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Current Schedule

This Week

Monday April 13 at 4:10 PM in SE 105 [Special Day]
Seymour Sherman Memorial Lecture: Mike Hill (Minnesota), “Group actions, algebra up to homotopy, and flavors of commutativity”

A group is just a set together with a multiplication with certain properties, and these are ubiquitous in modern mathematics. In homotopy theory, we often see less-rigid objects, with many of the familiar properties like associativity or commutativity only holding "up to homotopy". This part of the story has been well-understood since the 1970s. 

When we add in the action of a fixed finite group, the game changes wildly. Even what we mean by commutative becomes less clear. This is the heart of the evolving subject of equivariant algebra. I'll describe how to understand groups "up to homotopy", what we see when we put in a group action, and how a classification problem of what "commutative" means in this context connects to combinatorics.

Wednesday April 15 at 4:00 PM in RH 104
Morgan Opie (Northwestern), “Comparing methods for enumerating unstable topological vector bundles”

Given a finite CW complex X, fix reduced complex topological K-theory class h on X, and a positive integer r. It is classical that there are finitely many rank r unstable representatives for the K-theory class, so we can try to compute the number of such representatives. However, the problem is generally quite hard. We focus on the case of complex projective spaces. When h=0, i.e., for stably trivial bundles on complex projective spaces, there are some additionally tools from Weiss calculus that can be applied extremely successfully, as shown by Hu. One might hope that these methods are also useful for enumerating rank r vector bundles with fixed non-zero K-theory classes, but as of yet there is no computationally tractable way to relate the two problems. In this talk, I will survey some experimental evidence for a strong relationship between the h=0 and h\neq 0 enumerations, based on computations for small corank bundles. This talk is based joint work-in-progress with Yang Hu.

Upcoming Weeks

Wednesday April 22 at 4:00 PM in RH 104
Itamar Mor (UIUC), “The synthetic EHP sequence”

The lower central series is a classical tool that deforms any group into an abelian group (in fact, into a Lie algebra). In the 60s, Curtis and Rector used this to define a version of the unstable Adams spectral sequence, computing the homotopy groups of a space from the free (derived) Lie algebra on its homology. I'll describe a categorification of this construction in the spirit of synthetic spectra, which gives a way to deform a space into a derived Lie algebra. Many classical constructions in unstable homotopy theory may be deformed in this way: for example, I will discuss how this may be used to relate the classical EHP spectral sequence (which I will introduce) to Curtis' algebraic EHP spectral sequence, which can essentially be computed by machine.

Wednesday April 29 at 4:00 PM in RH 104
Ramyak Bilas (IU)