Special Meeting: Analysis in Number Theory Year 2005-6 at the CRM

特别会议：CRM 2005-6 年数论分析

• 批准号: 0531946
• 负责人: David Dummit
• 金额: $ 7万
• 依托单位: University of Vermont & State Agricultural College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0531946&HistoricalAwards=false
• 关键词: Special; Meeting; Analysis; Number; Theory

The 2005-2006 special year "Analysis in Number theory" at theCentre de Recherche en Mathematiques in Montreal will focus on two main areas: 1) The fall semester on p-adic analysis in arithmetic geometry, specifically i) A motivated definition of a p-adic Langlands correspondence; ii) Arithmetic intersection of cycles and modular forms. 2) The winter semester on analytic number theory, specifically i) "Hot" traditional areas, such as bounds for the size of L-functions, the use of higher dimensional L-functions, and understanding the "anatomy" of integers (distribution of prime divisors, multiplicative functions, smooth numbers, etc.); ii) Additive Combinatorics, an exciting new subject that has already led to several important breakthroughs (including that there are infinitely many k-term arithmetic progressions of primes, and non-trivial bounds on very short exponential sums).There will be a total of six workshops and two major "schools" to introduce junior mathematicians to the key exciting themes. Activities during the year are centered around several key participants, particularly early career researchers who are making an enormous impact such as Bhargava, Green, Soundararajan and Tao.In addition there will be extended visits by more than forty active researchers.This grant will help junior US mathematicians to take advantage of these opportunities.Number theory, a subject often motivated by the simplest and most basic of questions, is nevertheless breathtaking in its scope of techniques and breadth of applications. Certain number theoretic topics are very exciting at the moment, following some extraordinary recent breakthroughs in the understanding of some fundamental and longstanding questions. This special year in number theory will give more researchers the opportunity not only to learn about these breakthroughs, but to collaborate with a wide array of number theorists from around the globe. Of particular emphasis are the development of a "p-adic Langlands correspondence", which will tie together far flung topics in a surprising way, and the further development of "additive combinatorics", which has recently led to the proof that there are infinitely many k-term arithmetic progressions of primes,a famous old question. There will also be focus workshops on other topics that the organizers believe are primed for significant advances. One of these areas has seen the proof that there are gaps between primes that are far smaller than the average, another longstanding question. This special year gives U.S. mathematicians, especially young researchers, an unusual opportunity to interact with a large number of the world's leading number theorists, particularly those from Europe and Asia.

蒙特利尔数学研究中心2005-2006特别年度“数论分析”将集中在两个主要领域：1)秋季学期算术几何中的p进分析，特别是i) p进朗兰兹对应的动机定义；循环与模形式的算术交。2)冬季学期解析数论，特别是i)传统的“热点”领域，如l函数大小的界限，高维l函数的使用，理解整数的“解剖”（质因数分布，乘法函数，光滑数等）；ii)加性组合学，一个令人兴奋的新学科，已经导致了几个重要的突破（包括有无限多个k项的素数等差数列，以及非常短的指数和的非平凡界限）。总共将有六个研讨会和两个主要的“学校”，向初级数学家介绍关键的令人兴奋的主题。这一年的活动主要围绕几位关键参与者展开，尤其是具有巨大影响力的早期职业研究人员，如Bhargava、Green、Soundararajan和Tao。此外，还将有40多名活跃的研究人员进行长期访问。这项资助将帮助美国的年轻数学家利用这些机会。数论是一门经常被最简单和最基本的问题所激发的学科，然而它的技术范围和应用范围却令人惊叹。随着最近对一些基本和长期问题的理解取得了一些非凡的突破，某些数论主题目前非常令人兴奋。今年是数论的特殊年份，更多的研究人员不仅有机会了解这些突破，而且可以与来自世界各地的数论专家合作。特别强调的是“p进朗兰兹对应”的发展，它将以一种令人惊讶的方式将遥远的主题联系在一起，以及“加性组合学”的进一步发展，它最近导致了一个著名的老问题，即存在无限多个k项素数等差数列的证明。会议还将针对组织者认为有望取得重大进展的其他主题举办专题研讨会。其中一个领域已经证明质数之间的间隔远远小于平均值，这是另一个长期存在的问题。这个特殊的年份给了美国数学家，尤其是年轻的研究人员一个不同寻常的机会，可以与大量世界领先的数论专家，特别是来自欧洲和亚洲的数论专家进行交流。