Workshop
I will introduce polynomial functors, closed subsets thereof, and prove the descending chain condition for these. I will then discuss some applications, both of the DCC itself and of ideas that go into its proof. These applications include Stillman's conjecture (an algorithmic proof with Lason–Leykin that follows work by Erman–Sam–Snowden) and shorter proofs of earlier results with Kuttler and with Eggermont on (not necessarily bounded) Delta varieties and Plücker varieties.
What does it mean for a sequence of representations of different groups to "stabilize"? Which representation theoretic quantities should we expect to stabilize, and which should we expect to be eventually polynomial? What should we expect to happen in the modular setting (positive characteristic or quantum parameter a root of unity) where semisimplicity fails? In this minicourse we will discuss various instances of stability and polynomiality phenomena in representation theory, going into some of the underlying representation theory behind representation stability. The focus will be on giving combinatorial and categorical descriptions of the representation theory of symmetric groups and related families of groups and algebras, and using these descriptions to see how these stability, polynomiality, and periodicity phenomena arise.
In this minicourse, through lectures and exercises, we will develop the foundations of FI-modules and their variants. We will survey applications to topology, algebraic geometry, and combinatorics. The theory will draw on representation theory, category theory, homological algebra, and algebraic combinatorics, though prerequisites will be kept to a minimum.
Study of (homogeneous) ideals of a polynomial ring in countably many variables that are invariant under the action of the monoid of strictly increasing functions appeals to various areas. In this talk, we discuss the recent results about these ideals, and the rationality of their equivariant Hilbert series.
Every finitely generated FI-module may be described by a presentation matrix. I will explain how to think about these matrices, and how to perform some important operations. In particular, we will see how to compute the eventual behavior of an FI-module from its presentation matrix. Part of this talk is based on joint work with P. Patzt.
For fixed positive integers n and k, the Kneser graph KG(n,k) has vertices labeled by k-element subsets of {1,2,...,n} and edges between disjoint sets. Keeping k fixed and allowing n to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with all of the other actions. In this talk, we provide a framework for studying families of this kind using the FI-module theory of Church, Ellenberg, and Farb, and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequences span a range of topics including enumeration, concerning counting occurrences of subgraphs, topology, concerning Hom-complexes and configuration spaces of the graphs, and algebra, concerning the changing behaviors in the graph spectra. We will also hint at how this framework can be exploited to prove non-trivial facts about random walks on these families of graphs. This talk spans joint work with Graham White and David Speyer
Conference
Let V be a finite-dimensional (complex) vector space, and Sym(V) be the symmetric algebra on this vector space. We can consider the multiplication map Sym(V) ⊗ V → V as a complex of GL(V)-representations of length 2.
I this talk, I will describe how tensor powers of the above complex define interesting complexes of representations of the symmetric group S_n, which were studied by Deligne in the paper "La Categorie des Representations du Groupe Symetrique S_t, lorsque t n’est pas un Entier Naturel".
I will then explain how computing the cohomology of these complexes helps establish a relation between the Deligne categories and the representations of S_∞, which are two natural settings for studying stabilization in the theory of finite-dimensional representations of the symmetric groups.
Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan–Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength.
Through Bieri–Eckmann duality the top dimensional cohomology of the congruence subgroups of SL_n(Z) is identified with the coinvariants of the Steinberg module. We will give an introduction to these concepts. In joint work with Jeremy Miller and Andy Putman, we prove that cohomology classes conjectured by Lee and Szczarba in the 1970's exist, and find more for all primes p>5.
Borel–Serre duality relates high dimensional cohomology of arithmetic groups to the low dimensional homology of these groups with coefficients in the Steinberg representation. We recall Bykovskii’s presentation for the Steinberg representation and explain its connection to modular symbols. Next, we describe the Steinberg representation as an object in a symmetric monoidal category, and use its presentation to describe an action of the free skew commutative algebra. Finally, we perform a Gröbner-theoretic analysis of this action to obtain new information on the homology of certain arithmetic groups with coefficients in the Steinberg representation. For example, we show that the sequence of homology groups H_1(Gamma_n(3), St_n) exhibit representation stability. In other words, the codimension one cohomology of level three congruence subgroups is representation stable. This is an ongoing project with Jeremy Miller and Peter Patzt.
The space M_{g,n} is a compactification of the moduli space algebraic curves with marked points, obtained by allowing smooth curves to degenerate to nodal ones. We will talk about how the asymptotic behavior of its homology, H_i(M_{g,n}), for n >> 0 can be studied using the category of finite sets and surjections.
Consider the following classical question: What is the minimum k, such that one can write the roots of a general degree n polynomial using only functions of k variables? In joint work with Benson Farb, we broaden the context to define this as an intrinsic invariant, the "resolvent degree", of a finite group G. In this talk, I'll explain the basics of resolvent degree, its ties to representation theory, and I'll discuss ongoing joint work with Benson Farb and Mark Kisin in which we have made some progress on computing resolvent degree using the geometry of the moduli of principally polarized abelian varieties.