Free Probability and Cohomology in von Neumann Algebra Theory.

数学中的一个基本主题是，在许多情况下，复杂的有限维结构的某些方面随着这种结构的维度变得无限而变得简单。这种非常矛盾的行为构成了物理学中热力学理论的基础，在热力学理论中，当粒子数量变大时，复杂的粒子系统允许更简单的全局描述(只涉及几个变量，如压力和温度)。沃库列斯库在20世纪80年代创立的自由概率数学理论表明，冯·诺伊曼代数可以被视为随机矩阵渐近行为的热力学变量系统。该理论的一个显著特点是，变量不再互换：它们相乘的顺序很重要。这导致了一个极其丰富的理论，它导致了经典对象的自由概率推广，如偏微分方程、布朗运动等。本项目涉及通过构造不变量更好地理解随机矩阵的渐近行为以及由此产生的渐近极限的问题，不变量本质上是代数的和解析的。该项目的目的是继续研究与代数的子因子和更一般的包含有关的上同调不变量，特别是L^2不变量及其类似物。对其中一些不变量的研究涉及到许多由自由概率理论提供的分析工具，例如布朗运动和相关的类拉普拉斯算子的自由概率模拟。对这些算子的研究-出现在随机矩阵的渐近极限-也是随机矩阵理论的重要兴趣。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A fundamental theme in mathematics is that, in many cases, certain aspects of complicated finite-dimensional structures become simpler as the dimension of such structures goes to infinity. This very paradoxical behavior underlies the theory of thermodynamics in physics, in which complicated systems of particles admit a simpler global description (involving just a few variables, such as pressure and temperature) when the number of particles becomes large. A mathematical theory of free probability, initiated by Voiculescu in the 1980s, has shown that von Neumann algebras can be viewed as systems of 'thermodynamic variables' for asymptotic behavior of random matrices. A distinctive feature of the theory is that the variables no longer commute: the order of their multiplication matters. This results in an extremely rich theory that leads to free probability generalizations of classical objects such as partial differential equations, Brownian motion, and so on. The present project concerns questions of better understanding of asymptotic behavior of random matrices and of the resulting asymptotic limit through construction of invariants, which are both algebraic and analytic in nature. The project's aim is to continue the study of cohomological invariants associated to subfactors and more general inclusions of algebras, in particular L^2 invariants and their analogs. The study of some of these invariants turns out to involve many analytical tools supplied by free probability theory, such as free probability analogs of Brownian motion and of the associated Laplace-like operators. The study of these operators - arising in the asymptotic limit of random matrices - is of significant interest in random matrix theory as well.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.