Topology seminar - Fall 2013
September 5
Bert Guillou - Introduction to computational stable homotopy theory
I will discuss the May and Adams spectral sequences, which are machines for computing the stable homotopy groups of spheres. Using these tools, we will determine the 2-primary stable homotopy groups in dimensions less than 14.
September 19
Jonathan Thompson - On the Steenrod Algebra and its Dual
Following closely a 1957 paper of Milnor, I will introduce the Steenrod algebra and discuss some of its properties and structure.
September 26
Bert Guillou - Finite subalgebras of the Steenrod Algebra
We will explore the algebra structure of the Steenrod algebra at the prime 2. We will see that every element in positive degree is nilpotent, and we will consider certain finite subalgebras.
October 3
Jonathan Thompson - (Even) More on the Steenrod Algebra and Its Dual
Continuing my presentation of Milnor's paper, I will prove a result on the structure of the dual of the Steenrod algebra and give some consequences of this result for the Steenrod algebra.
October 10
Scott Bailey (Clayton State University) - Modules and splittings
In this talk, we will discuss past, present, and future work in the classification of stable isomorphism classes of B-modules (where B is a sub-Hopf algebra of the Steenrod algebra). Past, present, and future applications to the splitting of the Tate spectra of v_n-periodic cohomology theories will also be discussed.
October 18
Carolyn Yarnall (Wabash College) - The Slice Tower of Suspensions of HZ
The slice filtration is a filtration of equivariant spectra
developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire
invariant one problem. I will begin by recalling the definition of the
slice filtration along with some of its basic properties. Then I will
discuss some computational methods for determining slice towers. Finally,
I will present the general form of the slice tower for a suspension of the
Eilenberg-MacLane Spectrum associated to the constant Mackey functor for a
cyclic p-group and highlight the patterns that arise by showing a few key
examples.
October 31
Clinton Hines - Combinatorial Formulae for the \Chi_y Genus of Quasitoric Manifolds.
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.
November 7
Josh Roberts (Piedmont College) - Persistent Homology - An Introduction to Applied Algebraic Topology
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.
November 14
Ben Braun - Eulerian Idempotents and Hodge-type decompositions of Hochschild homology
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.
November 21
John Mosley - The Enriques-Kodaira Classification and $\Omega^{SU}_{4}$.
In this talk we will discuss the Enriques-Kodaira Classification of (minimal) compact complex surfaces, and how that helps us understand the problem of representing SU cobordism classes in dimension 4.
