the only dimension where there may exist infinitely many such `exotic’ smooth structures on the underlying topological manifold. After briefly describing the history of this problem, we will provide new techniques for constructing new smooth 4-manifolds. These methods rely on constructing symplectic manifolds, as this better allows for one to differentiate the smooth structures.
October 8 Jeff Slye An introduction to Operads
Operads give a concrete way to encode n-ary operations in a symmetric monoidal category. Of course, we could encode an operation which gives us an associative or commutative algebra directly. However, things get more interesting when we consider situations such as existed with the fundamental group of a space. There, for example, associativity only existed up to homotopy classes of maps. How can we encode something such as ``associativity up to homotopy?'' We cover the motivations and basic definitions of operads and algebras over operads in order to build up to just such an operad.
October 22 Ian Barnett The Yoneda Lemma
The Yoneda lemma is a result in category theory that completely describes natural transformations out of hom-functors, also known as represented functors. In this talk we will assume no prior knowledge of category theory, and so will spend time defining and giving examples of basic notions in category theory. We will then present as much of the proof of the Yoneda lemma a time allows.
October 29 Shane Clark Homology via Homotopy: The Infinite Symmetric Product
In algebraic topology, consulting various homology theories can provide different insights to a given problem. In this talk we will construct a new homology theory by considering the infinite symmetric product of a space X and its corresponding homotopy groups. This talk involves objects found in Topology I & II, but is open to all graduate students.
November 5 Deborah Vicinsky (Wabash College) Categories with trivial associated stable categories
I will construct the suspension functor in two categories. The first is the category of small categories with the canonical model structure, in which the weak equivalences are equivalences of categories and the cofibrations are injective on objects, and the second is the category of directed graphs with the Bisson-Tsemo model structure. Then I will show that the categories of spectra for these two categories are homotopically trivial. Finally, I will discuss why this result is interesting (or at least odd) and give a method for identifying other categories in which this occurs.
November 12 McCabe Olsen Lie Groups and Lie Algebras
A differentiable manifold which exhibits a group structure compatible with differentiability is known as a Lie group. In this talk, we provide a basic introduction to Lie groups including some examples and properties. We will also define, discuss, and provide examples of Lie algebras. We will then discuss the algebraic and topological relationship between Lie groups and Lie algebras. No prior topological knowledge will be assumed.
November 19 Sarah Yeakel (UIUC) A chain rule for Goodwillie calculus
In the homotopy calculus of functors, Goodwillie defines a way of assigning a Taylor tower of polynomial functors to a homotopy functor and identifies the homogeneous pieces as being classified by certain spectra, called the derivatives of the functor. Michael Ching showed that the derivatives of the identity functor of spaces form an operad, and Arone and Ching developed a chain rule for composable functors. We will review these results and show that through a slight modification to the definition of derivative, we have found a more straight forward chain rule for endofunctors of spaces.
December 3 John Mosley A Friendly Categorical Talk About Cobordism
Cobordism is an equivalence relation on manifolds. We can also define a category Bord(n), for a dimension n, whose objects are closed manifolds and whose morphisms are cobordisms between the manifolds. In this talk we will introduce cobordism, cobordism categories, and some reasons a person could be interested in such things.
December 10 Katie Paullin An Introduction to Normal Surface Theory and 3-Manifold Algorithms
A nice property of manifolds is that locally, a manifold appears like Euclidean space and for 3-manifolds specifically, many properties are determined by the surfaces they contain, which is helpful in the writing of algorithms. In this talk, I will discuss 3-manifold algorithms and we will see how normal and almost no
February 5 Bert Guillou An introduction to equivariant homotopy theory
If G is a topological group, we can make sense of a (continuous) action of G on a space. If X and Y are two spaces equipped with G-actions, an equivariant map f:X-->Y is a map that is compatible with the G-actions on X and Y. We shall explore what it should mean for this to be an equivariant (weak) homotopy equivalence, and we will look at equivariant cohomology theories.
February 12 Chad Linkous An Introduction to Differentiable Manifolds
This talk introduces the idea of a differentiable, or smooth, manifold. We will give a few examples and compare with other types of manifolds before moving on to look at one of the most important tools in the theory of smooth manifolds-tangent spaces. After constructing the tangent space to a smooth manifold at a given point, we will see that every smooth map between differentiable manifolds induces a linear map between the corresponding tangent spaces. Then, time permitting, we will conclude by revealing some well known theorems of differential topology.
March 12 Cary Malkiewich (UIUC) Coassembly in algebraic K-theory
Algebraic K-theory provides a rich set of invariants for rings, spaces, and many kinds of objects. It is hard to compute, so we often study easier approximations such as topological Hochschild homology (THH). In this talk we study a "dual" K-theory of topological spaces, and find that its linear approximation does not behave as expected; a certain variant of the Novikov conjecture is false. In a different setting, we are able to compose two of these linear approximations to K-theory, and the resulting map is the equivariant norm, which is well understood. This gives us a new tool for understanding the K-theory of BG when G is a finite group. We end with applications: a splitting theorem after K(n)-localization, and a surprising connection between the Whitehead group and Tate cohomology.
March 26 Dustin Hedmark Spectra and Generalized (co)homology
In this talk we will explore how we can use homotopy classes of maps from X to K(G,n) to define a cohomology theory. We will use this example to develop spectra, as well as discuss some common examples of spectra. This talk will align closely with what we have been doing in Dr. Guillou's topology class.
April 2 Luis Sordo Vieira Level of a Topological Space
The level of a Z/2Z-space X is defined to be the minimum n such that there is a Z/2Z equivariant map from X to Sn-1 (with the antipodal action). The level of a unital ring is defined to be the minimum n such that -1=e12+e22+...+en2 where en are elements of the ring. We explore an intimate relationship between the level of a Z/2Z space and the level of commutative R algebras. We also compute the level of even dimensional real projective spaces, spheres, and mention results on the computations of odd dimensional projective spaces.
