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

Betti numbers of maximal Cohen--Macaulay modules over the cone of a Calabi--Yau variety
by Alexander Pavlov (University of Wisconsin - Madison)
The coordinate ring of the cone of a Calabi-Yau variety is an isolated singularity. We will present a way to compute Betti numbers of maximal Cohen-Macaulay modules over this ring using two ingredients: Orlov’s equivalence of triangulated category of singularity and derived category of coherent sheaves over the Calabi-Yau variety, and box-product type resolution of the diagonal of the Calabi-Yau variety.

On the defining equations of Rees algebras
by Claudia Polini (University of Notre Dame)
Rees rings of ideals and modules are ubiquitous in commutative algebra and play a major role in equisingularity theory. Lately applied mathematicians, more precisely in geometric modeling and computer aided design, have turned their attention to Rees rings as well. Computing the defining equations of Rees algebras is a major problem in elimination theory. Since it is a very difficult task, it is important to gather at least partial information such as degree bounds. In joint work with Bernd Ulrich we obtain such bounds for ideals of low codimension.

Newton Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
by Lauren Williams (University of California - Berkeley)
We use the cluster structure on the Grassmannian to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. From a given plabic graph G we have two coordinate systems: we have a positive chart for our A-model Grassmannian, and we have a cluster chart for our B-model (Landau-Ginzburg model) Grassmannian. On the A-model side, we use the positive chart to associate a corresponding Newton-Okounkov (A-model) polytope. On the B-model side, we use the cluster chart to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a B-model polytope. Our first main result is that these two polytopes coincide. Our second main result is an explicit formula for the lattice points of the polytopes, in terms of Young diagrams, which suggests a connection to quantum cohomology. This is joint work with Konstanze Rietsch.

Frobenius powers of ideals
by Emily Witt (University of Kansas)
Given an ideal and a positive real parameter, there are many ways to construct a new ideal. For example, when the parameter is an integer, one may simply take the corresponding power of the ideal. Similarly, in prime characteristic, if the parameter is an integer power of the characteristic, then one may take the Frobenius power of the ideal. Further examples of this include the multiplier ideal construction from birational geometry, and the test ideal construction in prime characteristic. These constructions are known to be useful tools in measuring the singularities of the original ideal, and have recently been the subject of intense study. In this talk, we discuss a new construction in prime characteristic that "mimics" the usual Frobenius powers of an ideal. We will relate these "generalized Frobenius powers" to test ideals and multiplier ideals, and describe how they measure the singularities of generic polynomials. This is joint work with Daniel Hernández and Pedro Teixeira.

Random Flag Complexes and Asymptotic Syzygies
by Jay Yang (University of Wisconsin - Madison)
We use the probabilistic method to construct examples of conjectured phenomenon about asymptotic syzygies. In particular, we use the Stanley-Reisner ideals of random flag complexes to construct new examples of Ein and Lazarsfeld's nonvanishing for asymptotic syzygies and of Ein, Erman, and Lazarsfeld's conjectural on the asymptotic normal distribution of Betti numbers.

Vanishing of Littlewood-Richardson polynomials is in P
by Alexander Yong (University of Illinois at Urbana-Champaign)
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming. This is joint work with Anshul Adve and Colleen Robichaux.

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