Fall 2024

• 11/20/2024 - No seminar this week
  
  

• 11/13/2024 - Jack Jeffries (University of Nebraska-Lincoln)

Title:  Differential operators on singular spaces and sandwich Bernstein-Sato polynomials

Abstract: The ring of differential operators on a polynomial ring of characteristic zero is the well-studied Weyl algebra. This ring exhibits a number of striking finiteness properties with a wide range of applications. For rings with singularities, the ring of differential operators can be poorly behaved and the useful finiteness properties fail in general; however, for rings with sufficiently nice singularities, some of the theory can be salvaged. In this talk, we will consider Bernstein's inequality, which is a fundamental result for the Weyl algebra. We discuss some extensions of this classical result to various singularities. This is based on joint work with Àlvarez Montaner, Hernández, Núñez-Betancourt, Teixeira, and Witt, and with Lieberman.

• 11/06/2024 - Luis Ferroni (Institute for Advanced Study)

Title: Chow functions for partially ordered sets

Abstract: In a landmark paper in 1992, Stanley developed the foundations of what is now known as the Kazhdan--Lusztig--Stanley (KLS) theory. To each kernel in a graded poset, he associates special functions called KLS polynomials. This unifies and puts a common ground for i) the Kazhdan--Lusztig polynomial of a Bruhat interval in a Coxeter group, ii) the toric g-polynomial of a polytope, iii) the Kazhdan-Lusztig polynomial of a matroid. In this talk I will introduce a new family of functions, called Chow functions, that encode various deep cohomological aspects of the combinatorial objects named before. In the three settings mentioned before, the Chow function describes i) a descent-like statistic enumerator for paths in the Bruhat graph, ii) the enumeration of chains of faces of the polytope, iii) the Hilbert series of the matroid Chow ring. This is joint work with Jacob P. Matherne and Lorenzo Vecchi.

• 10/30/2024 - Erik Bates (NC State University)

Title: A new combinatorial interpretation of the (sum of (generalized)) Fibonacci numbers

Abstract: The sum of Fibonacci numbers, i.e. the sequence 2, 4, 7, 12, 20, 33, 54, 88, ... has many combinatorial interpretations.  For instance, the n-th term in this sequence is the number of length-n binary strings that avoid 001.  In this talk, I will describe a related (but to my knowledge, new) interpretation: given a length-3 binary string---called the keyword---we say two length-n binary strings are equivalent if one can be obtained from the other by some sequence of substitutions: each substitution replaces an instance of the keyword with its negation, or vice versa.  It turns out that the number of induced equivalence classes is again the n-th term in the aforementioned sequence.  What makes this result surprising is that it does not depend on the keyword, despite the fact that the sizes of the equivalence classes do.  If the keyword has length m, then we instead use the sum of (m-1)-step Fibonacci numbers.  This is joint work with undergraduates (present and past): Blan Morrison, Patrick Revilla, Mason Rogers, Arianna Serafini, Anav Sood.

• 10/23/2024 - Rafael S. González D'León (Loyola University Chicago)

Title:  On Whitney numbers of the first and second kind, or is it the other way around?

Abstract:  The Whitney numbers of the first and second kind are a pair of poset invariants that are relevant in various areas of mathematics. One of the most interesting appearances of these numbers is as the coefficients of the chromatic polynomial of a graph. They also appear as counting regions in the complement of a real hyperplane arrangement.   In this talk, I will introduce these concepts and will share a very curious phenomenon: sometimes the Whitney numbers of the first and second kind of a poset happen to be also the Whitney numbers of the second and first kind but of a different poset. To find examples of this phenomenon we rely on the poset technique of edge labelings. Some recent results about this phenomenon could shed light on longstanding questions in the subject.

• 10/16/2024 - Ritvik Ramkumar (Cornell University), Notice different room: COX 306

Title: Hilbert scheme of points on a threefold

Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will introduce this object and review some well-known results when X is a curve or a surface. The main focus of this talk will be on the case when X is a threefold. Specifically, I will present some tantalizing open questions and delve into describing the structure of the smooth points on Hilb^d(X). If time permits, I will also discuss the mildly singular points of this Hilbert scheme. This is all joint work with Joachim Jelisiejew and Alessio Sammartano.