April 9 David Royster Cobordism from Poincare to apres Thom
We define the unoriented cobordism groups of smooth manifolds, compute a few of them, see how the rest can be done and see what the heck an involution has to do with anything.
April 16 John Mosley The Image of SU Cobordism Under the Witten Genus
Last week, Dr. Royster gave us an introduction to the equivalence relation on manifolds of cobordism. In this talk, we will discuss one way topologists study cobordism: through the notion of genera. In particular, we will work toward computing the image of one flavor of cobordism, SU cobordism, under the Witten genus.
April 23 Michael Andrews (MIT) The v1-periodic part of the Adams spectral sequence - dancers to a discordant system
Algebraic topologists are interested in the class of spaces which can be built from spheres. For this reason, when one tries to understand the continuous maps between two spaces up to homotopy, it is natural to restrict attention to the maps between spheres first. The groups of interest are called the homotopy groups of spheres.
Topologists soon realized that it is easier to work in a stable setting. Instead, one asks about the stable homotopy groups of spheres or, equivalently, the homotopy groups of the sphere spectrum. Calculating all of these groups is an impossible task but one can ask for partial information. In particular, one can try to understand the global structure of these groups by proving the existence of recurring patterns; this is analogous to the fact that we cannot find all the prime numbers, but we can prove theorems about their distribution. These patterns are clearly visible in spectral sequence charts for calculating \pi*(S0) and my thesis came about because of my desire to understand the mystery behind these powerful dots and lines, which others in the field appeared so in awe of.
In this talk, we'll begin by examining why Ext groups show up in topology. After discussing the purpose of a spectral sequence, we'll play with the Adams spectral sequence. We'll get a feel for how algebraic relations are displayed by the charts, how to read off homotopy groups, and we'll make precise to what extent the algebra reflects the topology. Then we'll take a step back to see repeating patterns in the charts for odd primes and we'll describe a theorem that completely describes the Adams spectral sequence at an odd prime p above a line of slope 1/(p^2-p-1).
September 11 Agnes Beaudry (University of Chicago) Chromatic Levels in the Homotopy Groups of Spheres
Understanding the homotopy groups of spheres πn(Sk) is one of the great challenges of algebraic topology. One of the fundamental theorems in this field is the Freudenthal suspension theorem. It states that πn+k(Sk) is isomorphic to πn+k+1(Sk+1) when k is large. Homotopy theorists call this phenomena stabilization. The stable homotopy groups of spheres are defined to be these families of isomorphic groups. They form a ring, commonly denoted by π*S. Despite its simple definition, this ring is extremely complex; there is no hope of computing it completely. However, it carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. There are many structural conjectures about the chromatic filtration. In this talk, I will give an overview of chromatic theory and talk about one of the structural conjectures, the chromatic splitting conjecture.
September 25 John Mosley The Jones Polynomial
Knot Theory is a subject in topology that studies embeddings of S1 in R3. We call these embeddings 'knots' (hence 'Knot Theory'). In this talk, we will discuss some of the basic ideas in the subject of Knot Theory. We will then discuss and give a construction for a useful knot invariant called the Jones Polynomial.
October 9 The Other Signature Theorem
The celebrated Hirzebruch’s Signature Theorem expresses the signature of an oriented 4k-dimensional manifold as a characteristic number in cohomology with rational coefficients. I will discuss a similar result for the Kervaire invariant of a spin manifold that involves characteristic numbers in real K-theory. The presentation will be non-technical and will require very little knowledge of algebraic topology.
October 23 Dan Ramras (IUPUI) Homotopy groups of character varieties
Given a discrete group Γ and a (complex reductive or compact) Lie group G, the character variety Xr (G) is the quotient for the conjugation action of G on Hom(Γ, G). When G is complex reductive, this quotient should be interpreted in the sense of Geometric Invariant Theory. When G = GL(n) or SL(n), the subspace of irreducible representation coincides with the smooth locus of Xr (G). The rational homology of these spaces has been studied in various cases by a number of authors, and when G = U(n) or SU(n), the homotopy type of the stable moduli spaces Xr (U) and Xr (SU) are explicitly known. In this talk I'll discuss recent progress on understanding low-dimensional homotopy (and integral homology) of character varieties and of their subspaces of irreducible representations. This is joint work with Indranil Biswas, Carlos Florentino, and Sean Lawton.
October 30 Luis Sordo Vieira Eilenberg-Mac Lane Spaces
A space X is a K(G,n) if πn(X)=G and πi(X)=0 if i not equal to n. An interesting aspect is that the homotopy type of a CW comples K(G,n) is uniquely determined by G and n. We will investigate the construction of K(G,1), otherwise known as BG, for an arbitrary (discrete) group G, the homology of K(G,1) spaces, and the infinite symmetric product SP(X).
November 6 Wesley Hough Limits, Colimits, and Homotopy . . . Oh, my!
Given maps f: X --> Y and g: X --> Z of topological spaces, we obtain a unique map h: X --> Y x Z that respects the appropriate projections. This property corresponds more generally to the limit of a diagram of spaces. In this talk, we will define the limit, colimit, and their homotopy analogs and discuss their universal properties and relative merits/uses. No prior topological knowledge is assumed.
November 15 Dustin Hedmark Introduction to vector bundles and their classifications
We will introduce the definition of a vector bundle and look at a few examples. Next we will look at how to make new vector bundles from old bundles using familiar algebraic operations like direct sum, tensor product, and the pullback. Finally we will discuss classifying isomorphism classes of bundles over a topological space X, and time permitting, we will show these these isomorphism classes are in bijection with homotopy classes of maps from X to Grassmanians on R infinity.
