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 in 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 Mel Hochster, University of Michigan

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Speaker Ryan Watson, University of Nebraska-Lincoln

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Speaker Adam LaClair, University of Nebraska-Lincoln

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Reception 

Day 2 (all times are in CDT)

October 26

Speaker Nil Şahin, Bilkent University

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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.

Speaker Debjit Basu, University of Kansas

Title: 

Vanishing of weight-one syzygies on projective varieties

Abstract: 

In this talk we study conditions under which weight one Koszul cohomology vanishes on projective varieties. As corollary of more general results, we obtain statements on the so-called property-(Mq) reflecting on the higher syzygies of minimal surfaces and higher dimensional projective varieties. We also provide lower bounds on multiples m such that multiple L = mB and adjoint L' = K_{X}+mB line bundles satisfy property-(Mq) for base point free and ample line bundle B on a projective variety X. Property-(Mq) is complementary to property-(Np) though it is far less studied. Notably, our bounds for property-(Mq) for both L and L' are equal to or better than existing bounds for property (Np) as established by other authors—particularly in relation to Mukai’s conjecture and its higher-dimensional analogues. By considering both properties -(Mq) and (Np), we gain a significantly deeper understanding of the minimal free resolution.