CA+

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

• About CA+
• 2025: Ames, IA
  March 1-2, 2025
• 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

Standard monomials and Gröbner bases for positroid varieties
by Ayah Almousa (University of South Carolina/Fields Institute)
Abstract: One way to probe the structure of an algebraic variety is to understand the anatomy of its coordinate ring at each degree, which can be achieved by constructing a Gröbner basis. Influential work of Hodge in the 1940s paved the way for using Gröbner bases to combinatorially study the Grassmannian. In this talk, we will follow Hodge’s approach in order to investigate certain subvarieties of the Grassmannian called positroid varieties. Introduced by Knutson–Lam–Speyer in 2013, positroid varieties provide a stratified decomposition of the Grassmannian into subvarieties which enjoys many advantages over other previously studied decompositions. We will see that a Gröbner basis approach to investigating positroid varieties has powerful applications in algebra, geometry, and combinatorics. This is joint work with Shiliang Gao and Daoji Huang.

Truncating multigraded modules
by Lauren Cranton Heller (University of Nebraska, Lincoln)
Abstract: On a single projective space truncations of the coordinate ring are powers of the maximal homogeneous ideal and have linear resolutions given by Eagon-Northcott complexes. I will propose a definition of truncation for the coordinate rings of smooth projective toric varieties and exhibit explicit cellular resolutions of the resulting ideals. My construction agrees with the recent proof by Hanlon, Hicks, and Lazarev of the existence of short line bundle resolutions on smooth projective toric varieties.

Tate algebras and some questions in prime characteristic
by Rankeya Datta (University of Missouri)
Abstract: We will introduce the class of Tate algebras over a non-Archimedean field. A non-Archimedean field is a field equipped with an absolute value that satisfies a stronger 'non-Archimedean' triangle inequality. Tate algebras are to classical rigid analytic geometry what polynomial rings over a field are to algebraic geometry/commutative algebra. We will highlight the many pleasing properties that Tate algebras share with polynomial rings over a field. However, there are surprising differences between the two classes of rings in prime characteristic when it comes to investigating notions of singularities defined using the Frobenius map. We will mention some fundamental questions in prime characteristic commutative algebra that are open for homomorphic images of Tate algebras because of these differences. Time permitting, we will then discuss how to tackle these open questions using non-Archimedean functional analysis. This talk is based on joint works with (subsets of) Takumi Murayama, Neil Epstein, Kevin Tucker and Karl Schwede.

Curves, surfaces and applied algebraic geometry
by Jose Israel Rodriguez (University of Wisconsin, Madison)
Abstract: Applied Algebraic Geometry is an inclusive community involving industry, scientists, and mathematicians. It combines centuries worth of AG ideas with modern computing power to drive research into applications from new angles. The success and growth in this area comes from the fact that many problems can be restated in terms of solving systems of polynomial equations. The solution sets to these equations are algebraic sets, which we know as algebraic varieties. In this talk I will (1) present motivating examples of algebraic varieties appearing in applications, (2) illustrate the core ideas of symbolic and numerical methods to describe curves, and if time permits (3) outline a new numerical algorithm (u-generation) for intersecting a curve with a hypersurface for maximum likelihood estimation.

Ideal closure operations in characteristic zero via resolution of singularities
by Karl Schwede (University of Utah)
Abstract: A big story in commutative algebra over the past 45 years has been a connection between the measures of singularities associated to Frobenius and tight closure theory in characteristic p with those coming out of birational algebraic geometry via resolution of singularities and associated vanishing theorems in characteristic zero. However, there was no known "closure operation" in characteristic zero whose tight-closure-like properties were derived from such resolutions. In this talk, I will introduce such an operation, which we call Koszul-Hironaka (KH) closure. We show it has many similar properties to tight closure (colon capturing, a version of the Briancon-Skoda theorem, etc) but at the same time differs in other key ways (in some ways that are quite advantageous). This is joint work with Neil Epstien, Peter McDonald, and Rebecca R.G.

Stable cohomology and Koszul duality
by Keller VandeBogert (University of Notre Dame/Fields Institute)
Abstract: The cohomology of certain families of vector bundles on projective space exhibit a striking stability phenomenon, the proof of which was motivated by similar phenomena for line bundles on flag varieties. In this talk, I'll talk about how this "stable" cohomology is secretly computing derived invariants on the category of polynomial functors, a connection which we can exploit to prove a conjectured representation stability phenomenon for line bundle cohomology. On the other hand, this bridge allows us to apply commutative-algebraic techniques to tackle (and solve) problems in polynomial functor theory. This is based on joint work with Claudiu Raicu.

Weak order on alternating sign matrix varieties
by Anna Weigandt (University of Minnesota)
Abstract: Alternating sign matrices (ASMs) form the MacNeille completion of the strong Bruhat order on the symmetric group. There is a natural interpretation of this poset as the containment order on ASM varieties, which are generalized determinantal ideals. In 2018, Hamaker and Reiner defined weak Bruhat order on ASMs, which when restricted to the symmetric group, is the usual weak order. We initiate a geometric study of weak order on ASMs varieties, focusing on how combinatorial properties of this poset describe geometric properties of ASM varieties. This is joint work with Laura Escobar and Patricia Klein.