TAPIRS

June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence. But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group? The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points. I will present an equivariant log concavity conjecture, and show that one can use representation stability to prove infinitely many cases of this conjecture for configuration spaces.

This talk is based on joint work with Jacob Matherne, Dane Miyata, and Eric Ramos.

The algebraic objects in this talk are motivated by applications to algebraic statistics. Toric ideals of hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations. Lastly, the method provided in this work allows one to define Segre languages of algebraic objects in a more general framework. This is based on joint work with Uwe Nagel.

This talk is about the strength of homogeneous polynomials. The strength is a subadditive invariant determined by the convention that a nonzero polynomial has strength 1 exactly when it is reducible. This invariant has been defined by Ananyan and Hochster in their paper proving Stillman's conjecture and has appeared in various works since.

• Why look at the strength of polynomials?
• How do you compute it?
• Is bounded strength a closed condition?
• What is the strength of a generic polynomial?

I will answer some of these questions.

The category OI has objects the finite totally ordered sets and morphisms the order-preserving injections. There is a shift functor on the category of OI-modules. I will discuss what we know about an OI-module under iterated shifts and how it can be used to find an explicit bound for the Castelnuovo-Mumford regularity of the OI-module in terms of its generating degree and presentation degree. (This is a joint work with Liping Li.)

I will present some results from a work in progress joint with Th. Heidersdorf on the Deligne categories for the family of groups GL_n(F_q), n ≥ 0. The Deligne categories interpolate the tensor categories of complex representations of GL_n(F_q), and have been previously constructed by F. Knop and E. Meir. I will describe some properties of these categories as well as their relation to the category of algebraic representations of the infinite group GL_∞(F_q).