### It’s All in the Symmetries

**Analytic and Geometric Number Theory**

The tools used in modern number theory range over many fields from algebra and algebraic geometry to harmonic analysis, representation theory, ergodic theory and combinatorics. Central to the study of many problems concerning prime numbers, diophantine equations and number fields, are sieve methods, L-functions (and with these the theory of automorphic forms), diophantine geometry and homogeneous spaces. All of these topics have for some time, and still do, enjoy substantial progress, especially when used in combination with each other as well as with novel inputs from unexpected sources such as ergodic theory and logic (related to transcendence). These were the topics and the theme of the special year in the School of Mathematics at the Institute for Advanced Study during 2009-10.

Some of the highlights were the mini-courses and lecture series connected with these topics. These included Soundararajan’s lectures on weak subconvexity for general automorphic L-functions and its application to the solution of the “holomorphic QUE” conjecture in the theory of modular forms, Pila’s and Zannier’s lectures on the proof of the Andre’-Oort and related conjectures using ideas that go back to Bombieri’s and Pila’s real analytic/geometric bounds for points on curves and Bhargava’s lectures on upper bounds for average ranks of the universal family of elliptic curves. Another direction in which major progress has been achieved, both at the Institute and elsewhere, during the last year and which was reported in many talks and in the workshop is the use of ergodic theory (Furstenberg theory and the inverse Gowers norm problem) and combinatorics in the form of spectral gaps (expanders) in the study of prime numbers. Specifically the Green-Tao work on simultaneous linear equations in primes and the affine linear sieve.

The developments mentioned which combine various topics rely heavily on developments in each of these separately. Lecture series introducing some of these as well as recent breakthroughs connected with them were presented in many of the seminars. These include new developments on the subconvexity problem for higher degree L-functions (a problem whose complete solution would have striking applications both in diophantine equations and in the analytic theory of automorphic forms), counting rational points on homogeneous varieties and advanced methods in combinatorial sieving.