• 10/09/2024 - John Graf (NC State University)

Title: Symmetric Functions, Plethysm, and Schur’s Q-functions

Abstract: The Schur functions are an important basis of the ring of symmetric functions, and Schur’s Q-functions enjoy many analogous properties as a basis of the subring Gamma. We will begin by discussing various properties and bases of symmetric functions, before moving on to the comparisons between Schur functions and Schur’s Q-functions. In particular, plethysm is a type of composition of two symmetric functions, and we will discuss the stability property of plethysm that both bases have in common.

• 10/02/2024 - Hugh Thomas (Université du Québec à Montréal)

Title: Cyclic actions on noncrossing and nonnesting partitions

Abstract: Noncrossing partitions and nonnesting partitions are both counted by Catalan numbers. Noncrossing partitions on [n] admit a natural cyclic action of order 2n, induced by the Kreweras complement. Nonnesting partitions admit a natural toggle-based action; in fact, they admit one such action for each choice of Coxeter element of the symmetric group. We prove that the latter actions all have order 2n by constructing a family of bijections between noncrossing and nonnesting partitions, equivariant with respect to the cyclic actions on either side. This talk is based on arXiv:2212.14831, joint with Benjamin Dequêne, Gabriel Frieden, Alessandro Iraci, Florian Schreier-Aigner, and Nathan Williams. Our results were presented at FPSAC in July; our extended abstract (and Nathan Williams's slides) are available from the FPSAC website.

• 09/25/2024 - Nick Mayers (NC State University)

Title: The quantum k-Bruhat order

Abstract: Finding combinatorial interpretations for the structure constants of Schubert polynomials is a long-standing open problem in algebraic combinatorics. In the case where one of the Schubert polynomials is a Schur polynomial, the structure constants are encoded in a poset called the “k-Bruhat order”. In studying the k-Bruhat order, Bergeron and Sottile were led to introduce a monoid which encodes the chain structure of the k-Bruhat order. Using the monoid structure, the authors were able to establish properties and descriptions of certain structure constants. In this talk, after outlining the developments discussed above, we discuss ongoing work concerning an analogous story for quantum Schubert polynomials and an associated quantum k-Bruhat order. This is joint work with Laura Colmenarejo.

• 09/18/2024 - Sean Thompson (NC State University)

Title: Quiver connections and bimodules of basic algebras

Abstract: Motivated by the problem of classifying quantum symmetries of non-semisimple, finite-dimensional associative algebras, we define a notion of connection between bounded quivers and build a bicategory of bounded quivers and quiver connections. We prove this bicategory is equivalent to a bicategory of basic algebras,  bimodules, and intertwiners with some additional structure.

• 09/11/2024 - Philip Tosteson (UNC Chapel Hill)

Title: Representations of categories of finite sets

Abstract: A representation of the category of finite sets is a linear algebraic object, which roughly consists of a sequence of representations V_n of the symmetric group S_n related by transition maps.  These representations occur naturally in several places including in the study of Kazhdan-Lusztig polynomials of braid matroids, the homology of moduli spaces of curves, and representations of the Witt Lie algebra of polynomial vector fields. I will discuss examples and applications of these representations and their associated generating functions and symmetric functions.

• 09/04/2024 - Yairon Cid-Ruiz (NC State University)

Title: Multiplicities and integral dependence

Abstract: The theory of integral closure of ideals, originating in the early twentieth century with work of Krull, Zariski, Rees, and others, remains a vibrant area of research in algebraic geometry, commutative algebra, and singularity theory. This theory's significance partly stems from its connections with numerical invariants such as multiplicities. During the 1950s, significant advances by Samuel, Serre, and Rees brought multiplicities into the spotlight, spurring extensive development in various directions. During the 1970s, significant advances by Teissier, Briançon, and Skoda brought multiplicities and integral closure into the forefront of singularity theory. This talk will review the history and key results in these areas, while also highlighting some open problems.

• 08/28/2024 - Corey Jones (NC State University)

Title: An algebraic approach to discrete quantum field theories

Abstract: We will give an introduction to nets of associative algebras over discrete metric spaces, which arise in mathematical physics as axiomatizations of the observables content of quantum field theories over discrete spaces. We will present examples arising naturally from combinatorics and representation theory, and discuss some recent structural results about these objects. 

• 08/21/2024 - Beginning of the semester meeting