Away from Independence: Probability in Geometry, Topology, Number Theory and Mathematical Physics

贯穿此处描述的各个子项目的共同主线是，利用几何和函数分析观点，结合定量概率分析工具，研究独立不是自然条件的数学领域中的概率现象。第一个子项目是PI正在进行的对一大类高维概率分布的“典型的高斯边缘”现象的继续调查。在这项工作的最新部分中，PI发现对于一类非常一般的分布，发生这种现象的投影维度有一个明显的截止点。提出的未来工作的一个方向是通过研究附加几何条件对大多数投影看起来像高斯的最大维度上的测量的影响来丰富这一理论。第二个子项目涉及某些类型的随机矩阵的谱测量，并且与几何、数学物理和数论都有联系。在最近与M.Meckes的合作中，PI证明了随机酉阵的任意幂的经验谱分布的强浓缩结果。这项工作很可能有助于详细分析紧经典群中随机矩阵的最大特征值间隙的渐近行为。第三个项目是与P.Albin合作研究紧致流形上的条件，在该条件下，拉普拉斯算子的本征函数在高本征值极限下具有近似的高斯值分布。这种近似有时成立的猜想大多来自量子混沌，而且这些猜想存在一些数字证据。将PI的早期工作与伪微积分技术相结合似乎是有希望的。其他处于早期阶段的项目包括：与C.Hughes合作，在L函数的零点上寻找新的模式，灵感来自随机酉阵上的雨的深刻结果；与K.Kirkpatrick的联合项目，使用新的非对易版本的Stein方法证明多体量子动力学中的中心极限定理；和V.de Silva的一个联合项目，将PI早期关于随机单纯复形的Betti数的极限定理的工作扩展到持久同调的背景下。贯穿这些项目的一个主要目标是摆脱经典概率论传统上关注的问题，以便与广泛的其他数学领域相联系。似乎越来越多的领域，从黎曼几何到数论，到算法设计和其他领域，都发现概率观点可以打开解决以前难以理解的问题的大门，并提出以前从未考虑过的富有成效的新研究路线。这种探索的核心是独立的作用。经典概率论关注的是独立性发挥关键作用的情况--当一个人可以描述多个实验并假设它们的结果不会相互影响时，就可以非常详细地进行非常精确的分析。然而，在数学和科学的其他领域，概率以许多引人入胜和重要的方式出现，在这些领域中，独立性并不是自然发生的，而是有其他一些特殊的结构。例如，假设传感器网络需要一定程度的连通性才能正常运行。如果传感器从飞机上投放到荒野地区，重要的是要知道可能需要多少传感器才能获得足够的连接，网络才能正常运行。一种经典的概率方法可能是假设每对传感器独立地以规定的概率进行通信，但更现实的方法是结合空间数据，并假设传感器之间如果足够近就进行通信。由于缺乏独立性，理解这样一个随机网络发挥作用的可能性变得更加困难，但最终却是一个更有用的问题。这个项目的目标是调查这样的情况，在这种情况下，独立性假设已经消失，但可以利用几何或其他考虑因素。

The common thread running through the various sub-projects described here is to make use of geometric and functional-analytic perspectives in conjunction with tools of quantitative probabilistic analysis in order to investigate probabilistic phenomena in areas of mathematics in which independence is not a natural condition. The first sub-project is a continuation of the PI's ongoing investigation of the phenomenon of "typically Gaussian marginals'' of a large class of high-dimensional probability distribution. In the most recent part of this work, the PI found a sharp cut-off for the projection dimensions for which this phenomenon occurs for a very general class of distributions. One direction of proposed future work is to enrich this theory by investigating the effect of additional geometric conditions on the measure on the largest dimension onto which most projections look Gaussian. A second sub-project deals with the spectral measures of certain types of random matrices, and has connections to both geometry, mathematical physics and number theory. In very recent joint work with M. Meckes, the PI proved strong concentration results for the empirical spectral distributions of arbitrary powers of random unitary matrices. It is likely that this work will enable a detailed analysis of the asymptotic behavior of the maximal eigenvalue gap of random matrices from the compact classical groups. A third proposed project is a collaboration with P. Albin to investigate conditions on a compact manifold under which the eigenfunctions of its Laplacian have approximately Gaussian value distributions in the high-eigenvalue limit. Conjectures that such an approximation sometimes holds have mostly come out of quantum chaos, and some numerical evidence for these conjectures exists. Combining earlier work of the PI with techniques of pseudo-differential calculus appears promising. Other projects in early stages include a collaboration with C. Hughes to find new patterns in the zeroes of L-functions, inspired by a deep result of Rains on random unitary matrices; a joint project with K. Kirkpatrick to prove central limit theorems in many-body quantum dynamics using a new non-commutative version of Stein's method; and a joint project with V. de Silva to extend the PI's earlier work on limit theorems for Betti numbers of random simplicial complexes to the context of persistence homology.A main goal running through these projects is to move away from the problems that classical probability theory has traditionally focused on, so as to interface with a wide variety of other areas of mathematics. It seems that more and more fields, from Riemannian geometry to number theory to algorithm design and beyond, are finding that a probabilistic viewpoint can open doors to solutions to previously impenetrable problems, and suggest fruitful new lines of research that had not previously been considered. Central to such exploration is the role of independence. Classical probability theory focuses on situations in which independence plays a key role -- when one can describe multiple experiments and assume that their outcomes cannot affect each other, very precise analyses can be carried out in tremendous detail. However, probability arises in many fascinating and important ways in other areas of mathematics and science in which independence does not naturally occur, but where there is some other special structure. Consider, for example, a sensor network that needs a certain degree of connectivity in order to function properly. If sensors are dropped from an airplane over a region of wilderness, it is important to know how many are probably needed to gain enough connectivity for the network to function. A classical probabilistic approach might be to assume that each pair of sensors communicates with some prescribed probability, independently, but a more realistic approach is to incorporate spacial data and say that sensors communicate if they are close enough together. Understanding the likelihood that such a random network will function becomes a harder problem because of the lack of independence, but an ultimately more useful one. The goal of this project is to investigate situations like this, in which independence assumptions are gone but geometric or other considerations can be exploited instead.