## September 22-23, 2017, Minneapolis, MN

Our next meeting is September 22-23, 2017, at the University of Minnesota in Minneapolis, MN. Below you'll find a schedule, titles and abstracts, and more, as they become available.

### Speakers

**Alexander Pavlov**(Wisconsin)**Claudia Polini**(Notre Dame)**Emily Witt**(Kansas)**Lauren Williams**(UC Berkeley)**Jay Yang**(Wisconsin)**Alexander Yong**(UIUC)

### Registration

Please register for the upcoming meeting here.

### Funding?

We encourage graduate students and postdocs to apply for funding as you register by August 15, 2017.

### Schedule

- Friday:
- 2:00-2:30: Registration and coffee (Vincent 120).
- 2:30-3:20: Claudia Polini. "On the defining equations of Rees algebras" (Vincent 016).
- 3:35-4:25: Jay Yang. "Random Flag Complexes and Asymptotic Syzygies" (Vincent 016)
- 5:15: Conference dinner at Little Szchuan (304 Oak St. SE)

- Saturday:
- 8:45-9:30: Bagels and coffee (Vincent 120).
- 9:30-10:20: Lauren Williams. "Newton Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians" (Vincent 016).
- 10:30-11:00: Bagels and coffee (Vincent 120).
- 11:00-11:50: Alexander Pavlov. "Betti numbers of maximal Cohen--Macaulay modules over the cone of a Calabi--Yau variety" (Vincent 016).
- 12:00-1:15: Lunch on your own.
- 1:15-2:05: Emily Witt. "Frobenius powers of ideals" (Vincent 016).
- 2:15-2:45: Coffee break (Vincent 016).
- 2:45-3:15: Discussion (Vincent 120).
- 3:25-4:15: Alexander Yong. "Vanishing of Littlewood-Richardson polynomials is in P" (Vincent 016).

### 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: Orlovâ€™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.