Past Events

An alternating permutation w=a1⋯an of 1,2,…,n is a permutation such that ai>ai+1 if and only if i is odd. If En (called an Euler number) denotes the number of alternating permutations of 1,2,…,n, then ∑n≥0Enxnn!=secx+tanx. We will discuss such topics...

A substantial part of extremal combinatorics studies relations existing between densities with which given combinatorial structures (fixed size ``templates'') may appear in unknown (and presumably very large) structures of the same type. Using basic...

Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$. We discuss a recent generalization of both Derksen's recent analogue of the Skolem-Mahler-Lech theorem in positive...

In a previous lecture I waved my hands at the subject of period integrals for automorphic forms on SL_2. If that lecture had a point, it was that the phenomenon of non-vanishing of quadratic twists of L-functions was somehow related to the non...

Diophantine equations, one of the oldest topics of mathematical research, remain the object of intense and fruitful study. A rational solution to a system of algebraic equations is tantamount to a point with rational coordinates (briefly, a "rational...

Abstract: Diophantine equations, one of the oldest topics of mathematical research, remain the object of intense and fruitful study. A rational solution to a system of algebraic equations is tantamount to a point with rational coordinates (briefly, a...

How do simple local interactions combine to produce complex large-scale structure and patterns? The abelian sandpile model provides a beautiful test case. I'll discuss a pair of conjectures about the scale invariance and dimensional reduction of the...

We study the probability that the origin is connected to the sphere of radius R in critical percolation in high dimensions, namely when the dimension d is a large fixed constant, or when d>6 and the lattice is sufficiently spread out. I will present...

In this talk we present an algebraic context for knot theory. Knotted trivalent graphs (KTGs) along with standard operations defined on them form a finitely presented algebraic structure which includes knots, and in which many topological knot...

In the maximum likelihood mixture density estimation problem one is given a parametric family of densities and a data set. The question is then to find the regular Borel probability measure over the family's parameter space that yields the maximum...