Algecom-XXII

Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University, IUPUI, Washington University at St. Louis, Loyola University Chicago, DePaul University, University of Notre Dame, the University of Michigan and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).

Algecom Committee:

Hal Schenck Alexander Yong David Speyer Uli Walther Saugata Basu Evgeny Mukhin Peter Tingley Chris Drupieski Laura Escobar Kyle Petersen

Date: Saturday October 22nd, 2022

Location: Altgeld Hall 245, Univeristy of Illinois at Urbana-Champaign.

Parking: Free parking on Saturday is available in lot C7 (here).

Local Organizers (questions related to Algecom XXII):

Alex Yong, Elizabeth Kelley, Avery St. Dizier

Registration is free, but please use the registration link

For funding consideration, please register by 9/21/2022.

Tentative Schedule:

(All times are in the Central Time Zone)

9:00-9:30am Coffee and Registration

9:30-10:30am Patricia Hersh (University of Oregon)

"Generalized recursive atom ordering and equivalence to CL-shellability":
When Bjoerner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. We generalize the notion of recursive atom ordering, and we prove that any such generalized recursive atom ordering may be transformed via a reordering process into a recursive atom ordering. We also prove that a finite bounded poset admits a generalized recursive atom ordering if and only if it is ``CC-shellable'' by way of a CC-labeling which is self-consistent in a certain sense. This allows us to conclude that CL-shellability is equivalent to self-consistent CC-shellability. As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable. During this talk, we will review plenty of background on poset topology and specifically regarding the technique of lexicographic shellability. This is joint work with Grace Stadnyk

10:45-11:45am Sunita Chepuri (Lafayette College)

"Electrical networks and Lagrangian Grassmannians":
In this talk, we will first introduce the mathematical theory of electrical networks. We will then discuss Lam's generalization of circular planar networks to cactus networks. He showed that there was an embedding of these networks, up to electrical equivalence, into the totally nonnegative Grassmannian. However, not every point in the totally nonnegative Grassmannian is realized by such a network. We will see that the elements that are realized by cactus networks turn out to be those that are isotropic with respect to a particular bilinear form. In addition, we will revisit the connection between this Grassmannian embedding and the response network of a matrix. This is joint work with Terrence George and David Speyer.

11:45am-1:15pm Lunch

1:15-2:15pm Yibo Gao (University of Michigan)

"Symmetric structures in the strong Bruhat order":
The Bruhat order encodes algebraic and topological information of Schubert varieties in the flag manifold and possesses rich combinatorial properties. In this talk, we discuss three interrelated stories regarding the Bruhat order: self-dual Bruhat intervals, Billey-Postnikov decompositions and automorphisms of the Bruhat graph. This is joint work with Christian Gaetz.

2:30-3:30pm Ayah Almousa (University of Minnesota - Twin Cities)

"GL-Equivariant resolutions over Veronese Subrings":
We construct explicit GL-equivariant minimal free resolutions of certain (truncations of) modules of relative invariants over Veronese subrings in arbitrary characteristic. The free modules in the resolution correspond to certain skew Schur modules corresponding to "ribbon" or "skew-hook" diagrams, and the differentials at each step are surprisingly uniform. We then utilize the uniformity of these resolutions to explicitly compute information about tensor products, Hom, and Tor between these modules and show that they also have rather simple descriptions in terms of ribbon skew-Schur modules. I will emphasize the hidden role of symmetric function theory in detecting the answer to this question and guiding our intuition to build new tools to prove our results in arbitrary characteristic. This is joint work with Mike Perlman, Sasha Pevzner, Vic Reiner, and Keller VandeBogert.

3:45-4:45pm Jessica Striker (North Dakota State University)

"Algebraic, Geometric, and Combinatorial aspects of alternating sign matrices":
In this talk, we will discuss my favorite combinatorial objects, alternating sign matrices, from each of the perspectives listed in the name of this conference. In addition, we'll peruse lurking dynamics embedded in each of these perspectives.

4:45-5:45pm Poster Session

"Twists of Grassmannian Cluster Variables": Moriah Elkin
TBA: Shiliang Gao
"Noetherian Property of Invariant Rings": Annie Giokas
TBA: Owen Goff
"Oriented Matroid Circuit Polytopes": Jodi McWhirter
"On the Lefschetz property for quotients by monomial ideals containing squares of variables": Ritika Nair
"Bender-Knuth Involutions on Tableaux and Linear Extensions of Posets": Son Nguyen
"Quantum Modular Inclusion": Som Sadashiv Phene

6:00pm Dinner: Pizza in Altgeld

Lodging:

The designated hotel for the conference is the Illini Union Hotel. We have reserved a room block for participants, just use the booking code agc22 when you make your reservation. Reservations include a daily breakfast voucher and parking.

NOTE: The Illini Union Hotel is now fully booked for October 21st and 22nd.

Illini Union Hotel

1401 West Green Street, Urbana, IL, 61801

217-333-1241

https://illiniunionhotel.illinois.edu/

Other options for nearby lodging include the Hampton Inn and the Hyatt:

Hampton Inn Champaign/Urbana

1200 W University Ave, Urbana, IL, 61801

217-337-1100

https://www.reservations.com/hotel/hampton-inn-champaign-urbana

Downtown Champaign Hyatt

217 N Neil St, Champaign, IL, 61820

217-531-2800

https://www.hyatt.com/en-US/hotel/illinois/hyatt-place-champaign-urbana/cmizc