Future students

Title: Constructing broken SIDH parameters: a tale of De Feo, Jao, and Plut's serendipity

Abstract:

This talk is motivated by analyzing the security of the cryptographic key exchange protocol SIDH (Supersingular Isogeny Diffie-Hellman), introduced by 2011 by De Feo, Jao, and Plut. We will first recall some mathematical background as well as the protocol itself. The 'keys' in this protocol are elliptic curves, which are typically described by equations in x and y of the form y^2 = x^3 + ax + b.

Title: Packings of partial difference sets

Abstract:

Partial difference sets are highly structured group subsets that occur in various guises throughout design theory, finite geometry, coding theory, and graph theory. They admit only two possible nontrivial character sums and so are often studied using character theory.

Title: qRSt: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials

Abstract:

The Robinson--Schensted (RS) correspondence is a bijection between permutations and pairs of standard Young tableaux which plays a central role in the theory of Schur polynomials. In this talk, I will present a (q,t)-dependent probabilistic deformation of Robinson--Schensted which is related to the Cauchy identity for Macdonald polynomials.

Title: Pseuodrandom Cliquefree Graphs, Finite Geometry, and Spectra

Abstract:

A regular graph is called optimally pseudorandom if its second largest eigenvalue in absolute value is, up to a constant factor, as small as possible. Determining the largest degree of an optimally pseudorandom graph without a clique of size s is a well-known open problem in extremal graph theory.

Title: On the Theory of the Analytical Forms called Trees

Abstract:

Trees are among the most fundamental of combinatorial structures. Nowadays they appear all over mathematics and computer science, but this has not always been the case. Trees were first introduced, at least under that name, in an 1857 paper of Cayley by the same title as this talk.

Title: The Hepp bound of a matroid: flags, volumes and integrals

Abstract:

Invariants of combinatorial structures can be very useful tools that capture some specific characteristics, and repackage them in a meaningful way. For example, the famous Tutte polynomial of a matroid or graph tracks the rank statistics of its submatroids, which has many applications, and relations like contraction-deletion establish a very close connection between the algebraic structure of the invariant (e.g. Tutte polynomials) and the actual matroid itself.

Title: The Tutte Symmetric Function

Abstract:

The Tutte polynomial is one of the most celebrated and most well-studied graph functions, in part because it specializes to every graph polynomial with a linear deletion-contraction relation, such as the chromatic polynomial. In the 1990s, Stanley generalized the Tutte polynomial to a symmetric function, but at the cost of the deletion-contraction relation.

Title: Coxeter combinatorics and spherical Schubert geometry

Abstract:

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup.

Title: Pretty Good State Transfer and Minimal Polynomials

Abstract:

We examine conditions for a pair of strongly cospectral vertices to have pretty good quantum state transfer in terms of minimal polynomials, and provide cases where pretty good state transfer can be ruled out.