At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove

things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on

algebraic proof techniques. The PCP characterization of NP and the

Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples.

Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved

algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques.

This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings.