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

该奖项支持美国研究人员参加将在蒙特利尔举行的一系列关于数论各种主题的研讨会。这是一个在2013年取得重大突破和显著进展的课题。主要的突破是确定质数集合之间有无限频繁的小间隙。到目前为止，我们已经知道有无穷多个素数，其下一个素数最多比它大270个。这与其他自然发生的数字序列不同；例如，正方形没有无限频繁的小间隙。这一系列讲习班将总结该领域广泛的最新发展，并将促进进一步的进展。该奖项将帮助年轻的美国数学家（学生、博士后和其他资源有限的年轻研究人员）到蒙特利尔的数学研究中心（CRM）参加2014年秋季的四个研讨会，主题如下：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量的分布；——加性组合与展开，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元组和群展开性质方面取得了根本性突破；——计算算术对象，椭圆曲线的秩，2014年11月10日至14日（http://www.crm.umontreal.ca/act/theme/theme_2014_2_en/counting_e.php）： Manjul Bhargava思想最新发展的首次会议；——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）：这是一个令人兴奋的复兴领域，由最近在质数和碎片、黎曼ζ函数矩以及更好地理解算法运行时间方面的非凡工作所引领。这些课题是数论领域的热点问题。这些会议将为年轻的美国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。曼珠尔·巴尔加瓦方法的发展在很大程度上只有他的学生和少数其他人很好地理解；在这一重点年之前，CRM的暑期学校和这些研讨会，将有更多的人参与其中，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小型学校之后举行的，这个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，这些互动的目标是建立长期的工作关系。

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.