Speaker: Parikshit Gopalan, Georgia Tech. Title: Symmetric Polynomials mod 6 and Simultaneous Communication Protocols Abstract: Representations of Boolean functions as multivariate polynomials modulo $m$ are well studied in complexity theory. One of the surprising results in this area is that simple functions like OR can be represented by low degree polynomials when $m$ is composite but not when $m$ is prime. Such representations over composites are not as well understood as in the prime case. We study the problem of representing symmetric Boolean functions as symmetric polynomials over composites. We show an equivalence between such representations and simultaneous communication protocols. Functions represented by symmetric polynomials of degree $d$ modulo 6 correspond to functions computable by 2 player protocols where Player1 and Player2 are given only $log d$ lower order digits of the weight of the input in base 2 and 3 respectively. This equivalence allows us to use tools from communication complexity and number theory to show improved bounds on the degree of such representations. We will also describe connections between such polynomials and explicit constructions of combinatorial objects such as Ramsey graphs and set systems with restricted intersections. This is joint work with Nayantara Bhatnagar and Richard J. Lipton