November 20 Bert Guillou An introduction to operads
Operads first arose in the 60's and 70's for the study of loop spaces, but there was a large resurgence of interest in the 90's once connections with Koszul duality, moduli spaces, and representation theory were realized. I will discuss the definition and familiar examples in both topology and algebra. We will see Stasheff polyhedra in the context of loop spaces as well as examples related to moduli spaces.
December 4 Robert Cass The Freudenthal Suspension Theorem
The Freudenthal suspension theorem asserts that for an (n-1)-connected CW complex X the suspension map from πi(X) to πi+1(SX) is an isomorphism for i < 2n - 1 and a surjection for i = 2n - 1. We will introduce relative homotopy groups and the long exact sequence in homotopy groups for a space X and a subspace A. With these tools we will show how the Freudenthal suspension theorem follows from the homotopy excision theorem. Time permitting, we will examine some consequences for homotopy groups of spheres.
December 11 Kate Ponto My preferred proof of the Lefschetz fixed point theorem
There are many different proofs of the Lefschetz fixed point theorem. The most familiar approach uses simplicial approximation and is often a first example of the power of simplicial homology. I'll talk about a very different proof that I find much more useful. This proof requires more input, but it generalizes easily.
Februar 6 Bert Guillou Real division algebras and the Hopf invariant one problem
I will discuss the existence of division algebra structures on R^n. This is closely related to the famous Hopf invariant one problem that was solved by Frank Adams in the 1950's.
February 13 Jonathan Thompson An introduction to topological K-theory
K-theory is a cohomology theory for spaces that arises from consideration of vector bundles. We will discuss this theory and some important properties, including the Bott periodicity phenomenon and the existence of Adams operations.
February 20 Bert Guillou The Hopf invariant one problem via K-theory
I previously sketched Adams' original approach to the Hopf invariant one problem via secondary operations in singular cohomology. In this talk, I will present the simpler solution using Adams operations in K-theory.
February 27 John Mosley A Different Friendly Talk About Cobordism
In this talk I will present some slightly less basic ideas about cobordism. In particular, we will discuss cobordism as a homology theory and its relationship to K-Theory. This talk might be a little less friendly than the talk on Wednesday.
March 6 Ryan Curry Hirzebruch's proof of his Signature Theorem
We will prove Hirzebruch's signature theorem and show its utility in a computation.
March 13 David White (Wesleyan University) A Characterization for Monoidal Localizations of Equivariant Spectra
The proof of the Kervaire Invariant One Theorem demonstrates conclusively the value of equivariant spectra, and in particular the computational strength of equivariant commutative ring spectra. A key step in the proof relied on the commutativity of a particular localization of a commutative ring. However, a recent example due to Mike Hill demonstrates that it is possible for localizations (even very nice ones) to break commutativity. In this talk we will characterize the localizations which preserve G-equivariant commutativity, and we will investigate the phenomenon of localizations which destroy some, but not all, of the commutative structure (where equivariant commutativity is measured by the presence of multiplicative norm functors corresponding to subgroups of G). Our results are in fact much more general, and hold in the language of model categories and operads, and we will say a word about this general setting if there is time.
March 27 John Mosley Yet Another Friendly Talk About Cobordism
In this talk we will continue our discussion of Cobordism as a Homology Theory. Then, we will talk about the relation of Cobordism to K-Theory.
April 3 Sean Tilson Power operations and the Kunneth spectral sequence
Power operations have been constructed and successfully utilized in a variety of spectral sequences. Such constructions arise from highly structured ring spectra. In this talk, we show that the Kunneth Spectral Sequence enjoys some nice multiplicative properties and use old computations of Steinberger's with our current work to compute operations in the homotopy of some relative smash products. We will end with an application of these computations to give a non-existence result for $E_{\infty}$ complex orientations of certain ring spectra.
April 10 Clinton Hines Wedge Quasitoric Manifolds and Spin Cobordism
Quasitoric manifolds are smooth 2n-manifolds admitting a "nice" action of the compact n-torus so that the quotient of this action yields a (combinatorially) simple polytope. They are a generalization of smooth projective toric variaties and much is known about these manifolds in terms of complex cobordism theory. In fact they were used by Buchstaber and Panov to show that every cobordism complex class contains a (connected) quasitoric manifold. Far less is known about spin quasitoric manifolds and spin cobordism which requires the calculation of KO-characteristic classes. We consider a procedure developed to investigate topological data for spin quasitoric manifolds which utilizes a wedge polytope operation on the quotient polytope. We'll discuss a list of results concerning these "wedge" quasitoric manifolds, including such topics as Bott manifolds, the connected sum, the Todd genus and lastly specific criteria in terms of combinatorial data allowing for the calculation of KO-characteristic classes of spin quasitoric manifolds.
April 17 Kate Ponto Cobordism and Thom Spectra
Building on previous seminar talks, we will show that Thom spectra are the spectra that correspond to the homology theory of cobordism.
April 24 Josh Roberts Algebraic K-theory and crossed objects
After reviewing the classical lower K-groups, Milnor's K_2, and Quillen's plus construction (stopping for examples along the way), we will look at definitions of crossed modules and crossed complexes. After showing that certain K-groups can be regarded as these crossed objects, we will see how this might give insight into explicit descriptions of the plus construction in terms of generators and relations of the Steinberg group.
September 5 Bert Guillou Introduction to computational stable homotopy theory
I will discuss the May and Adams spectral sequences, which are machines for computing the stable homotopy groups of spheres. Using these tools, we will determine the 2-primary stable homotopy groups in dimensions less than 14.
