Topology seminar - 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 `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 normal surfaces are the basis to many of these algorithms.
