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.

For a fibration (with a connected base space) the Euler characteristic of the total space is the product of the Euler characteristics of the base and the fiber.  The Euler characteristic is also additive on subcomplexes.  The generalizations of the Euler characteristic to fixed point invariants, primarily the Lefschetz number and Reidemeister trace, are similarly additive and multiplicative.   Classically these results were proven using a variety of techniques.

Recently, Mike Shulman and I have shown that all of these results are consequences of a simple formal observation and some specific topological input.  We think of the Euler characteristic as an endomorphism rather an integer.  With this change in perspective, the product of integers becomes a composite of functions and the topological results follow from a more general theorem about composites of traces.

Quasitoric manifolds (QTMs) are smooth compact manifolds admitting a well-behaved action of the compact torus so that the quotient of this action is diffeomorphic (as a manifold with corners) to a combinatorially simple polytope.  We'll develop a procedure to attempt to view any QTM as a codimension 2 subquasitoric manifold of an "ambient" wedge QTM.  We formulate these wedge QTMs on the level of polytopes from the wedge polytopal construction.  The existence of such wedge QTMs in the general case is still unknown but we'll demonstrate a proof for the existence of such constructions for any Bott tower and discuss a similar conjecture concerning Bott manifolds and connected sums of the aforementioned.  We will focus on small dimensional examples to view these constructions.