A Panoramic Account of Sieve Theory by Prof. Louis Gaudet, UMass Amherst

A Panoramic Account of Sieve Theory by Prof. Louis Gaudet, UMass Amherst, Friday March 15, 1:00 – 1:50pm, North Science Building 015, Wachenheim

Abstract:
Sieve methods in analytic number theory are used to detect prime numbers within (often arithmetically interesting) subsets of the integers. Although the basic inclusion-exclusion process to find primes goes back to Eratosthenes (3rd century BCE), sieve theory began in earnest in 1915 with the work of Viggo Brun. Since then, sieve methods have flourished: they have both been honed as tools in their own right, and they have been applied to much greater effect with the input of various other tools—the theory of L-functions, harmonic analysis, and homogeneous dynamics, just to name a few.

I will try to give a sort of panoramic account of the theory—the basic principles of how it works, some key results, and its limitations, most notably the infamous “parity barrier.” I will discuss some of the most impressive applications of sieves, such as “bounded gaps between primes,” due independently to Yitang Zhang and James Maynard. Lastly, I will muse on some open questions within the theory and possible avenues of research, including formulating the sieve process as a linear programming problem.

This talk is for colloquium credit.