Schedule for: 16w5155 - Free Resolutions, Representations, and Asymptotic Algebra

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

It is well-known by now that homological stability for a sequence of moduli spaces $X_1, X_2, X_3$, over a finite field $F_q$ sometimes translates into the existence of a limit for the point-count $$q^{-dim X_n} |X_n(F_q)|.$$ Sometimes you can tell a similar story about sequences of moduli spaces whose cohomology is representation (or: naturally forms a finitely generated module for some FI-like category.) This provides, for example, a geometric way of thinking about the distribution of irreducible factors of random polynomials and of irreducible factors of random maximal tori in algebraic groups over $F_p$. I will also talk about work of Vlad Matei on the function-field Landau problem — how often is a random polynomial a sum of two squares? — and explain what this has to do with the category FI_B introduced by J. Wilson.

Twisted commutative algebras are a generalization of commutative algebras in which the commutativity axiom has been modified. These algebras, and their modules, appear often when studying stability phenomena: for example, modules over the simplest tca are equivalent to FI-modules, while modules over another specific tca govern the stable representation theory of the orthogonal group. In this talk, I will introduce tca's, explain what we know and don't know about them, and how they connect to other topics of interest (such as FI-modules).

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Let $F_k(M)$ denote the ordered k-point configuration space of a connected open manifold M. For a given manifold, as $k$ increases, this family of spaces exhibits a phenomenon called "representation stability" with respect to the natural symmetric group actions. In this talk I will discuss what this means and describe a new method for establishing this form of stability, extending previous results of Church and others to include non-orientable manifolds. This project is work in progress, joint with Jeremy Miller.

I will survey some results and questions concerning the asymptotic behavior of the syzygies of algebraic varieties embedded by complete linear series as the positivity of the embedding line bundle grows.

The geometry of a Cartier divisor $L$ on an algebraic variety $X$ is encoded by its section ring $R(X,L)$. Such rings can be quite complicated; for example, they need not be finitely generated. Nevertheless, useful information can be extracted by studying the asymptotic behavior of the graded pieces. I'll discuss the first steps toward an analogous theory for cycles of higher codimension. The key perspective is that enumerative constructions should play the role of sections for divisors. The talk will focus on the surprising relationships with convex analysis and the theory of convex bodies.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The object of "generic representation theory" is to describe families of $\text{GL}_n(\mathbb{F}_q)$-representations in characteristic $p$, similar to the way that an FI-module captures a whole family of $S_n$-representations. However a basic finiteness property of generic representations, the Lannes-Schwartz conjecture of 1994, was never resolved. Putman, Sam, and Snowden proved this conjecture last year by understanding VI-modules and VIC-modules, which are $\text{GL}_n(\mathbb{F}_q)$ analogues of FI-modules. At the same time, the methods they introduced provide the strongest tools we have for proving finiteness properties for twisted commutative algebras like FI, FI$_d$, VI, etc. I'll give an accessible overview of generic representations and describe the innovations of Putman, Sam, and Snowden, which show us that the Lannes-Schwartz conjecture was not really about characteristic-$p$ representations at all. I'll also explain how their methods provide "user-friendly" tools usable by non-experts.

Families of polynomial ideals in high dimension but with symmetry often exhibit certain stabilization even as the dimension grows, for example being generated by the orbits of a short list of polynomials. Similarly an equivariant Gröbner basis of an ideal is a set of polynomials whose orbits form a Gröbner basis, which is a useful computational tool for working with these families of ideals. We describe the current state of equivariant Gröbner basis algorithms including criteria for guaranteeing termination and strategies for speeding up computation. This is joint work with Chris Hillar and Anton Leykin.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

There has been a wealth of work done over several decades regarding the rank-selected homology and Whitney homology of the partition lattice by Hanlon, Stanley, Sundaram, Wachs, Welker, and numerous others. This talk will focus on new results and new questions that came out of taking a representation theoretic stability perspective. The $S_n$-module structure for the ith cohomology group for the complement of a type A complexified braid arrangement (and hence for the configuration space for n distinct points in the plane) can be rephrased as Whitney homology of the partition lattice via an $S_n$-equivariant version of the Goresky–MacPherson formula. Thus, results from combinatorics regarding the partition lattice translate to that setting. This is joint work with Vic Reiner.

