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.

Abstract: In knot theory, a rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a nonalternating form and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued  fraction expansion of the rational number in which all partial denominators are even (called all-even form}). In this talk we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or equivalently the nonalternating form of the corresponding rational number.    



This is joint work with Yuanan Diao and Claus Ernst. The talk will define and explain all terms, with a general audience in mind.

Title: The Universal Valuation of Coxeter Matroids 



Abstract: Valuations on a family of polytopes are functions which behave nicely with respect to subdivisions in this family. One important question is the determine the structure of the set of all valuations on a certain family. This can be done by constructing a "universal valuation" which is a valuation that can be specialized to any other valuation on this family. Coxeter matroids are a generalization of matroids to an arbitrary root system. As with usual matroids, we can interpret Coxeter matroids as polytopes. In this talk, we will construct a universal valuation for the family of Coxeter matroid polytopes.