The Kauffman bracket skein algebra was originally defined as a generalization of the Jones polynomial for knots and links on a surface and is one of the few quantum invariants where the connection to hyperbolic geometry is fairly well-established.  Explicating this connection to hyperbolic geometry requires an understanding of the non-commutative structure of the skein algebra, especially at roots of unity.  We'll present some of the known (and not known) properties of the skein algebra.  Highlights include the Chebyshev polynomials, quantum tori,…

A Belyi map $\beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$.  Replacing $\mathbb{P}^1$ with an elliptic curve $E: \ y^2 = x^3 + A \, x + B$, there is a similar definition of a Belyi map $\beta: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$.  Since $E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R})$ is a torus, we call $(E, \beta)$ a Toroidal \Belyi pair. There are many…

The complete homogeneous symmetric (CHS) polynomials can be used to define a  family of norms on Hermitian matrices. These 'CHS norms' are peculiar in the sense that they depend only on the eigenvalues of a matrix and not its singular values (as opposed to the Ky-Fan and Schatten norms). We will first give a general overview behind the construction of these norms (as well as their extensions to all n x n complex matrices). The construction and validation of these norms will…

It is widely believed that Weierstrass ignored Eisenstein's theory of elliptic functions and developed an alternative treatment, which is now standard, because of a convergence issue. In particular, the Eisenstein series of weight two does not converge absolutely while Eisenstein's theory assigned a value to this series. It is now well-known that the quantity which Eisentsein assigned to this series is not only correct, but it has interesting interpretations and attracted much attention. It has been proved by Damerell in 1970…

We consider the problem of comparing the number of discrete points that belong to a set with the measure (or volume) of the set, under circumstances where we expect these two numbers to be approximately equal. We start with a locally compact, abelian, topological group G. We assume that G has a countably infinite, torsion free, discrete subgroup H. But to make the talk easier to follow we will mostly consider the case G = R^N and H = Z^N.…

Hassett spaces in genus 0 are moduli spaces of weighted pointed stable rational curves; they are important in the minimal model program and enumerative geometry. We compute the Chow ring of heavy/light Hassett spaces. The computation involves intersection theory on the toric variety corresponding to a graphic matroid, and rests upon the work of Cavalieri-Hampe-Markwig-Ranganathan. This is joint work with Siddarth Kannan and Shiyue Li.

We will discuss the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the 0-norm of the vector. Our main results are new improved bounds on the minimal 0-norm of solutions to systems Ax=b, where A is an integer matrix, b is an integer vector and x is either a general integer vector (lattice case) or…

In this talk we discuss some problems related to finding large induced subgraphs of a given graph G which satisfy some degree-constraints (for example, all degrees are odd, or all degrees are j mod k, etc). We survey some classical results, present some interesting and challenging problems, and sketch solutions to some of them. This is based on joint works with Michael Krivelevich, and with Liam Hardiman and Michael Krivelevich.