CA+

A two-day conference on commutative algebra and related fields like algebraic geometry, number theory, combinatorics, and more.

• About CA+
• 2023: Minneapolis, MN
  April 1, 2023
• 2022: Ames, IA
  April 30 - May 1, 2022
• 2021: Minneapolis, MN
  October 30, 2021
• 2017: Minneapolis, MN
  September 22-23, 2017
• 2016: Madison, WI
  November 4-5, 2016
• 2015-16 Meetings
  November 2015 and April 2016

Titles and Abstracts

New Questions for Commutative Algebraic Stemming from Theoretical Computer Science
by Hang (Amy) Huang (Auburn University)
Abstract: For the past few years, the new tool border apolarity has resolved a bunch of open problems in tensor geometry, shedding light on a lot more open problems in theoretical computer science. I will briefly explain the new development and discuss how the knowledge of commutative algebraists (on multi-graded Hilbert schemes in particular) could bring us further. I will also briefly report on the current progress in classifying the vector space of matrices of low rank: why we care about them and how we get the new progress using tools from syzygies.

Singularities in Positive Characteristic
by Janet Page (North Dakota State University)
Abstract: Algebraic geometry aims to understand the shapes defined by polynomial equations, called algebraic varieties, and commutative algebra often provides the tools with which to do so. One particular focus of this study is singularities, or points where the variety is not smooth (or more precisely, not a manifold). For example, a curve is not smooth if it has a sharp corner or if it crosses over itself. In this talk, I’ll discuss some of the ways in which commutative algebraic tools can be used to study singularities, with a special focus on what can happen in positive characteristic. In fact, we’ll see that singularities in positive characteristic can be much “worse” than in characteristic 0.

Local cohomology, Schubert varieties, and Dyck patterns
by Michael Perlman (University of Minnesota)
Abstract: Given a closed subvariety Z in a smooth complex variety X, the local cohomology modules with support in Z are holonomic D-modules, and thus have finite filtration with simple composition factors. Via the example of the Grassmannian, we will explain how Kazhdan-Lusztig theory may be used to determine the D-module structure on local cohomology in the case when X is a Hermitian symmetric space and Z is a Schubert variety, including combinatorial formulas describing the composition factors and the weight filtration in the sense of mixed Hodge modules. Upon restriction to the opposite big cell, these calculations recover several previously known results concerning local cohomology with support in determinantal varieties, and provide new insights regarding the structure of local cohomology with support in symmetric determinantal and Pfaffian varieties.

Short resolutions of the diagonal and a Horrocks-type splitting criterion in Picard rank 2
by Mahrud Sayrafi (University of Minnesota)
Abstract: In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Then in 2015, Eisenbud-Erman-Schreyer used the BGG correspondence for products of projective spaces to prove a version of this criterion under an additional hypothesis. This talk is about the key ingredient for proving a Horrocks-type splitting criterion for vector bundles over a smooth projective toric variety X of Picard rank 2: a short resolution of the diagonal sheaf consisting of finite direct sums of line bundles. I'll discuss the construction via a variant of Weyman's "geometric technique," as well as additional properties and applications. This is joint work with Michael Brown.

The total rank conjecture in characteristic 2
by Mark Walker (University of Nebraska - Lincoln)
Abstract: This is joint work with Keller VandeBogert. Let R be a Cohen-Macaulay local ring of dimension d, and assume M is a non-zero R-module of finite length and finite projective dimension. The Total Rank Conjecture predicts that the sum of the Betti numbers of M (i.e., the ranks of the free modules appearing in its minimal free resolution) is at least 2^d. This conjecture was known previously for all rings of odd prime characteristic and certain rings whose residue field has characteristic other than 2. In this talk I will describe a new proof that applies to rings of characteristic 2. There is also a generalization of this result that applies to "tiny complexes", which give the appropriate analogue of resolutions of modules as above for non-Cohen-Macaulay rings.