Title:  Chromatic Levels in the Homotopy Groups of Spheres

Abstract:  Understanding the homotopy groups of spheres $\pi_nS^k$ 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 $\pi_{n+k}S^k$ is isomorphic to $\pi_{n+k+1}S^{k+1} $ when $k$ is large. Homotopy theorists call this phenomena \emph{stabilization}. The stable homotopy groups of spheres are defined to be these families of isomorphic groups. They form a ring, commonly denoted by $\pi_*S$. 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 \emph{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 \emph{chromatic splitting conjecture}.

Title:  Algebraic K-theory and crossed objects

Abstract:  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.

Title:  Cobordism and Thom Spectra

Abstract:  Building on previous seminar talks, we will show that Thom spectra are the spectra that correspond to the homology theory of cobordism.

Title:  Power operations and the Kunneth spectral sequence

Abstract:  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.

Title:  Yet Another Friendly Talk About Cobordism

Abstract:  TBA

Title:  Hirzebruch's proof of his Signature Theorem

Abstract:  We will prove Hirzebruch's signature theorem and show its utility in a computation.

Title:  A Different Friendly Talk About Cobordism

Abstract:  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.

Title:  The Hopf invariant one problem via K-theory

Abstract:  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.

Title:  An introduction to topological K-theory

Abstract:  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.