Claremont Center for the Mathematical Sciences

Classical singular moduli are special values of the modular j-function at imaginary quadratic irrationalities and have well-known importance for class field theory. I will describe some recent joint work with O. Imamoglu and A. Toth on certain real quadratic analogues of traces of singular moduli and their relation to mock modular forms.

Artin conjectured that if b is an integer other than -1 or a square, then b is a primitive root modulo infinitely many primes. Although still unproven, this conjecture is known to be true under the assumption of the generalized Riemann hypothesis. I will discuss the history and some motivation behind studying this conjecture, and survey some known results. I will introduce a generalization of the primitive root conjecture to number fields and discuss some applications including a direct connection to the determination of number rings with a Euclidean Algorithm. This is joint work with Ram Murty.

Let $\cal L$ be a finite family of graphs. We describe the typical structure of $\cal L$-free graphs, improving our earlier results on the Erdos-Frankl-Rodl theorem, by proving our conjecture from our earlier paper. Let

$$p=p({\cal L})=\min_{L\in {\cal L}}\chi(L)-1.$$

We shall prove that the structure of almost all $\LL$-free graphs is very similar to the Turan graph $T_{n,p}$, where ``similarity'' is measured in terms of graph theoretical parameters of $\LL$. Some more recent developments, including extensions for induced containment, bipartite graphs, hypergraphs, will be discussed as well.

Partially joint work with: Alon, Bollobas, Butterfield, Morris, Mubayi, Samotij, and Simonovits.

A continuation of an ANTC seminar I gave last year, this talk will be self contained. The natural multiplicative structure of the norm group of local field extensions can sometimes be supplemented with an unexpected additive one, transforming the norm group into a local field of characteristic $p$. This so-called field-of-norms construction induces an equivalence of categories between certain local field extensions and local fields of characteristic $p$. In this talk, I will present a simple example of the power of the field of norms to answer questions from $p$-adic dynamical systems.

The Waring rank of a polynomial of degree d is the least number of terms in an expression for the polynomial as a sum of d-th powers. The problem of finding the rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to statistics, signal processing, and computational complexity. Other than a well-known lower bound for rank in terms of catalecticant matrices, there has been relatively little progress on the problem of determining or bounding rank for a given polynomial (although related questions have proved very fruitful). I will describe new upper and lower bounds. The improved lower bound is especially interesting, dealing with the geometry of catalecticants. The talk should be accessible to all. This is joint work with J.M. Landsberg.

In this talk I will consider the problem of determining the maximum density of sets in Euclidean space which avoid some given distances, in the sense that there are no two points in the set at the given distances. To find upper bounds for the maximum density we use the Fourier coefficients of the autocorrelation function of the characteristic function of a distance avoiding set together with linear programming. This method is a continuous analog of the sphere packing bound due to Cohn and Elkies.

I will show two applications of the bound: Computing new upper bounds for the density of sets avoiding the unit distance in dimension 2,...,24. Giving an alternative, quantitative, and extremely simple proof of a result of Furstenberg, Katznelson, Weiss, proved by ergodic theoretic methods: Every planar set of positive density does not avoid arbitrary large distances.

This is joint work with Fernando M. de Oliveira Filho
(http://arxiv.org/abs/0808.1822).

In this talk, I'll present some ongoing work I'm doing with Anna Bargagliotti (Univ. of Memphis) concerning linear tests of uniformity for rank data. In particular, I'll show how the representation theory of the symmetric group plays a natural role when trying to understand why different tests of uniformity might react differently to the same data set. I'll also say something about the intriguing appearance of "group frames" in the computation of a couple of popular linear test statistics.

One approach to integer programming (maximizing an objective function across all “feasible” integer points satisfying a set of linear inequalities) is to start at some feasible integer point and continue stepping to a “nearby” feasible integer point that is better according to our objective function. If the set of possible nearby points is chosen correctly, using Herbert Scarf’s definition of neighbors, we will eventually arrive at the desired optimum integer point.

Consider the integer points in a polytope and connect with an edge every pair who are neighbors. This gives the set of edges of a useful simplicial complex called the neighborhood complex. Using the fact that the neighborhood complex is contractible, we can compute generating functions for the following types of sets: those defined as the projection of the set of integer points of a polyhedron.

As a concrete example of this sort of set, suppose we are given relatively prime positive integers a_1,…,a_d, and define S to be the set of integers that can be written as a nonnegative integer combination of these a_i. We’d like to answer questions like “What is the largest integer not in S?” and “How many integers are not in S?” These questions can be attacked via generating functions.

This talk builds on joint work with Herbert Scarf.

In 1986, Vaughan Jones discovered the Jones Polynomial, a powerful invariant of knots and links. In 1998, Mikhail Khovanov took it one step further with a categorification of the Jones polynomial we now call Khovanov homology, which determines the Jones polynomial and is a stronger invariant. In this talk we will review the Jones polynomial and see the basics of Khovanov homology.

The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over sparse random graphs, the phase structure of the graph bisection problem displays certain familiar properties, but also some surprises. It is known that when the mean degree is below the critical value of 2 log 2, the bisection width (or cutsize) is zero with high probability. We study how the minimum bisection width increases with mean degree above this critical threshold, finding a new analytical upper bound that improves considerably upon previous bounds. Combined with recent results on expander graphs, our bound suggests the unusual scenario that random graph bisection is replica symmetric up to and beyond the critical threshold, with a replica symmetry breaking transition possibly taking place above the threshold. An intriguing algorithmic consequence is that although the problem is NP-hard, we can conceivably find near-optimal bisection widths (whose ratio to the optimal value approaches 1 asymptotically) in polynomial time for typical instances near the phase transition. This is joint work with Gabriel Istrate, Bruno Goncalves, Robert Sumi and Stefan Boettcher.