Interactions Between Random Matrix Theory, Number Theory and Combinatorics

首席研究员计划进行两个项目，专门研究随机矩阵理论、数论和组合学之间的相互作用。在第一个项目中，以他最近与Persi Diaconis和Brian Conrey的合作为基础，首席研究员将探索随机矩阵的长期系数分布、Conrey、Farmer、Keating、Rubinstein和snaith对l函数矩的猜想以及与计数幻方相关的枚举组合学中的一些经典问题之间的联系。项目二从统一的角度研究展开图理论中的一个主要问题和量子混沌理论中的一个基本猜想。由Lubotzky和Weiss提出的展开图理论中的一个基本问题是，Cayley图族的展开族在多大程度上是群的性质，与生成元的选择无关。对于许多自然群族，特别是对于特殊的二阶线性群，数值实验表明它可能是一般生成器选择的扩展族（独立性猜想）。由Bohigas， Giannoni和Shmit提出的量子混沌中的一个基本猜想断言，量子化混沌哈密顿量的特征值的行为类似于适当的随机矩阵集合的典型成员的谱。这两种猜想都可以被看作是断言确定性构造的谱一般表现得像一个大随机矩阵的谱：在体（量子混沌猜想）和在谱的边缘（独立性猜想）。首席研究员将在群环元素光谱的背景下证明这些猜想。随机矩阵理论起源于维格纳在五十年代早期提出的一个建议，即重核的共振谱可以用一个大的随机矩阵的谱来模拟。在随后的五十年中，随机矩阵理论的范围和深度得到了极大的发展；在过去的十年里，这门学科经历了爆炸式的增长。首席研究员的第一个项目旨在建立随机矩阵理论两种最新发展路线之间的联系。一是发现并利用了特征值统计与列举组合中最长增减子序列问题之间的联系；另一个是对随机矩阵的特征多项式和相关的全局统计的兴趣爆发，特别是与黎曼ζ函数的矩有关，黎曼ζ函数在数论中是一个重要的函数。主要研究人员的第二个项目致力于研究随机矩阵和扩展图之间的连接——高度连接的稀疏图可以沿着短路径有效地将信息快速传播到许多节点。最近这种图的显式构造引起了人们对它们在网络设计、复杂性理论、编码理论和密码学中的潜在应用的兴趣。这些结构本质上是代数的，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、看似无关的主题优雅地结合起来，以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.