Title: The Geometry and Combinatorics of Convex Union Representable Complexes  



Abstract: The study of convex neural codes seeks to classify the intersection and covering patterns of convex sets in Euclidean space. A specific instance of this is to classify "convex union representable" (CUR) complexes: the simplicial complexes that arise as the nerve of a collection of convex sets whose union is convex. In 2018 Chen, Frick, and Shiu showed that CUR complexes are always collapsible, and asked if the converse holds: is every collapsible complex also CUR? We will provide a negative answer to this question, and more generally describe the combinatorial consequences arising from the geometric representations of CUR complexes. This talk is based on joint work with Isabella Novik.

Title: Determinantal formulas with major indices

Abstract: Krattenthaler and Thibon discovered a beautiful formula for the determinant of the matrix indexed by permutations whose entries are q^maj( u*v^{-1} ), where “maj” is the major index. Previous proofs of this identity have applied the theory of nonsymmetric functions or the representation theory of the Tits algebra to determine the eigenvalues of the matrix. I will present a new, more elementary proof of the determinantal formula. Then I will explain how we used this method to prove several conjectures by Krattenthaler for variations of the major index over signed permutations and colored permutations. This is based on joint work with Donald Robertson and Clifford Smyth.

Title: Quotients of Lattice Path Matroids 



Abstract: Matroids are a combinatorial object that generalize the notion of linear independence. One way to characterize matroids is via polytopes, as shown in the work of Gelfand, Goresky, MacPherson, Serganova. In this talk we will focus on a particular class of matroids called Lattice Path Matroids (LPMs). We will show when a collection M_1,...,M_k of LPMs are a flag matroid, using their combinatorics. Part of our work will show that the polytope associated to such a flag can be thought as an interval in the Bruhat order, and thus provides a partial understanding of flags of LPMs from a polytopal point of view. We will not assume previous knowledge on matroids nor quotients. This is joint work with Kolja Knauer.

Title: Three combinatorial applications



Abstract: This is not one seminar talk. This is three small seminars.  First, we prove a geometric result of Dirichlet using combinatorics. Second, as an application of posets we obtain Sylvester's two coin result.  Finally, we present a counting proof when 2 is a quadratic residue in a finite field.



The third topic is joint work with Frits Beukers and Karthik Chandrasekhar.

Title: Limit Laws for q-Hook Formulas



Abstract:  Various asymptotic aspects of the Hook Length Formula for standard Young tableaux have been studied recently in combinatorics and probability.   In this talk, we study the limiting distributions that come from random variables associated to Stanley's q-hook-content formula for semistandard tableaux and q-hook length formulas of Bj\"orner-Wachs related to linear extensions of labeled forests.  We show that, while these limiting distributions are "generically'' asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the moduli space of distributions of these statistics in several regimes.  The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded 2-norm. The closure of the moduli space of these distributions in the L\'evy metric gives rise to the moduli space of DUSTPAN distributions.  As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.



This talk is based on joint work with Joshua Swanson at UCSD.

Title: Increasing and Invariant Parking Sequences 



Abstract: The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, the cars are allowed to have different sizes and each takes up a number of adjacent parking spaces after a trailer that was parked at the start of the street. A preference sequence in which all the cars are able to park is called a parking sequence. In this talk, we will look at increasing parking sequences and their connections to lattice paths. We will also discuss two notions of invariance in parking sequences and present various characterizations and enumerative results. This is joint work with Catherine Yan.

Title: Linear recurrences indexed by Z

Abstract: We consider a system of equations in variables indexed by the integers, in which each variable is equal to a linear combination of the previous variables. We will show a number of general results about these systems, including an analog of Gaussian elimination, a parametrization of solutions, and (time-permitting) a characterization of systems whose solutions are periodic.

Title: Quasi-cluster algebras and triangulations of the Möbius strip



Abstract: In this talk, we will first define triangulations of marked surfaces, then use it to define quasi-cluster algebras (the equivalent of cluster algebras for non-orientable surfaces). We will list a few properties of these algebras, then we will count the number of triangulations of the Möbius strip, the only surface with a finite number of triangulations.