September 19 Jonathan Thompson On the Steenrod Algebra and its Dual
Following closely a 1957 paper of Milnor, I will introduce the Steenrod algebra and discuss some of its properties and structure.
September 26 Bert Guillou Finite subalgebras of the Steenrod Algebra
We will explore the algebra structure of the Steenrod algebra at the prime 2. We will see that every element in positive degree is nilpotent, and we will consider certain finite subalgebras.
October 3 Jonathan Thompson (Even) More on the Steenrod Algebra and Its Dual
Continuing my presentation of Milnor's paper, I will prove a result on the structure of the dual of the Steenrod algebra and give some consequences of this result for the Steenrod algebra.
October 10 Scott Bailey (Clayton State University) Modules and splittings
In this talk, we will discuss past, present, and future work in the classification of stable isomorphism classes of B-modules (where B is a sub-Hopf algebra of the Steenrod algebra). Past, present, and future applications to the splitting of the Tate spectra of v_n-periodic cohomology theories will also be discussed.
Ocotober 18 Carolyn Yarnall (Wabash College) The Slice Tower of Suspensions of HZ
The slice filtration is a filtration of equivariant spectra developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire invariant one problem. I will begin by recalling the definition of the slice filtration along with some of its basic properties. Then I will discuss some computational methods for determining slice towers. Finally, I will present the general form of the slice tower for a suspension of the Eilenberg-MacLane Spectrum associated to the constant Mackey functor for a cyclic p-group and highlight the patterns that arise by showing a few key examples.
October 31 Clinton Hines Combinatorial Formulae for the \Chi_y Genus of Quasitoric Manifolds.
We recall the definition of a quasitoric manifold as any smooth 2n-manifold admitting a nice action of the compact torus. We then consider an equivalent formulation in terms of combinatorial data and its related stably complex structure. Next we'll demonstrate Panov's proof for calculating the \Chi_y-genus of quasitoric manifolds in terms of this combinatorial description and elicit an explicit formula for the Todd genus. Lastly, we'll work through a couple of small dimensional examples and postulate some related conjectures concerning "wedge" quasitoric manifolds.
November 7 Josh Roberts (Piedmont College) Persistent Homology - An Introduction to Applied Algebraic Topology
Given a filtration of a simplicial complex we can construct a series of invariants called the persistent homology groups of the filtration. In this talk we will give a basic introduction to the theory of persistence and explain how these ideas can be used in data analysis.
November 14 Ben Braun Eulerian Idempotents and Hodge-type decompositions of Hochschild homology
The Eulerian idempotents are fascinating elements of the group algebra of the symmetric group. They were first investigated in the 1980's, arising in multiple contexts including topology, representation theory, and combinatorics. In this talk, we will survey how Eulerian idempotents can be used to produce splittings of Hochschild homology. If time permits, we will also discuss type B Eulerian idempotents and splittings of Hochschild homology for algebras with an involution.
November 21 John Mosely The Enriques-Kodaira Classification and $\Omega^{SU}_{4}$
In this talk we will discuss the Enriques-Kodaira Classification of (minimal) compact complex surfaces, and how that helps us understand the problem of representing SU cobordism classes in dimension 4.
January 16 Justin Noel (Mathematics Institute at the University of Bonn and Max-Planck Institute for Mathematics) Equivariant homology of representation spheres and computations indexed by Picard groups.
We extend computations of Lewis and Ferland of the Bredon cohomology of G-representation spheres. Their work gives a complete computation of the RO(C_p) graded groups of the Burnside Mackey functor. We extend their computations to other groups and also identify the Pic(S_{C_n}) groups through a range. The first half of the talk should be rather elementary and suitable for graduate students.
January 31 Nat Stapleton (MIT) The Morava E-theory of Centralizers
We will discuss recent work in progress towards providing an algebro-geometric interpretation for the Morava E-theory of centralizers of tuples of commuting elements in symmetric groups. We will begin with an introduction to the inertia groupoid functor and attempt to say something about its significance in chromatic homotopy theory. Then we will introduce Morava E-theory and discuss its associated formal group. After this we will explain work in progress relating the Morava E- theory of centralizers to schemes that classify very particular subgroup schemes in a p-divisible group built out of the formal group associated to E_n.
February 7 Andrew Wilfong Projective Toric Varieties in Cobordism
Toric varieties are fascinating objects that link algebraic geometry and convex geometry. They make an appearance in a wide range of seemingly disparate areas of mathematics. In this talk, I will discuss the role of projective toric varieties in one facet of topology called cobordism theory. Generally speaking, cobordism is an equivalence relation on smooth manifolds. After an introduction to projective toric varieties and cobordism, I will address the question of when an equivalence class in cobordism contains a projective toric variety, providing results in low dimensions. I will also discuss the role that toric varieties play in the algebraic structure on the set of these equivalence classes.
February 14 Jonathan Thompson A brief introduction to ordinary K-theories
I will discuss some results from a paper of Jack Morava explaining the existence of cohomology theories whose topological indices have interesting arithmetical properties.
February 21 Anna Marie Bohmann (Northwestern University) Graded Tambara functors
10 AM Let G be a finite group. We can consider G-equivariant cohomology theories on G-spaces, which are given by G-equivariant spectra. These spectra don't just have homotopy groups, but rather homotopy "Mackey functors," and this extra structure has proved useful in calculations. If our spectrum has a G-ring structure, then recent work of Strickland and Brun shows that its zeroth homotopy groups form a "Tambara functor." I will discuss current work with Vigleik Angelveit about including the higher homotopy groups: this gives the notion of a graded Tambara functor. I will begin with a discussion of Mackey and Tambara functors before tackling the graded version.
