## Spring 2023, Minneapolis, MN

Our next meeting is April 1, 2023, at the University of Minnesota in Minneapolis, MN. Below you'll find a schedule, titles and abstracts, and more, as they become available.

### Speakers

**Hang (Amy) Huang**(Auburn)**Janet Page**(NDSU)**Michael Perlman**(Minnesota)**Mahrud Sayrafi**(Minnesota)**Mark Walker**(Nebraska - Lincoln)

### Registration

Please register for the upcoming meeting here.

### Funding?

Funding is available for mileage and lodging. Please apply when you
register by February 20, 2023, to be considered for
funding.
We gratefully acknowledge NSF funding from
grant number DMS-2000390.

### Schedule

Please use the SOUTH entrance of Vincent Hall, as it is the only entrance that will be unlocked.

- Saturday:
- 8:30-9:00: Registration (Vincent 120)
- 9:00-9:50:
**Janet Page**Singularities in Positive Characteristic (Vincent 16) - 10:00-10:30: Coffee and Refreshments (Vincent 120)
- 10:30-11:20:
**Michael Perlman**Local cohomology, Schubert varieties, and Dyck patterns (Vincent 16) - 11:30-12:00:
**Graduate Student Turbo Talks**(Vincent 16) - 12:00-1:30: Catered Lunch and Organized Discussions (Vincent 120/Courtyard -- Bring appropriate outdoor clothing!)
- 1:30-2:20:
**Mahrud Sayrafi**Short resolutions of the diagonal and a Horrocks-type splitting criterion in Picard rank 2 (Vincent 16) - 2:30-3:20:
**Hang (Amy) Huang**New Questions for Commutative Algebraic Stemming from Theoretical Computer Science (Vincent 16) - 3:30-4:00: Coffee and Refreshments (Vincent 120)
- 4:00-4:50:
**Mark Walker**The total rank conjecture in characteristic 2 (Vincent 16)

### 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.

### Lodging and Travel Info

Those participants who are granted funding will be housed at a hotel within walking distance of our meeting. The conference organizers will arrange rooms for funded participants.

Recommended parking is in the Church Street Garage and the Washington Avenue Ramp. Here is a campus map.

We expect that out-of-town participants will drive, arriving on Friday, March 31, and depart on the evening of Saturday, April 1.