We consider sums in which an additive character of a finite field F is applied to a binomial whose individual terms (monomials) become permutations of F when regarded as functions.  These Weil sums characterize the nonlinearity of power permutations of interest in cryptography.  They also tell us about the correlation of linear recursive sequences over finite fields that are used […]

The classical Sperner -  KKM (Knaster - Kuratowski - Mazurkiewicz) lemma has many applications  in combinatorics, algorithms, game theory and mathematical economics. In this talk we consider generalizations of this lemma as well as Gale's colored KKM lemma and Shapley's KKMS theorem. It is shown that spaces and covers can be much more general and […]

We begin with a review of the Bateman—Horn conjecture, which sheds light on the intimate relationship between polynomials and prime numbers.  In this expository talk, we survey a host of applications of the conjecture.  For example, Landau’s conjecture, the twin prime conjecture, and the Green—Tao theorem are all consequences of the Bateman—Horn conjecture.  Moreover, the […]