Random matrices, arithmetic combinatorics, and incidence geometry

这个提议寻求在数学的三个不同的（只是松散相关的）领域取得进展。首先，在随机矩阵理论领域，我们寻求建立在理解这些矩阵迷人的普遍性现象的最新进展的基础上，通过推广可以建立普遍性的集合，并寻找导致这种普遍性的潜在机制。其次，在算术组合领域，我们寻求改进非交换条件下的近似群和相关结构的理论，对几何群论和有限群论有潜在的影响。最后，我们希望扩展和理解代数几何、代数拓扑和关联组合之间令人兴奋的新联系，这些联系通过应用代数方法导致了后者的显着突破。物理系统通常非常复杂，由这些系统中无数组件之间的相互作用驱动。然而，在许多情况下，一种被称为普遍性的显著现象发生了，在这种现象中，各种系统的宏观统计数据都受单一的普遍数学定律的支配，几乎不考虑这些系统的组成部分如何在微观层面上相互作用。例如，数学中著名的中心极限定理断言，许多统计数据（例如人口中身高的分布）都可以用一条被称为“钟形曲线”或“高斯分布”的曲线来描述。虽然中心极限定理现在已经被很好地理解了，但还有一些其他的普遍规律仍然是神秘的，比如戴森正弦定律，它已经被经验地发现可以控制诸如核散射、公共汽车到达时间和数论中zeta函数的间隔等各种统计数据。然而，我们最近已经开始理解这一规律的一个简单的模型类称为随机矩阵模型，尽管即使是这些模型仍有很多工作要做。我们计划进一步了解随机矩阵模型和相关模型的普适性的潜在原因，希望对其他模型的普适性现象也有所了解。研究计划的第二个方面涉及各种网络的扩展现象，最著名的是米尔格拉姆的“六度分离”实验，该实验断言世界上任何两个人最多有六个熟人联系在一起。对于这种现象的数学模型，即所谓的Cayley图，膨胀的存在与不存在某种数学对象密切相关，这种数学对象被称为近似群。在了解近似群体的确切样子方面已经取得了很大的进展，但我们对这些群体的控制仍然没有我们希望的那么精确。我们计划进一步研究近似群理论，着眼于将其应用于其他数学领域，如理解群的几何。组合入射几何是研究简单的几何对象，如线、圆和点如何尽可能有效地排列在一起，以实现各种目的（例如，尝试排列给定数量的点和线，使尽可能多的点位于尽可能多的线上）。了解如何有效地设计这种配置的限制是一个基本的数学问题，也有实际应用（例如，在设计手机的频率配置，以最大限度地提高容量和减少干扰）。近年来，通过引入代数（特别是代数几何和代数拓扑）的方法，该学科取得了一些突破。这种意想不到的发展仍然没有得到很好的理解；代数方法几乎可以完全解决一些问题，而对其他表面上类似的问题却毫无用处。我们计划进一步研究这些方法，并确切地了解它们的优势和局限性。

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.