February 28 Clinton Hines Wedge Quasitoric Manifolds
Quasitoric manifolds (QTMs) are smooth compact manifolds admitting a well-behaved action of the compact torus so that the quotient of this action is diffeomorphic (as a manifold with corners) to a combinatorially simple polytope. We'll develop a procedure to attempt to view any QTM as a codimension 2 subquasitoric manifold of an "ambient" wedge QTM. We formulate these wedge QTMs on the level of polytopes from the wedge polytopal construction. The existence of such wedge QTMs in the general case is still unknown but we'll demonstrate a proof for the existence of such constructions for any Bott tower and discuss a similar conjecture concerning Bott manifolds and connected sums of the aforementioned. We will focus on small dimensional examples to view these constructions.
March 21 Kate Ponto Additivity and multiplicativity of traces
For a fibration (with a connected base space) the Euler characteristic of the total space is the product of the Euler characteristics of the base and the fiber. The Euler characteristic is also additive on subcomplexes. The generalizations of the Euler characteristic to fixed point invariants, primarily the Lefschetz number and Reidemeister trace, are similarly additive and multiplicative. Classically these results were proven using a variety of techniques.
Recently, Mike Shulman and I have shown that all of these results are consequences of a simple formal observation and some specific topological input. We think of the Euler characteristic as an endomorphism rather an integer. With this change in perspective, the product of integers becomes a composite of functions and the topological results follow from a more general theorem about composites of traces.
April 4 John Mosley Emulating a Theorem of Stong
In this talk I will briefly describe a theorem of Stong, discuss a similar theorem in a different ring, and discuss a number theory conjecture necessary for the proof of the similar theorem.
April 18 Jonathan Thompson A brief introduction to ordinary K-theories
I will discuss some results from a paper of Jack Morava explaining the existence of cohomology theories whose topological indices have interesting arithmetical properties.
October 8 Jonathan Thompson On the Rigidity of the Elliptic Genus
In the mid 1980's Ochanine made a conjecture concerning genera in the theory of equivariant cohomology. In this talk, we will prove this conjecture in a special case.
October 11 Niles Johnson (Ohio State Newark) Ecological Niche Topology
The ecological niche of a species is the set of environmental conditions under which a population of that species persists. This is often thought of as a subset of "environment space" -- a Euclidean space with axes labeled by environmental parameters. This talk will explore mathematical models for the niche concept, focusing on the relationship between topological and ecological ideas. We also describe applications of machine learning to develop empirical models from data in the field. These lead to novel questions in computational topology, and we will discuss recent progress in that direction. This is joint with John Drake in ecology and Edward Azoff in mathematics.
October 22 Andrew Wilfong Smooth Toric Varieties in Complex Cobordism
In 1960, Milnor and Novikov proved that the complex cobordism ring is a polynomial ring with one generator in each even dimension. However, convenient choices for these generators are still unknown. In this talk, I will discuss the role that smooth projective toric varieties play in this polynomial ring structure. More specifically, I will present evidence supporting the conjecture that the cobordism class of a smooth projective toric variety can be chosen for each polynomial generator.
October 29 John Mosley Representing Cobordism Classes by Non-Singular Algebraic Varieties
A well-known theorem of Stong states that every complex cobordism class contains a non-singular algebraic variety. In this talk, we will discuss his proof of this theorem. We will then discuss the work of Connor and Floyd, connecting complex cobordism to SU cobordism, and consider a similar question for SU cobordism classes.
November 12 Bill Robinson Sheaves, Stalks, and Germs: Turning Your Presuppositions About Sheaves Into Suppositions
Sheaves are constructions that relate ideas of topology, algebraic geometry, and number theory. In topology, they encode distinctions between local and global properties of a space. In this talk, we will introduces sheaves and sheaf cohomology and mention a few reasons why you might care about such a thing. After the talk, the audience will be qualified to use the word "sheafification."
November 19 Bert Guillou Infinite loop spaces
We will discuss loop spaces and infinite loop spaces, which play the roles of groups and abelian groups in homotopy theory. Infinite loop spaces in particular are of interest, as they correspond to (connective) cohomology theories. There are several approaches to the subject, and we will focus on that of G. Segal.
November 29 John Mosely Representing Cobordism Classes by Non-Singular Algebraic Varieties (Qualifying Exam)
A well-known theorem of Stong states that every complex cobordism class contains a non-singular algebraic variety. In this talk, we will discuss his proof of this theorem. We will then discuss the work of Connor and Floyd, connecting complex cobordism to SU cobordism, and consider a similar question for SU cobordism classes.
January 19 Jonathan Tompson A cell structure on the real Grassmannian manifold
In this talk, I will use the notion of a Schubert symbol to define a cell structure on the Grassmannian.
February 1 John Mosley Generalizing a Lemma of Kharlamov
A lemma of Kharlamov states that for a Kähler manifold, V, of complex dimension 2n, χ(V2n )= (-1)n σ(V2n ) mod 4 (where χ is the Euler Characteristic and σ is the signature). In this talk I will present a generalization of this lemma, given by Ochanine, using the Ty-genus. I will also present some background information, regarding formal groups and multiplicative sequences, that is necessary for the generalization.
February 22 Alissa Crans (Loyola Marymount University) A Survey of Quandle Theory
A quandle is a set equipped with two binary operation satisfying axioms that capture the essential properties of the operations of conjugation in a group and algebraically encode the three Reidemeister moves from classical knot theory. This notion dates back to the early 1980's when Joyce and Matveev independently introduced the notion of a quandle and associated it to the complement of a knot. We will focus on an introduction to the theory of quandles by considering examples, discussing quandle (co)homology and applications, and introducing recent work in this area.
