Interactions of Computational Complexity Theory and Mathematics

TL;DR

This paper surveys diverse interactions between computational complexity theory and various fields of mathematics, highlighting the extensive and surprising ways in which computational ideas influence mathematical research and understanding.

Contribution

It provides a broad, accessible overview of how computational complexity intersects with multiple mathematical disciplines, emphasizing the depth and potential of these interdisciplinary connections.

Findings

Connections span number theory, geometry, operator theory, group theory, physics, analysis, lattice theory, and invariant theory.

Highlights the natural and surprising ways computation influences mathematical fields.

Encourages further exploration of computational methods in mathematics.

Abstract

$ $[This paper is a (self contained) chapter in a new book, Mathematics and Computation, whose draft is available on my homepage at https://www.math.ias.edu/avi/book ]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: $\bullet$ Number Theory: Primality testing $\bullet$ Combinatorial Geometry: Point-line incidences $\bullet$ Operator Theory: The Kadison-Singer problem $\bullet$ Metric Geometry: Distortion of embeddings $\bullet$ Group Theory: Generation and random generation $\bullet$ Statistical Physics: Monte-Carlo Markov chains $\bullet$ Analysis and Probability: Noise stability $\bullet$ Lattice Theory: Short vectors $\bullet$ Invariant Theory: Actions on matrix tuples