When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially […]
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 […]
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 […]
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 […]
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 […]
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 […]
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. […]
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 […]
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 […]
Let d >= 2 be a natural number. We determine the minimum possible size of the difference set A-A in terms of |A| for any sufficiently large finite subset A […]
The set of subsets {1, 3}, {1, 3, 4}, {1, 3, 4, 6} is a symmetric chain in the partially ordered set (poset) of subsets of {1,...,6}. It is a […]
A power permutation of a finite field F is a permutation of F whose functional form is x -> x^d for some exponent d. Power permutations are used in cryptography, […]