March 1 Andrew Wilfong Quasitoric Representatives in Complex Cobordism
In this talk, I will present a result of Buchstaber and Ray which states that every complex cobordism class can be represented by a quasitoric manifold. After a brief introduction to quasitoric manifolds, I will describe a collection of quasitoric manifolds which are projectivizations of line bundles over bounded flag manifolds. These quasitoric manifolds multiplicatively generate the ring of complex cobordism. After proving this, I will discuss the connected sum of quasitoric manifolds, and I will show how this construction can be used to represent every complex cobordism class with a quasitoric manifold.
March 22 Clinton Hines Quasitoric Manifolds and Generalized Bott Manifolds
Based on the works of Choi, Masuda and Suh, we will discuss necessary and sufficient conditions for a quasitoric manifold (over a product of simplices) to be equivalent to a generalized Bott manifold. The argument is formulated around the bundle structure and cohomology rings of the manifolds.
March 29 Teena Gerhardt (Michigan State University) Computations in Algebraic K-Theory
In this talk I will introduce algebraic K-theory and describe an approach to K-theory computations using equivariant stable homotopy theory. I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups using these equivariant methods.
April 5 Doug Ravenel (University of Rochester) The Arf-Kervaire problem in algebraic topology
Hayden Howard Lecture
In 2009 Mike Hill, Mike Hopkins and I solved the Kervaire invariant problem in stable homotopy theory. I will describe the history of the problem beginning with Pontryagin's work on the homotopy groups of spheres in the 1930s and Kervaire-Milnor's work on exotic spheres in the 1960s and give a very brief outline of the proof.
April 12 Kate Ponto Multiplicativity of fixed point invariants
The Euler characteristic of the total space of a fibration is the product of the Euler characteristics of the base and fiber (as long as the base is connected). If the fibration satisfies restrictive additional hypotheses this extends to generalizations of the Euler characteristic such as the Lefschetz number and Nielsen number.
Thinking of the Euler characteristic as an endomorphism rather an integer, the multiplicativity result becomes a factoring result. Recently Mike Shulman and I have shown that this factoring generalizes to the Lefschetz number and Reidemeister trace. This extends the classical multiplicativity results and does not require any hypotheses beyond those needed to define the invariants.
April 19 Bill Robinson Brown Representability
Given a particular sequence of spaces (called an omega spectrum) and homotopy classes of maps into those spaces, it is possible to construct a theory that satisfies the axioms of a general cohomology theory. In this talk I will present a theorem of Brown which is essentially the converse: every cohomology theory can be constructed (''represented'') in this way. The proof will be given for a general contravariant functor satisfying certain axioms, and then for the assembly of these functors into a cohomology theory. I will then give some examples of well-known cohomology theories and the spectra representing them.
April 26 Bruce Hughes (Vanderbilt University) Approximate fibrations and neighborhoods of manifolds
Chapman's 1981 Memoirs of the American Mathematical Society showed the importance of approximate fibrations in the theory of topological manifolds and established some of their most important properties. I will survey some of the developments that took place over the next 30 years. These include a classification theorem from the point of view of controlled topology and a description of germs of neighborhoods of submanifolds of manifolds and stratified spaces. I will then discuss recent joint work with Stacy Hoehn Fonstad, which uses this theory to show how mapping cylinder neighborhoods appear after crossing with the real line.
September 7 and 14 Kate Ponto A little bit of homotopy theory: cohomology and homotopy
I'll talk about connections between cohomology and homotopy (Eilenberg- MacLane spaces and Brown representability). While this is certainly important by itself, it also uses many important ideas and results along the way.
September 21 Andrew Wilfong Stolz's Conjecture and Bott Towers
Stolz's conjecture proposes that the Witten genus vanishes for string manifolds which admit a metric of positive Ricci curvature. This conjecture has only been verified for several special classes of manifolds. Among these are complete intersections in products of complex projective spaces. In this talk, I will discuss the conjecture with regard to Bott towers. These are stacks of projective toric varieties in which each level can be viewed as a generalization of a product of complex projective lines. I will describe how the combinatorial structure of Bott towers can be used to prove the vanishing of the Witten genus for each level of the Bott towers and for string hypersurfaces within them.
September 28 and October 5 Ben Braun Why is a Cohen-Macaulay complex Cohen-Macaulay? And why do we care if we just want to count stuff?
Cohen-Macaulay simplicial complexes are ubiquitous in topological combinatorics. The defining condition for a complex X to be Cohen-Macaulay is typically first given as a vanishing condition on the homology of links of faces of X; this is relatively simple to understand, in that one only needs to know the basics of simplicial homology to parse the definition. However, this definition is poorly motivated and, as a result, is often confusing when first encountered.

In this pair of talks, we will outline the connection between Cohen-Macaulay complexes defined in the above manner and Cohen-Macaulay complexes defined as complexes whose face rings, aka Stanley-Reisner rings, are Cohen-Macaulay. Along the way we will introduce the tool of local cohomology for modules over polynomial rings. Using the ring-theoretic interpretation of what a Cohen-Macaulay complex is, we will give a general framework explaining why Cohen-Macaulay complexes are so useful when studying enumeration problems.

We will assume that the audience knows what a simplicial complex is, and knows at least a little about simplicial homology, but we will begin Part 1 with a brief review of these concepts anyway.

October 12 Bill Robinson Homology of a Relation
Given a relation between two sets, we can form a chain complex that encodes information about that relation. In this talk I will introduce the homology of a relation and present several examples. I will also discuss a duality theorem and briefly sketch the theorem's role in showing that the ÄŒech and Vietoris homology theories are isomorphic.