The space of $m \times n$ matrices admits a natural action of the group $GL_m \times GL_n$ via row and column operations on the matrix entries. The invariant closed subsets are the determinantal varieties defined by the (reduced) ideals of minors of the generic matrix. The minimal free resolutions of these ideals are well-understood by work of Lascoux and others. There are however many more invariant ideals which are non-reduced, and they were classified by De Concini, Eisenbud and Procesi in the 80s. I will explain how to determine the syzygies of a large class of these ideals by employing a surprising connection with the representation theory of general linear Lie superalgebras. This is joint work with Jerzy Weyman.

Inspired by results of Ein, Lazarsfeld, Erman, and Zhou on the non-vanishing of Betti numbers of high Veronese subrings, we describe the behaviour of the Betti numbers of Stanley-Reisner rings associated with iterated barycentric or edgewise subdivisions of a given simplicial complex. Our results show that for a simplicial complex $\Delta$ of dimension $d-1$ and for $1 \leq j \leq d-1$ the number of $0$'s in the $j$-th linear strand of the minimal free resolution of the $r$-th barycentric or edgewise subdivision is bounded above only in terms of $d$ and $j$ (and independently of $r$). This is joint work with Aldo Conca and Volkmar Welker.

I'll explain a connection between Hopf rings and secant schemes of Veronese embeddings of arbitrary projective schemes and how this can be used to prove the existence of a uniform bound on the degrees of the minimal generators of the ideal of the rth secant scheme independent of the Veronese embedding. This is based on arXiv:1510.04904 [math.AC].

In the same way that a matrix of homogeneous polynomials gives rise to a graded module, a matrix over a category gives rise to a "graded module" where the objects of the category provide the degrees, and the arrows provide the monomials (appearing in linear combinations as entries in the matrix). When the category is combinatorial in nature, such a matrix may be entered into a computer. Using examples from combinatorics, geometry, and topology, I will demonstrate a computer program that takes a matrix over the category of finite sets and returns its Hilbert series.

I will discuss representation-theoretic patterns in the cohomology of congruence subgroups as the rank goes to infinity. This is joint work with Steven Sam.

Much of the work in homological invariants of FI-modules has been concerned with properties of certain right exact functors. One reason for this is that the category of finitely generated FI-modules over a Noetherian ring very rarely has sufficiently many injectives. In this talk we consider the (left exact) torsion functor on the category of finitely generated FI-modules, and show that its derived functors exist. Properties of these derived functors, which we call the local cohomology functors, can be used in reproving well known theorems relating to the depth, regularity, and stable range of a module. We will also see that various facts from the local cohomology of modules over a polynomial ring have analogs in our context. This is joint work with Liping Li.

Many statistical models come in families of algebraic varieties parameterised by combinatorial data, such as a finite graph or a natural number. The question is whether their defining equations look alike for all sufficiently large models in the family. I will present older and newer positive results of this kind.

MCM approximations, introduced by Auslander and Buchweitz, provide interesting invariants of modules over a complete intersection (or more generally a Borenstein ring). I'll explain the (simple) computation of them, and some new observations concerning complete intersections that I've been discussing with Sasha Pavlov and Irena Peeva; we have some conjectures that M2 has played a role in formulating and testing!

In this talk we will consider some families of varieties with actions of certain finite reflection groups -- varieties such as the hyperplane complements or complex flag manifolds associated to these groups. Beautiful results of Grothendieck–Lefschetz and Lehrer relate the topology of these complex varieties with point-counting over finite fields. Church, Ellenberg and Farb noticed that, in this context, representation stability in cohomology corresponds to asymptotic stability of various point counts over finite fields. The cohomology rings of the families considered have the structure of a graded FIW-module. We will discuss what this means, and how asymptotic stability is a direct consequence of the finite generation for FIW-algebras generated in degree at most one. We will also see that this is not necessarily the case when some of the generators are in degree two or higher. This is joint work with Jennifer Wilson.

I will discuss work related to strengthening homological tools over Cox rings of smooth toric varieties. In short, it appears that minimal free resolutions are not always the best complexes to capture geometry. This is joint work in progress with Daniel Erman and Gregory G. Smith.

Ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid of strictly increasing functions arise in various contexts. We study such an ideal using an ascending chain of invariant ideals. We establish that the associated equivariant Hilbert series is a rational function in two variables. This is used to prove that the Krull dimensions and multiplicities of ideals in such an invariant filtration grow eventually linearly and exponentially, respectively. Furthermore, we determine the terms that dominate this growth. This may also be viewed as a method for assigning new asymptotic invariants to a homogenous ideal in a noetherian polynomial ring. This is based on joint work with Tim Roemer.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.