Report on "Mathematical Aspects of P vs. NP and its Variants."

TL;DR

This report summarizes recent developments and ideas discussed at a 2011 workshop on mathematical approaches to computational complexity, focusing on geometric, representation-theoretic, and number-theoretic methods related to P vs. NP.

Contribution

It provides an overview of current research directions and preliminary results in geometric complexity theory and Blum-Shub-Smale models from a major workshop.

Findings

Recent progress in geometric complexity theory

New insights into the Blum-Shub-Smale model

Emerging ideas for resolving P vs. NP

Abstract

This is a report on a workshop held August 1 to August 5, 2011 at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University, Providence, Rhode Island, organized by Saugata Basu, Joseph M. Landsberg, and J. Maurice Rojas. We provide overviews of the more recent results presented at the workshop, including some works-in-progress as well as tentative and intriguing ideas for new directions. The main themes we discuss are representation theory and geometry in the Mulmuley-Sohoni Geometric Complexity Theory Program, and number theory and other ideas in the Blum-Shub-Smale model.