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

Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials
by Patricia Klein (Minnesota)
Knutson and Miller established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially-defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt conjectured a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono. We prove this conjecture in full generality. The proof provides tools for assessing the Cohen--Macaulayness of equidimensional unions of matrix Schubert varieties, of which alternating sign matrix varieties are an important example. This talk will include sufficient geometric and combinatorial background to situate our main result as well as a description of open problems in homological commutative algebra motivated by this geometry and combinatorics.

The closed support problem for local cohomology
by Monica Lewis (Minnesota)
Local cohomology modules are (typically) very large algebraic objects that encode rich geometric information about the structure of a commutative ring. It is an open question whether the local cohomology of a Noetherian ring must have a finite set of minimal primes. A positive answer to this and to many stronger questions has been given for large classes of regular rings by endowing the local cohomology of R with the action of a suitable noncommutative ring of endomorphisms: either the ring of differential operators over R in characteristic 0, or the ring generated by R and its Frobenius homomorphism in characteristic p > 0. In this talk, we will review the classic finiteness theory over a regular ring, will discuss what is known to hold (and known to fail) in the singular setting, and will present recent results for certain classes of positive characteristic singularities.

Chow rings of matroids are Koszul
by Matthew Mastroeni (Iowa State)
The Chow ring of an algebraic variety is an algebro-geometric analog of the cohomology ring of a smooth manifold that encodes important information about the intersections between its subvarieties. Feichtner and Yuzvinsky computed a presentation for the Chow ring of a smooth toric variety associated to a matroid (and some other data) which is now called the Chow ring of the matroid. These rings have garnered significant attention in recent years thanks in part to their role in the resolution of Rota’s Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. From a commutative algebra standpoint, Chow rings of matroids are very nice graded Artinian Gorenstein rings defined by quadratic relations, and so, a natural conjecture posed by Dotsenko is that the Chow ring of a matroid is always Koszul. In this talk, we will discuss various techniques for determining when a ring is Koszul and how the combinatorics of a matroid influences algebraic properties of its Chow ring, culminating with very recent work joint with Jason McCullough giving an affirmative answer to Dotsenko’s conjecture.

Invariant theory of cyclic permutations: a test case
by Vic Reiner (Minnesota)
Invariant theory studies, for a group G acting on a polynomial ring S, both the internal structure of the G-invariant ring R, and the structure of S as an R-module. These are best understood when G is finite, and S has characteristic zero, but mysteries remain. This talk will review the very well-understood case where G is a reflection group, such as the symmetric group on n letters, and then pivot to a thornier test case, when G is generated by an n-cycle inside the symmetric group. Although these cyclic groups have several features in their favor (e.g., they are abelian and permutation groups), their invariant theory still contains mysteries. In particular, we will discuss some results and conjectures about them arising from work with summer 2021 REU students.

Singular Hodge theory of matroids
by Botong Wang (Wisconsin)
The de Bruijn--Erdos theorem states that the pairwise intersection of n lines in the projective plane is either a single point or consists of at least n points. As its higher dimensional generalization, Dowling and Wilson proposed the Top-heavy conjecture, which states that in a rank d matroid, the number of rank k flats is always less or equal to number of rank d-k flats for k ≤ d/2. I will give an overview of a proof of the conjecture in the realizable case, which is a joint work with June Huh. The proof uses the singular Hodge theory of certain algebraic varieteis called the Schubert varieties of hyperplane arrangements. I will also give a brief summary of a more recent work, further joint with Tom Braden, Jacob Matherne and Nick Proudfoot, which proves the Top-heavy conjecture in full generality.