School of Mathematics

The Stepanov Method

Avi Wigderson
Institute for Advanced Study
May 25, 2010

The Stepanov method is an elementary method for proving bounds on the number of roots of polynomials. At its core is the following idea. To upper bound the number of roots of a polynomial f(x) in a field, one sets up an auxiliary polynomial F(x) , of (magically) low degree, which vanishes at the roots of f with high multiplicity. That appropriate F exits is usually proved by a dimension argument.

Reductions Between Expansion Problems

Madhur Tulsiani
Institute for Advanced Study
May 18, 2010

The small-set expansion conjecture introduced by Raghavendra and Steuerer is a natural hardness assumption concerning the problem of approximating edge expansion of small sets (of size $\delta n$) in graphs. It was shown to be intimately connected to the well-known Unique Games Conjecture.

Pursuing this line of research further, we obtain the following results: