When: Sunday, April 30,
10am
Where: Schreiber 309
Speaker: Shachar Lovett,
UCSD
Title:
The Monomial Structure of Boolean Functions
Let $f:\{0,1\}^n \mapsto \{0,1\}$ be a boolean function. It can be uniquely represented as a multilinear polynomial. What is the structure of its monomials? This question turns out to be connected to some well-studied problems, such as the log-rank conjecture in communication complexity and the union-closed conjecture in combinatorics. I will describe these connections, and a new structural result, showing that set systems of monomials of boolean functions have small hitting sets, concretely of poly-logarithmic size. The proof uses a combination of algebraic, probabilistic and combinatorial techniques.
Based on joint work with Alexander Knop, Sam McGuire, and Weiqiang Yuan.