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°Õ¾±³Ù±ô±ð:ÌýAn algebraic framework for twualities of embedded graphs

Abstract:

We develop algebraic tools to identify and generate new surface embeddings of graphs with various forms of self-twuality including geometric duality, Petrie duality, Wilson duality, and both forms of triality (which is like duality, but of order three instead of two).  These operations are of particular interest because of their interplay with graph symmetries and graph polynomials.

Title: What is the degree of a Grothendieck polynomial?

Abstract:

Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees.

°Õ¾±³Ù±ô±ð:ÌýOn the approximability of the maximum cardinality stable matching problem with ties  

Title: Enumerating hereditary classes of chord diagrams

Abstract:

A class of combinatorial structures is hereditary if membership in the class is closed under taking substructures. Hereditary classes have been extensively studied for a variety of objects, notably graphs and permutations. A central problem is to determine the number of objects of size n in a given hereditary class. We discuss this problem for chord diagrams, perfect matchings of [2n].

Title: Counting Antichains in the Boolean Lattice

Abstract:

How many antichains are there in the Boolean lattice P(n)? Sperner's theorem (1928) tells us that the largest antichain in P(n) has size A = (n choose n/2). A subset of an antichain is an antichain, so there are at least 2^A antichains in P(n). Interestingly, it turns out that this is close to the total, as Kleitman (1969) showed that the number of antichains is 2^(A(1+x)) where x goes to zero as n goes to infinity.

°Õ¾±³Ù±ô±ð:ÌýThe discrete logarithm problem in finite fields

Abstract:

The security of currently deployed public key protocols relies on the presumed hardness of problems often coming from number theory, such as factoring a large integer or solving the discrete logarithm problem in some group.

In this talk we focus on discrete logarithms in finite fields. We explain what is a discrete logarithm, why cryptographers need them, and we focus then on algorithms to solve the related problem, together with open questions in this area.

°Õ¾±³Ù±ô±ð:ÌýIdentities for ninth variation Schur Q-functions

Abstract:

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity.

Title: The spectral radius of graphs with no odd wheels

Abstract:

The odd wheel W_{2k+1} is the graph formed by joining a vertex to a cycle of length 2k. In this talk, we will investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W_{2k+1}.

°Õ¾±³Ù±ô±ð:ÌýLattice Walk Enumeration: Analytic, algebraic and geometric aspects

Abstract:

This talk will examine the rich topic of lattice path enumeration. A very classic object of combinatorics, lattice walks withstand study from a variety of perspectives. Even the simple task of classifying the two dimensional walks restricted to the first quadrant has brought into play a surprising diversity of techniques from algebra to analysis to geometry. We will consider walks under a few different lenses.

Title: Average Mixing Matrices of Trees and Stars

Abstract:

We define the average mixing matrix (AMM) of a continuous-time quantum walk on a graph using the orthogonal projections onto the eigenspaces of the adjacency matrix A. From there, one of the properties that has been studied is the rank of the AMM. This is easiest to do if the eigenvalues of A are simple, and we’ll review some of the results on this from Coutinho et. al. (2018).