C.A.T.S.:

Rankeya Datta: Applications of Mittag-Leffler modules to prime characteristic

Abstract: A central conjecture in prime characteristic commutative algebra is that test elements exist for the class of excellent domains. By the pioneering works of Hochster and Huneke, the conjecture holds for F-finite noetherian domains and domains that are essentially of finite type over excellent local rings. However, the problem remains elusive in general, even in interesting geometric settings. For example, due to recent constructions of Murayama and myself, we do not know if the rigid analytic analog of the coordinate ring of an affine variety admits test elements over an arbitrary non-Archimedean field of prime characteristic. Sharp proposed an approach to tackle the conjecture for rings of this latter type via a technical notion called Frobenius intersection flatness. This notion was first introduced by Hochster and Huneke in their efforts to show the existence of test elements. In unrelated seminal work on the descent of projectivity, Raynaud and Gruson introduced the notion of a Mittag-Leffler module. In this talk we will connect the intersection flatness condition to the Mittag-Leffler condition. Consequently, we will show how the existence of test elements in the rigid analytic and related settings reduces to a simple question about the openness of pure loci for certain maps of modules. This talk is based on joint work with Neil Epstein and Kevin Tucker and is also related to work with Takumi Murayama and Karl Schwede.

Florian Enescu: Rational twist in positive characteristic

Abstract: Motivated by the definition of the Frobenius complexity of a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of affine semigroup rings for which these generating functions are rational. We will discuss this class of rings and aspects of the rationality of the complexity generating function. This work is joint with Yongwei Yao.

Tài Hà Newton-Okounkov bodies and algebraic invariants of graded families of monomial ideals.

Abstract: We shall discuss how to use combinatorial information of the Newton-Okounkov body associated to a graded family of monomial ideals to investigate the Noetherian property of the corresponding Rees algebra, and to understand and bound the analytic spread, asymptotic regularity, symbolic relation type and Veronese degree of the given family of ideals.

Patricia Klein: Alternating sign matrix varieties

Abstract: Matrix Schubert varieties, introduced by Fulton in the '90s, are affine varieties that "live above" Schubert varieties in the complete flag variety. They have many desirable algebro-geometric properties, such as irreducibility, Cohen--Macaulayness, and easily-computed dimension. They also enjoy a close connection with the symmetric groups. Alternating sign matrix (ASM) varieties, introduced by Weigandt just several years ago, are generalizations of matrix Schubert varieties in two senses: (1) ASM varieties are unions of matrix Schubert varieties and (2) the defining equations of ASM varieties are determined by ASMs, which are generalizations of permutation matrices. ASMs have been important objects of study in enumerative combinatorics since at least the '80s and appear in statistical mechanics as the 6-vertex lattice model. Although ASMs have a robust combinatorial underpinning and although their irreducible components are matrix Schubert varieties, they are nevertheless much more difficult to get a handle on than matrix Schubert varieties themselves. In this talk, we will define ASMs, compare and contrast with matrix Schubert varieties, and state some open problems.

Paolo Mantero: Symbolic powers and matroids

Abstract: Matroids are ubiquitous objects in Mathematics generalizing the concept of linear independence from Linear Algebra. Their theory has a rich history and it is a vibrant area of research. Since they can be viewed as special simplicial complexes, one can study their Stanley--Reisner ideals I_Delta. In fact, in 2011, Minh and Trung, Terai and Trung, and Varbaro gave a new, elegant characterization of matroids in terms of the Cohen--Macaulay property of the symbolic powers of I_Delta. We will discuss a short, elementary proof their theorem and some questions arising naturally from this new approach. The talk is based on joint work with J. Lyle.

Frank Moore: Non-commutative invariant theory

Abstract: Let k be a field, and let G be a finite subgroup of GL_n(k). Then G acts naturally on the polynomial ring S = k[x_1,\dots,x_n]. The ring of invariants of this action, denoted S^G, is the set of polynomials on which G acts trivially. The study of such rings inspired many of the early researchers in commutative algebra such as Emmy Noether and David Hilbert. In noncommutative algebraic geometry, one is interested in replacing S with a noncommutative analogue, called an AS-regular algebra. This leads one to wonder which results from classical invariant theory remain true in this broader context. In this talk, I will survey both the commutative and noncommutative landscape of invariant theory, with mention of some of our recent work on the noncommutative side of the story.

Christopher Manon: Cox rings of toric vector bundles

Abstract: I'll give an overview of some recent work on the Cox rings of projectivized toric vector bundles. A toric vector bundle is a vector bundle over a toric variety equipped with an action by the defining torus of the base. Toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization. I'll begin with a classification result which shows that a toric vector bundle can be captured by an arrangement of points on the Bergman fan of a certain matroid and describe how this data leads to a presentation of the Cox ring. Then I'll describe how these properties interact with natural operations on toric vector bundles. This involves the geometry of the closely related class of toric flag bundles and links some combinatorial questions about multilinear operations on matroids to the topic of representation stability. This is joint work with Kiumars Kaveh and Courtney George.