October 19 Serge Ochanine Fagniano, Euler, Jacobi, Lazard, Hirzebruch, Quillen, Landweber, and so many others
This will be a brief introduction to the interaction of the formal group theory and topology.
October 26 Brad Fox An Extended Euler-Poincare Theorem
The Euler-Poincare Formula is a well-known linear relation involving the f-vector and Betti sequence of a finite simplicial complex. In this talk I will present a theorem by Bjö rner and Kalai that extends this relation as well as characterizes possible f-vector/Betti sequence pairs.
November 2 and 16 John Mosley A congruence between the signature and the Euler characteristic
A lemma of Kharlamov states that for a Kä hler manifold, V, of complex dimension 2n,
χ(V2n )= (-1)n σ(V2n ) mod 4
(where χ is the Euler Characteristic and σ is the signature). In this talk I will present a generalization of this lemma, given by Ochanine, using the Ty-genus. I will also present some background information, regarding formal groups and multiplicative sequences, that is necessary for the generalization.
November 9 and 30 Clinton Hines Toric Varieties and Quasitoric Manifolds in Cobordism
Toric varieties are rich mathematical objects that connect several subjects: algebraic geometry, combinatorics, and topology to name a few. Quasitoric manifolds though similar in several aspects to smooth complete toric varieties, are preferable when dealing with cobordism theory. Indeed, in dimensions greater than 2, every complex cobordism class contains a quasitoric manifold. We will take a look at some of the properties that make quasitoric manifolds agreeable under cobordism, including the connected sum and a quick look at Hirzebruch's T-y genus.
January 27 and February 3 Ben Braun A neat way to prove that K_{3,3} is not planar using the Borsuk-Ulam theorem
Simplicial complexes are a family of topological spaces built out of "nice" pieces of Euclidean space. We will consider the following question: If D is a simplicial complex, for which dimensions n can we embed D in n-dimensional Euclidean space? It is well-known that if D is the complete bipartite graph K_{3,3}, one cannot embed D in the plane; we thus say K_{3,3} is not a planar graph. We will outline a beautiful proof of this fact using the Borsuk-Ulam theorem. Along the way, we will introduce without proof the Borsuk-Ulam theorem and the related machinery of Z_2-spaces, the Z_2-index, and deleted joins of simplicial complexes.
February 10 Ben Braun Glory is fleeting, but a topological invariant is forever
We will discuss the fundamental question "what is an invariant?" In particular, if you have a topological invariant, what is varying? And what is unchanged? Why do we care? We will introduce the Euler characteristic of a space as an example of a topological invariant.

This talk will be accessible to undergraduate math majors. If you have had a topology or analysis course that is a plus, but everyone is welcome!
February 17 Clinton Hines Unit tangent vector fields on spheres, I
We discuss a lower bound to the number of orthonormal tangent vector fields to the n-sphere. This will be achieved via orthonormal multiplications as they relate to Clifford Algebras. We will talk about some concrete examples of Clifford Algebras and specifically how their structures generate these vector fields on spheres. This talk should be accessible to undergraduate math majors. If you've had a topology or analysis course that would be a plus, but everyone is welcome!
March 7 Kate Ponto A little bit of stable homotopy theory
As preparation for the topology seminar on March 24, I'll talk a little about generalized cohomology theories and some of the things they lead to (spectra, ring spectra, and highly structured ring spectra).
March 24 Niles Johnson (University of Georgia) Complex Orientations and p-typicality
In joint work with Justin Noel, we give computational results related to the structure of power operations on complex oriented cohomology theories (localized at a prime p), making use of the amazing connection between complex orientations and the theory of formal group laws. After introducing the relevant concepts, we will describe the main results: for primes p less than or equal to 13, orientations factoring non-trivially through the Brown-Peterson spectrum cannot carry power operations, and thus cannot provide MU_(p)-algebra structure. This implies, for example, that if E is a Landweber exact MU_(p)-ring whose associated formal group law is p-typical of positive height, then the canonical map MU_(p) --> E is not a map of H_infty ring spectra. It immediately follows that the standard p-typical orientations on BP, E(n), and E_n do not rigidify to maps of E_infty ring spectra. We conjecture that similar results hold for all primes.
April 7 Angelica Osorno (University of Chicago) 2-vector bundles and their classifying space
In recent work of Baas-Dundas-Richter-Rognes, the authors define 2-vector bundles and prove that their classifying spaces, K(Vect) is equivalent to the algebraic K-theory of the connective K-theory spectrum ku. In this talk we will give an introduction to bicategories and 2-vector spaces, explain the construction of the classifying space K(Vect). Finally we will explain how some extra structure in the bicategory of 2-vector spaces translate into an infinite loop space structure on K(Vect).
April 14 and 28 Clinton Hines and Beth Kirby The Ochanine Genus, Modular Forms, and the Brown-Kervaire Invariant
We consider a refinement of the universal elliptic genus, called the Ochanine or beta Genus. After a brief treatment of modular forms over graded rings, we examine certain modular forms over KO(lower star). We then show that the beta genus applied to a spin manifold is such a modular form. In a follow-up to this discussion, we will use these constructions to look at the Brown-Kervaire Invariant.
September 10 Andrew Wilfong The chi-y genus of quasitoric manifolds
Quasitoric manifolds can be viewed as a toplogical generalization of non-singular projective toric varieties. In this talk, I will define what a quasitoric manifold is, mainly focusing on its combinatorial structure. I will then present a formula for the chi-y gensu of a quasitoric manifold that utilizes this combinatorial stucture. Finally, I will illustarte how this formula is used with several basic examples.
November 5 Beth Kirby Computing the elliptic genus
We will discuss work by Ochanine and Corbounov on the elliptic genus of a complete intersection in a product of projective spaces. We will consider an example for the level 2 elliptic genus and discuss how this generalized to a level n elliptic genus. The computation is a result of a residue theorem for several complex variables. We may also discuss the challenges of computing the elliptic genus for a more general toric variety.
