The Schedule

Speaker Craig Huneke, University of Virginia

Title: The necessary, the sufficient or both?  A look at past and current attempts to understand licci ideals.

Abstract:

Since the 1970s, there have been many attempts to understand ideals in the linkage class of complete intersections (licci ideals).  At first, many properties were found that were necessary to be licci, showing both how special and powerful their structure can be.  Very few sufficient conditions were known.  This talk will review the current state of affairs, ending with a new sufficient condition to be licci.  All the new work is joint with Claudia Polini and Bernd Ulrich.

Speaker Daniel Katz, University of Kansas

Title: Joint reductions and mixed Buchsbaum-Rim polynomials of complete modules over a two-dimensional regular local ring.

Abstract: 

In this talk we present some recent work with V. Kodiyalam and J. Verma concerning complete modules over two dimensional regular local rings. We give a joint reduction number zero result and describe the multivariable Buchsbaum-Rim polynomial for a collection of complete modules. To establish our results, we introduce a new type of joint reduction and focus on a particular mixed Buchsbaum-Rim multiplicity. Most of the talk will recount the many antecedents for the topics addressed in this work.

Speaker Ryan Watson, University of Nebraska-Lincoln

Title: Cohomological Support Varieties Under Local Homomorphisms

Abstract: 

Given a finitely generated module M over a commutative noetherian local ring R, one may assign to it a conical affine variety called the cohomological support variety of M over R. This theory was first developed by Avramov for local complete intersection rings, and by the work of many has now been extended to encompass all commutative noetherian local rings. Geometric properties of this variety encode homological information of M and R. In this talk I will discuss what cohomological support varieties are, why they are useful, and some recent work on how they behave when restricting along a local homomorphism.

Speaker Adam LaClair, University of Nebraska-Lincoln

Title: Graded Möbius Algebras

Abstract:

The graded Möbius algebra of a matroid is a graded algebra which encodes the combinatorics of the lattice of flats of the matroid. It has featured prominently in recent work by Braden, Huh, Matherne, Proudfoot, and Wang in their proof of the Dowling–Wilson Top-Heavy Conjecture. In this talk, we will investigate algebraic properties of the graded Möbius algebra—such as Gröbner bases, presentation, and Koszulness—in terms of the combinatorial structure of the matroid. This is joint work with Matthew Mastroeni, Jason McCullough, and Irena Peeva.

Reception 

Day 2 (all times are in CDT)

October 26

Speaker Nil Şahin, Bilkent University

Title: Sally's Conjecture for 4-generated Numerical Semigroups

Abstract:

Characterizing numerical functions that might be Hilbert functions of one dimensional Cohen-Macaulay local rings is an open question. There is a conjecture of Judith Sally saying that "Hilbert function of a one dimensional Cohen-Macaulay local ring with small enough embedding dimension is nondecreasing". In this talk, we will focus on 4 generated symmetric and pseudo symmetric monomial curves and show that 4 generated pseudo symmetric monomial curves  satisfy Sally's conjecture.

Speaker Min Hyeok Kang, University of Missouri

Title: The Jordan Type of a Multiparameter Persistence Module

Abstract: 

In this talk, we present an approach to defining invariants of persistence modules over a poset inspired by the theory of modules of constant Jordan type. We define the Jordan type of a persistence module as the Jordan type of an associated nilpotent operator. Under this framework, we construct a new invariant and establish some completeness and stability results. In particular, we prove that the multirank invariants are complete for zigzag persistence modules. This is based on joint work with Calin Chindris and Dan Kline.

Speaker Christopher Wong, University of Kansas

Title: On the Bernstein-Sato Polynomial over a Numerical Semigroup Ring

Abstract:

Given a polynomial f, we can associate to it an invariant called the Bernstein-Sato polynomial (of f), whose roots possess many remarkable properties that encode information about the singularities of f. Recently there has been work to extend this theory to non-regular rings. In this talk we will focus on numerical semigroup rings, coordinate rings of one dimensional curves. We will discuss recent work with Hernández where we show that the Bernstein-Sato polynomial exists. Moreover, we can use techniques in positive characteristic to obtain explicit descriptions of the roots, which show that even in this non-regular case, the Bernstein-Sato polynomial reveals information not only about f but of the ambient numerical semigroup ring. 

Speaker Timothy Duff, University of Missouri

Title: Universal Gröbner Bases for (Universal) Multiview Ideals

Abstract: 

The universal multiview variety is the image of a rational map which models n arbitrary pinhole cameras and their n 2D projections of an arbitrary 3D point. If we fix a particular n-tuple of pinhole cameras, an analogous construction produces the so-called multiview variety associated to that n-tuple. Until recently, multiview varieties and their vanishing ideals have been much better studied than their universal counterparts. In joint work with Jack Kendrick and Rekha Thomas, we extend a result of Aholt-Sturmfels-Thomas concerning universal Gröbner bases (UGB) of multiview ideals to the setting of universal multiview ideals. Our result implies some nice commutative-algebraic consequences: for example, (universal) multiview ideals are Cartwright-Sturmfels, meaning that their (multigraded) generic initial ideals are always radical. In this talk, I will describe a key ingredient in our proof — a recent UGB criterion of Huang and Larson, which associates a simplicial complex to any set of nonzero squarefree polynomials in the vanishing ideal of a variety X. When X is irreducible, the Huang-Larson criterion states that such polynomials form a UGB for the vanishing ideal I(X) precisely when this simplicial complex is the algebraic matroid of X. Our result is then an application of their criterion: we determine the Huang-Larson complex for a candidate UGB, and show that it coincides with the algebraic matroid of the universal multiview variety.