Topology Seminar

## University of Kentucky Topology Seminar

#### During the semester the seminar usually meets at 3:30PM on Thursdays in POT 745

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

#### Spring 2018

 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 February 1 February 9 Clover May (University of Oregon) February 15 Ang Li February 22 March 1 Ben Riley March 8 Paul VanKoughnett (Northwestern University) March 22 Julie Vega March 29 Shane Clark April 5 April 10 Mike Hill (UCLA) April 19 Kalila Sawyer April 26

#### Fall 2017

 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 9 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!)

#### Spring 2017

 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).

#### Fall 2016

 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 Hedmark A 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 Yarnall Towers 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.

#### Spring 2016

 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 Î»-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 â€”> X be a smooth free involution. The triple (X, t, n) has the Borsuk-Ulam property if for every map f: Xâ€”> 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.

#### Fall 2015

 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 exoticâ€™ 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

#### Fall 2014

 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 Hirzebruchâ€™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.

#### Spring 2014

 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.

#### Fall 2013

 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.

#### Fall 2012

 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.

#### Spring 2012

 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 KÃ¤hler manifold, V, of complex dimension 2n, Ï‡(V2n )= (-1)n Ïƒ(V2n ) mod 4 (where Ï‡ is the Euler Characteristic and Ïƒ 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.

#### Fall 2011

 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.

#### Spring 2011

 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.

#### Fall 2010

 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.

#### Archive

 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 Dissertation. April 14, 2005 Elizabeth Stepp Large Whitney levels and closed antichains Dissertation. 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 Colloquium 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.