November 19 Kate Ponto Lefschetz fixed point theorem
Let f be an endomorphism of a finite simplicial complex X. Using the trace from linear algebra and the homology of X, we can assign a rational number to f. This invariant is called the Lefschetz number. Surprisingly, this invariant has connections to the fixed points of f (the points x in X where f(x)=x).

The Lefschetz fixed point theorem: If f has no fixed points the Lefschetz number of f is zero.

We can also associate an integer, called the index, to each fixed point. The index counts how essential the fixed point is. It is one way to get a partial answer the question: if we make a small change to f can we eliminate this fixed point?

The sum of the indices of all of the fixed points of f is called the index of f. If f has no fixed points its index has to be zero. Then the Lefschetz fixed point theorem is a consequence of the following:

Theorem: The Lefschetz number of f equals the index of f.

I will define the Lefschetz number and index and outline a standard proof of the Lefschetz fixed point theorem. If I have time, I will describe a nonstandard proof of the second theorem.
December 3 and 10 Kate Ponto A proof of the Lefschetz fixed point theorem.
I will describe the proof of the Lefschetz fixed point theorem that I find most useful. This proof uses category theory and has several advantages over the classical proof. One of the most significant strengths is that it generalizes easily.
April 19, 2007 Eric Kahn A nice connection between the Burnside and representation rings
This is part of a qualifying exam.
January 23, 2006 Erik Stokes Framed cobordism; the Pontryagin construction II
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
February 21, 2006 Vassily Gorbounov Introduction to the Langlands program
This program is a spectacular interplay between number theory, geometry, representation theory and analysis. Mostly we will concentrate on the geometric version of the program, which comes down to complex geometry and representation theory of infinite dimensional Lie algebras. The first introductory talk will be part of a series of talks based broadly on some work of Beilinson, Drinfeld, and others. Some lectures will be contributed by the other participants. Everybody is welcomed to attend!
February 28, 2006 Marian Anton Classfield from the topological viewpoint
We will explain how a topologist may look at classfield and then focus on the Frobenius element and its corresponding adele.
March 7, 2006 Marian Andon Elliptic curves and Galois representations
We will recall what is the Frobenius element and continue its study. Then we will construct two dimensional Galois representations by using elliptic curves.
April 6, 2006 Erin Militzer Exact couples: the algebraic theory
This is a talk for master degree.
April 11, 2006 Mohamed Elhamdadi Quandle Cohomology and Knot Invariants
Quandles, introduced by D. Joyce in 1982, are algebraic structures which model the Reidemeister moves in knot theory. These structures were discovered independently at the same time by S. Matveev under the name of distributive groupoids. Joyce associated a quandle to a knot, called knot quandle, and proved that it is a complete invariant of knots. Quandle cohomology was introduced by S. Carter et al. in early 2000 as a modification of rack cohomology theory of R. Fenn, C. Rourke, and B. Sanderson. We will give a survey of quandle cohomology and cocycle knot invariants, describe some of our recent joint work with S. Carter and M. Saito and conclude with some open problems.
April 13, 2006 Eric Kahn Relations between ordinary and extraordinary homology
This is a talk for master degree.
April 20, 2006 Joshua Roberts Hopf's Formula and Milnor's K2
This is a talk for master degree.
September 14, 2006 Matthew Wells Hopf Rinow Theorem on Completeness for Riemannian Manifolds
This is part of a qualifying exam.
March 1, 2005 Marian Anton Elliptic Curves I
This is part of a series of elementary lectures. Everybody is welcome.
March 8, 2005 Marian Anton Elliptic Curves II
This is part of a series of elementary lectures. Everybody is welcome.
March 22, 2005 Marian Anton Elliptic Curves III
This is part of a series of elementary lectures. Everybody is welcome.
April 12, 2005 Steve Elliott Simple homotopy theory for cell complexes
April 14, 2005 Elizabeth Stepp Large Whitney levels and closed antichains
April 26, 2005 Jacob Lurie A Generalization of the Character Theory of Hopkins, Kuhn, and Ravenel
In this talk, we will review the "higher" character theory of Hopkins, Kuhn, and Ravenel, which gives a description of the rational Morava E-theory of classifying spaces for finite groups. We will then describe a generalization of their result, and how it leads naturally to the idea of "higher equivariance".
April 27, 2005 Jacob Lurie Elliptic Cohomology and Derived Algebraic Geometry
We will give an overview of the classical approach to elliptic cohomology, leading up to the construction of the spectrum of topological modular forms (tmf) by Hopkins and Miller. We will then introduce the language of derived algebraic geometry, and explain how it can be used to give a new (and easier) construction of tmf.
September 27, 2005 Eric Kahn Smooth manifolds and smooth maps
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
October 4, 2005 Dave Watson The theorem of Sard and Brown
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
October 11, 2005 Jonathan Groves Proof of Sard's theorem
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
October 18, 2005 Jonathan Groves Proof of Sard's theorem (continued)
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint". (Time permitting, some additional comments will be made on regular values.)
October 25, 2005 Josh Roberts The degree modulo 2 of a mapping
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
November 1, 2005 Tricia Muldoon Oriented manifolds
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
November 8, 2005 Andrew Kirby Vector fields and the Euler number
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
November 15, 2005 Erik Stokes Framed cobordism; the Pontryagin construction
This is part of a series of presentations from Milnor's book "Topology from the Differentiable Viewpoint".
November 29, 2005 Matthew Bender Combinatorial homotopy
We discuss the point-set part of a paper by J.H.C. Whitehead.