This award supports the participation of US-based researchers in a series of workshops to be held in Montreal on various topics in the theory of numbers. This is a subject that has seen a major breakthrough and subsequent remarkable progress during 2013. The major breakthrough established that the set of prime numbers have small gaps between them infinitely often. To date it is now known that there are infinitely many primes for which the next prime is at most 270 larger. This is not the case for other naturally occurring sequences of numbers; for example, the squares do not have small gaps infinitely often.
The series of workshops will take stock of the broad range of recent developments in the field and will catalyze further progress. This award will help bring junior US mathematicians (students, postdoctoral fellows, and other young researchers with limited access to resources) to the Centre de Recherches Mathematiques (CRM) in Montreal to participate in four workshops in Fall 2014 on the following topics: -- Statistics and number theory, September 15-19, 2014 (www.crm.umontreal.ca/act/theme/theme_2014_2_en/stats_e.php): A better understanding of the distribution of important statistics in the study of arithmetic objects; -- Additive combinatorics and expanders, October 6-10, 2014 (www.crm.umontreal.ca/act/theme/theme_2014_2_en/combinatoire_e.php): A vibrant subject in pure mathematics today, with fundamental recent breakthroughs on prime k-tuples, and group expansion properties; -- Counting arithmetic objects, ranks of elliptic curves, November 10-14, 2014 (www.crm.umontreal.ca/act/theme/theme_2014_2_en/counting_e.php): The inaugural meeting on the latest developments stemming from the ideas of Manjul Bhargava; -- New approaches in probabilistic and multiplicative number theory, December 8-12, 2014 (www.crm.umontreal.ca/act/theme/theme_2014_2_en/nombres_e.php): An area with an exciting renaissance, led by extraordinary recent work on primes and fragmentation, moments of the Riemann zeta function, and better understanding of running times of algorithms.
These subjects are hot topics in number theory of a broad scope. The meetings will provide opportunities for junior US researchers to interact with the leading researchers in the world in their field. Developments in Manjul Bhargava's methods have largely been well understood only by his students and a few others; with a summer school at the CRM preceding this focused year and these workshops, many more people will be included and have an opportunity to learn these techniques. The meeting in additive combinatorics follows a mini-school at the Institute for Mathematics and its Applications, and this workshop will give junior researchers a chance to present their work in front of leaders in the field, with the goal of these interactions leading to long-term working relationships.
Number theory enjoys a privileged position within Mathematics as a fertile source of fundamental questions. Among the seven Millenium problems listed by the Clay Institute, no less than three--the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, and the Riemann hypothesis--were handed down by the Queen of Mathematics. Even by the standards of a subject that has remained vibrant since the days of Fermat and Gauss, the last two decades have witnessed a real golden age. Number theory has also spurred the growth of a wide array of new techniques in other areas of mathematics. In the academic year 2014-2015, a Thematic Year in number theory at the Centre de Recherches Mathematiques (CRM) in Montreal took stock of the most recent developments to emerge from this prolonged spate of activity (www.crm.umontreal.ca/Number2014/), with a focus on hot topics in the area. Support from NSF brought junior US mathematicians (students, postdoctoral fellows, and other young researchers with limited access to resources) to participate in four workshops in Fall 2014. These meetings gave the opportunity for junior US researchers to interact with the leading researchers in the world in their field. One goal for this series of conferences was to broaden participation by young participants, to encourage them to build connections with people from all over the world, and in particular to further existing relationships between researchers in the US and Canada. Talks and presentations were made accessible to students, building on knowledge gained at a summer school held at the CRM.
