Random matrices, arithmetic combinatorics, and incidence geometry

随机矩阵、算术组合和关联几何

• 批准号: 1266164
• 负责人: Terence Tao
• 金额: $ 75万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266164&HistoricalAwards=false
• 关键词: Random; matrices; arithmetic; combinatorics; incidence

This proposal seeks to make progress in three distinct (and only looselyrelated) areas in mathematics. Firstly, in the area of random matrix theory, we seek to build upon recent progress in understanding the fascinating universality phenomenon for such matrices, by generalising the ensembles for which universality can be established, and looking for the underlying mechanisms causing this universality. Secondly, in the area of arithmetic combinatorics, we seek to improve our theory of approximate groups and related structures in noncommutative settings, with potential impact on geometric group theory and finite group theory. Finally, we wish to extend and understand the exciting new connections between algebraic geometry, algebraic topology, and incidence combinatorics which have led to remarkable breakthroughs in the latter subject by applying algebraic methods.Physical systems are often incredibly complicated, being driven by interactions between countless components of those systems. However, in many cases, a remarkable phenomenon known as universality occurs, in which the macroscopic statistics of the a wide variety of systems are governed by a single universal mathematical law, almost without any regard to how the components of these systems interact with each other at the microscopic level. For instance, the famous central limit theorem in mathematics asserts that many statistics (e.g. the distribution of heights in a human population) are described by a single curve known as the "bell curve" or "Gaussian distribution". While the central limit theorem is now very well understood, there are other universal laws that are still mysterious, such as the Dyson sine law that has been empirically found to govern such diverse statistics as nuclear scattering, arrival times of buses, and the spacing of the zeta function in number theory. However, we have recently begun to understand this law for a simple class of models known as random matrix models, though even for these models there is still much work to be done. We plan to work on further understanding the underlying causes of universality for random matrix models and related models, with the hope of shedding insight on the universality phenomenon for other models as well.The second aspect of the research program concerns the phenomenon of expansion in various networks, most famously manifested by the "six degrees of separation" experiment of Milgram, that asserts that any two people in the world are linked by at most six degrees of acquaintance. For a mathematical model of this phenomenon known as a Cayley graph, the existence of expansion is closely tied to the absence of a certain type of mathematical object known as an approximate group. There has been a substantial amount of progress in understanding what exactly approximate groups look like, but the control we have on these groups is still not as precise as we would like. We plan to study the theory of approximate groups further, with an eye towards applications to other areas of mathematics such as understanding the geometry of groups.Combinatorial incidence geometry is the study of how simple geometric objects such as lines, circles, and points can be arranged together in as efficient a configuration as possible for various purposes (e.g. to try to arrange a given number of points and lines so that as many points lie on as many lines as possible). Understanding the limits of how efficiently one can design such configurations is a basic mathematical question which also has practical applications (e.g. in designing frequency configurations for cell phones in order to maximize capacity and minimize interference). Recently, there has been several breakthroughs in the subject by introducing methods from algebra (particularly algebraic geometry and algebraic topology). This unexpected development is still not well understood; the algebraic methods can solve some problems almost completely, while making no mark on other ostensibly similar problems. We plan to study these methods further and understand exactly what their strengths and limitations are.

该提议旨在在数学中的三个不同（并且仅是松散相关的）领域取得进展。 首先，在随机矩阵理论领域，我们试图在理解这种矩阵的迷人普遍性现象方面的最新进展，通过概括可以建立普遍性的合奏，并寻找导致这种普遍性的基本机制。 其次，在算术组合学领域，我们寻求在非交通设置中改善近似群体和相关结构的理论，对几何群体理论和有限群体理论产生潜在的影响。最后，我们希望扩展和了解代数几何形状，代数拓扑结构和发病率组合学之间令人兴奋的新连接，这些连接通过应用代数方法引起了后者主题的显着突破。 然而，在许多情况下，出色的现象被称为普遍性，其中各种系统的宏观统计量受单个通用数学定律的控制，几乎没有任何考虑这些系统的组件在微观层面相互相互作用。 例如，数学中著名的中心极限定理断言，许多统计数据（例如，人口中的高度分布）被称为“钟形曲线”或“高斯分布”的单个曲线描述。 尽管现在已经非常了解中心限制定理，但还有其他仍然是神秘的普遍定律，例如戴森正弦定律经验，这些定律已被发现控制了诸如核散射，公共汽车的到来时间以及Zeta功能在数量理论中的统计数字。 但是，我们最近开始为一种称为“随机矩阵”模型的简单模型理解这项定律，尽管即使对于这些模型，仍然有很多工作要做。 We plan to work on further understanding the underlying causes of universality for random matrix models and related models, with the hope of shedding insight on the universality phenomenon for other models as well.The second aspect of the research program concerns the phenomenon of expansion in various networks, most famously manifested by the "six degrees of separation" experiment of Milgram, that asserts that any two people in the world are linked by at most six degrees of熟人。 对于这种现象的数学模型，称为Cayley图，扩展的存在与缺少某种类型的数学对象密切相关。 在理解近似群体的样子方面取得了很大的进步，但是我们对这些组的控制仍然没有我们想要的那么精确。 我们计划进一步研究近似群体的理论，并着眼于应用于其他数学领域，例如了解群体的几何形状。CombinatorialIntigention几何形状的研究是研究如何将诸如线条，圆圈和点之类的简单几何对象一起排列在一起，以尽可能高效地进行配置，以便在许多方面（例如，都可以等地进行许多点，以便许多方案和线路都等级数量和线条。 了解一个基本的数学问题，它可以有效地设计这种配置的限制，它也具有实际的应用程序（例如，在设计手机的频率配置时，以最大程度地提高容量并最大程度地减少干扰）。 最近，通过引入代数（尤其是代数几何和代数拓扑）的方法，该主题取得了一些突破。 这种意外的发展仍然不太了解。代数方法几乎可以完全解决一些问题，同时对其他表面上类似的问题没有任何痕迹。 我们计划进一步研究这些方法，并准确理解它们的优势和局限性。