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.

Title:  The Slice Tower of Suspensions of HZ



Abstract:  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.



 

Title:  Modules and splittings

Abstract:  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.

 

Title:  (Even) More On the Steenrod Algebra and its Dual

Abstract:  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.

Title:  Finite subalgebras of the Steenrod Algebra

Abstract:  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.

Following closely a 1957 paper of Milnor, I will introduce the Steenrod algebra and discuss some of its properties and structure.

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.

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

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.