Number Theory

This program focuses on three connected areas of number theory: Analytic Number Theory, Number Theory and Probability, as well as Rational Points. The aim is to bring together researchers from these areas in order to push the boundaries of current central questions, building on fascinating recent advances in the individual areas.

Analytic Number Theory has seen a number of unexpected breakthroughs on very fundamental questions during recent years. Starting with Green and Tao’s work on progressions in the primes and Goldston, Pintz and Yıldırım’s work on prime gaps, progress in our understanding of the distribution of primes included breakthrough work by Zhang, Maynard and Tao on bounded gaps between primes. Subsequent progress on large gaps between primes includes work by Ford, Green, Konyagin, Maynard and Tao, who, building on arguments by Rankin, Maier and Pomerance together with more recent results on hypergraph coverings, resolve a long standing conjecture of Erdős on the maximum gap between consecutive primes (up to x). Apart from prime number theory, multiplicative number theory has seen major progress: the development of the pretentious approach to number theory by Granville and Soundararajan, and Matomäki and Radziwill’s breakthrough on multiplicative functions in short intervals, leading to the resolution of the Erdős Discrepancy problem.

The birth of Probabilistic Number Theory is often credited to Erdős and Kac, who showed that the distribution of the number of prime factors of large random integers is essentially Gaussian. Applications of classical probabilistic techniques in number theory and the study of L-functions are now abundant but the usage of modern probability techniques has not been so common. However, this has changed recently: random matrix theory, random fragmentation processes, Poisson branching processes, Stein’s method and martingales now play a key role in advancements on new and classical problems in number theory.

The study of Rational Points on varieties is connected to one of the oldest problems in number theory, namely that of understanding integral solutions to Diophantine equations. Central questions here involve both the existence and distribution of rational points. Our understanding of such questions has been considerably pushed forward not just by advances within Diophantine geometry (obstruction theory, descent, fibration method, torsors), but also by the interplay of algebraic geometry on one side and analytic methods from number theory (circle method, additive combinatorics), harmonic analysis, and ergodic theory / dynamical systems on the other side.

There will be two bridging workshops during the program, with talks equally divided between the adjacent thematic sections, aiming for discovery of new connections.

• Jan 11 – Feb 5: Section with focus on Number Theory and Probability
• Feb 8 – 12: Bridging workshop
• Feb 15 – Mar 19: Section with focus on Analytic Number Theory
• Mar 22 – 26: Bridging workshop
• Mar 29 – Apr 30: Section with focus on Rational Points

Participation in the program is by invitation only.