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.

I will discuss some results from a paper of Jack Morava explaining the existence of cohomology theories whose topological indices have interesting arithmetical properties.

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.

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.

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.

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.

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."

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.

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.