Topology seminar - Spring 2013

January 18

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 at 10AM

Anna Marie Bohmann - Northwestern University

Graded Tambara functors

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.