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.
Title: Eulerian Idempotents and Hodge-type decompositions of Hochschild homology
Abstract: 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.
Title: Persistent Homology - An Introduction to Applied Algebraic Topology
Abstract: 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.
Title: Combinatorial Formulae for the \Chi_y Genus of Quasitoric Manifolds.
Abstract: 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.
