Number theory: from Arithmetic statistics to Zeta elements, June 5-6, 2014

该奖项支持美国的研究人员参加在蒙特利尔举行的一系列关于数论中各种主题的研讨会。这是一个在2013年取得重大突破并随后取得显著进展的主题。这一重大突破确立了素数集之间经常有很小的差距。到目前为止，已知有无限多个素数，其下一个素数至多是270个大小。其他自然出现的数字序列则不是这样；例如，平方通常不会有无限小的间隙。这一系列讲习班将评估该领域最新的广泛事态发展，并将促进进一步的进展。该奖项将有助于将美国初级数学家(学生、博士后研究员和其他获取资源的年轻研究人员)带到蒙特利尔的数学研究中心，参加2014年秋季关于以下主题的四个研讨会：--统计和数论，2014年9月15日至19日，(http://www.crm.umontreal.ca/act/theme/theme_2014_2_en/stats_e.php)：更好地理解算术对象研究中重要统计数据的分布；--加性组合学和扩展器，2014年10月6日至10日(http://www.crm.umontreal.ca/act/theme/theme_2014_2_en/combinatoire_e.php)：当今纯数学中一个充满活力的学科，最近在素数k-字节组和群扩展性质方面取得了根本性的突破；--算术对象计数，椭圆曲线的阶数，2014年11月10日至14日(http://www.crm.umontreal.ca/act/theme/theme_2014_2_en/counting_e.php)：曼朱尔·巴尔加瓦思想产生的最新发展的成立会议；-概率和乘法数论的新方法，2014年12月8日至12日(http://www.crm.umontreal.ca/act/theme/theme_2014_2_en/nombres_e.php)：一个令人兴奋的复兴领域，由最近在素数和碎片化、Riemann Zeta函数的矩以及对算法运行时间的更好理解方面的非凡工作领导。这些主题是广泛范围的数论中的热门话题。这些会议将为美国初级研究人员提供与各自领域的世界领先研究人员互动的机会。Manjul Bhargava在很大程度上只有他的学生和其他几个人很好地了解了他的方法的发展；在今年重点关注的一年之前，CRM的暑期学校和这些研讨会，将有更多的人参加并有机会学习这些技术。加法组合学的会议是在数学及其应用研究所的一个迷你学校之后举行的，这次研讨会将给初级研究人员一个机会，在该领域的领导者面前展示他们的工作，目标是这些互动导致长期的工作关系。

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 (http://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 (http://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 (http://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 (http://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.