# NCSU Algebra & Combinatorics Seminar

The NCSU Algebra and Combinatorics Seminar is organized by Laura Colmenarejo, Corey Jones, and Nathan Reading.

We meet weekly on Mondays at

**1:00 PM**. In Fall 2021 we will be meeting jointly in-person (SAS 4201) and online. The Zoom link will be sent out to the mailing list along with talk announcements.

We will meet for an outdoor lunch in front of SAS Hall at 12:00 PM beforehand (weather permitting). All are welcome!

If you'd like to joining the mailing list, contact Corey Jones at cmjones6@ncsu.edu or any other organizer.

# Spring 2022

1/31/2022- Martin Helmer , NCSU:

**Conormal Spaces and Whitney Stratifications**

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. This is joint work with Vidit Nanda (Oxford).

# Fall 2021

8/30/2021- Laura Colmenarejo, NCSU:

**The Story of My Research**

In this talk, I will share with you what kind of problems I work on and what's my motivation. We will talk about the representation theory of finite groups and symmetric functions, and how algebraic combinatorics appears in the less expected places. This talk aims to be mostly informal and accessible for grad students.

9/13/2021-Corey Jones, NCSU:

**Fusion categories in Mathematics and Physics**

Fusion categories are algebraic structures that generalize the representation categories of finite groups. I will explain how fusion categories have become involved in diverse areas of mathematics and physics, from topologically ordered phases of matter in 2-dimensions to quantum symmetries of noncommutative spaces.

9/20/2021- Andy Manion, NCSU:

**Higher tensor products and Heegard floer homology**

I will discuss some algebraic aspects of recent work with Raphael Rouquier on a tensor product operation for categorified representations of U_q(gl(1|1)^+) and its connections to Heegaard Floer homology.

9/27/2021-Juan Villarreal Montoya, NCSU:

**Logarithmic vertex algebras**First, I will make a general introduction to vertex algebras. Then, I will mention some results of recent work with Bojko Bakalov on Logarithmic vertex algebras.

10/4/2021- Fall Break

10/11/2021-Jianping Pan, NCSU:

**Crystals, Stable Grothendieck Polynomials, and Putting Numbers In Boxes**

I will tell you about my dissertation work on two variants of stable Grothendieck polynomials and their combinatorics. Relevant combinatorial objects include crystals (edge-labeled directed digraphs from representation theory), tableaux (numbers in boxes with rules), decreasing factorizations (numbers in parentheses), and insertion algorithms (how to put numbers in boxes). Background in algebraic combinatorics is helpful but not necessary.

10/18/2021- Carlos M. Ortiz Marrero, Pacific Northwest National Lab and NCSU:

**Non-Local Games on Graphs**

In a non-local game, two non-communicating players cooperate to convince a referee about a strategy that does not violate the rules of the game. A quantum strategy for such a game enables players to determine their answers by performing joint measurements on a shared entangled state. In this talk we will concentrate on non-local games that come from problems in graph theory. We will survey some recent breakthroughs, explain why the study of such games is important, and introduce some connections to operator algebras and their representations.

10/25/2021- Steven Karp, UQAM:

**Wronskians, total positivity, and real Schubert calculus**

The totally positive flag variety is the subset of the complete flag variety Fl(n) where all Plücker coordinates are positive. By viewing a complete flag as a sequence of subspaces of polynomials of degree at most n-1, we can associate a sequence of Wronskian polynomials to it. I will present a new characterization of the totally positive flag variety in terms of Wronskians, and explain how it sheds light on conjectures in the real Schubert calculus of Grassmannians. In particular, a conjecture of Eremenko (2015) is equivalent to the following conjecture: if V is a finite-dimensional subspace of polynomials such that all complex zeros of the Wronskian of V are real and negative, then all Plücker coordinates of V are positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko (2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive.

11/01/2021- Joseph Pappe, UC Davis:

**Area-Depth Symmetric Catalan Polynomial**

We define two new symmetric q,t-Catalan polynomials in terms of area and a new statistic called depth and in terms of dinv and a new statistic called dinv of depth. We prove symmetry using an involution on plane trees from which we obtain another description of the usual q,t-Catalan polynomials. The same involution proves the symmetry of the Tutte polynomial for the Catalan Matroid. We also provide a combinatorial proof relating parking functions to the number of connected graphs on a fixed number of vertices.

11/08/2021- Tianyi Yu, UC San Diego:

**Grothendieck-to-Lascoux Expansions**

We establish the conjecture of Reiner and Yong for an explicit combinatorial formula for the expansion of a Grothendieck polynomial into the basis of Lascoux polynomials. This expansion is a subtle refinement of its symmetric function version due to Buch, Kresch, Shimozono, Tamvakis, and Yong, which gives the expansion of stable Grothendieck polynomials indexed by permutations into Grassmannian stable Grothendieck polynomials. Our expansion is the K-theoretic analog of that of a Schubert polynomial into Demazure characters, whose symmetric analog is the expansion of a Stanley symmetric function into Schur functions. This is joint work with Mark Shimozono.

11/15/2021- Cain Edie-Michell, UC San Diego:

**Algebras in Tensor Categories**

It is a classical result that the simple algebras in the category of finite-dimensional vector spaces are precisely the n x n matrix algebras. The notion of algebras in more general tensor categories is easy to formulate, and we can ask for classification results in these categories. Such classification results have broad applications to conformal field theory, topological quantum field theory, and subfactor theory. In this talk, I will present some progress on the classification of algebras in the categories coming from the quantum groups of type A. In particular, I will show the existence of a new family of exceptional examples.

11/22/2021-

11/29/2021-Nantel Bergeron, York University:

**From P-partition to bounded P-partitions**

The theory of P-partitions was developed by Stanley to understand/solve several enumerations problems and representations theory problems. Together with the work of Gessel, this led to the development of the space of quasisymmetric functions. Schur functions are naturally understood in the world of quasisymmetric functions as a sum over standard tableaux of Gessel fundamental functions.

Unrelated to this, Lascoux and Schutzenberger introduced a special class of polynomials related to Schubert varieties named flagged Schur functions. Much more recently Assaf and Searles developed the theory of slide polynomials that would decompose nicely Schubert polynomials. Together with Assaf, we noticed that these slide polynomials can be obtained as bounded P-partitions similar to Gessel Fundamental functions, and started to develop a more general theory of bounded P-partitions.

Our main theorem shows that if the bounds form a flag (only increase strictly at descents) then the bounded P partition polynomial enumerator is a positive sum of slide polynomials. This allows us to understand flag schur functions as a sum over standard tableaux of slide polynomials (completing the picture in a nice way.)