Interactions Between Random Matrix Theory, Number Theory and Combinatorics

Gamburd摘要 提案#0501245主要研究者计划开展两个专门研究随机矩阵理论、数论和组合学之间相互作用的项目。在第一个项目中，基于他最近与Persi Diaconis和Brian Conrey的联合工作，主要研究者将探索随机矩阵的长期系数的分布，Conrey，Farmer，Keating，Rubinstein和Snaithfor L-函数矩的猜想之间的联系，以及与计数幻方有关的枚举组合学中的一些经典问题。主要研究者的第二个项目致力于从统一的角度研究扩展图理论中的主要问题之一和量子混沌理论中的基本原理之一。Lubotzky和韦斯提出的扩张图理论中的一个基本问题是，在多大程度上Cayley图族的扩张族是群本身的性质，而与生成元的选择无关。数值实验表明，它可能是生成元一般选择的一个扩展族（独立猜想）。Bohigas、Giannoni和Shmit提出了量子混沌中的一个基本猜想，认为量子混沌哈密顿量的本征值的行为类似于随机矩阵适当系综的典型成员的谱。这两个猜想都可以被看作是断言一个确定性构造的谱一般表现得像一个大的随机矩阵的谱：在体（量子混沌猜想）和在谱的边缘（独立猜想）。随机矩阵理论起源于魏格纳在五十年代早期提出的重核共振线可以用一个大的随机矩阵的谱来模拟的建议。在随后的五十年里，随机矩阵理论的广度和深度都有了显著的提高，在过去的十年里，这门学科经历了爆炸式的增长.主要研究者的第一个项目旨在建立随机矩阵理论中两条最近发展路线之间的联系。 一个是发现和利用特征值统计之间的联系和最长增长的子序列问题的计数组合;另一个是兴趣的爆发，在特征多项式的随机矩阵和相关的整体统计，特别是与时刻的黎曼zeta函数，一个功能的根本重要性在数论。 第二个项目的主要作者是致力于研究随机矩阵和扩展图之间的连接-高度连接的稀疏图，有效地传播信息迅速到许多节点沿着短路径。最近，这种图的显式构造在网络设计、复杂性理论、编码理论和密码学等方面的潜在应用引起了人们极大的兴趣。这些结构是代数性质的，并提供了一个美丽的例子，说明如何将数论，群论和组合学中抽象的，不相关的主题优雅地结合起来解决一个重要的现实问题。

Gamburd Abstract Proposal #0501245The principal investigator plans to pursue two projects devoted tothe study of interactions between random matrix theory, numbertheory and combinatorics. In the first project, building on hisrecent joint work with Persi Diaconis and with Brian Conrey, the principal investigator will explore connections between thedistribution of the secular coefficients of random matrices, theconjectures of Conrey, Farmer, Keating, Rubinstein and Snaithfor moments of L-functions, and some classical problems in enumerative combinatorics related to counting magic squares. The second project of the principal investigator is devoted to studying from a unified point of view one of the main problems in the theory of expander graphs and one of the basic conjectures in the theory of quantum chaos. A basic problem in the theory of expander graphs, formulated by Lubotzky and Weiss, is to what extent being an expander family for a family of Cayley graphs is aproperty of the groups alone, independent of the choice of generators.For many natural families of groups, in particular, for special lineargroup of order two, numerical experiments indicate that it mightbe an expander family for generic choices of generators(Independence Conjecture). A basic conjecture in Quantum Chaos,formulated by Bohigas, Giannoni, and Shmit, asserts that theeigenvalues of a quantized chaotic Hamiltonian behave like thespectrum of a typical member of the appropriate ensemble of randommatrices. Both conjectures can be viewed as asserting that adeterministically constructed spectrum generically behaves likethe spectrum of a large random matrix: in the bulk (QuantumChaos Conjecture) and at the edge of the spectrum (IndependenceConjecture). The principal investigator will work on proving theseconjectures in the context of the spectra of elements in group rings.Random Matrix Theory originated in Wigner's suggestion in theearly fifties that the resonance lines of heavy nuclei might bemodelled by the spectrum of a large random matrix. In the ensuingfifty years the scope and depth of Random Matrix Theory hasdramatically increased; in the past decade the subject hasundergone explosive growth. The first project of the principalinvestigator aims at forging a link between two recent lines ofdevelopment in Random Matrix Theory. One is the discovery andexploitation of the connections between eigenvalue statistics andthe longest-increasing subsequence problems in enumerativecombinatorics; another is the outburst of interest incharacteristic polynomials of random matrices and associatedglobal statistics, particularly in relation with the moments ofthe Riemann zeta function, a function of fundamental importance innumber theory. The second project of the principal investigatoris devoted to studying connections between random matrices and expander graphs -- highly connected sparse graphs that efficientlypropagate information quickly to many nodes along short paths. Recentexplicit constructions of such graphs have created an explosion ofinterest in their potential applications to network design, complexitytheory, coding theory and cryptography. These constructions are algebraicin nature, and provide a beautiful example of how abstract, seeminglyunrelated topics in number theory, group theory and combinatorics can be elegantly combined to solve an important real-world problem.