Free Probability, Transport, and Applications

最优运输是研究将大量物品从一个地方移动到另一个地方，同时使成本最小化的方法。例如，1781年，法国数学家蒙日考虑了以最优方式移动成堆的泥土来填洞的问题。这个领域已经发展成为一个蓬勃发展的数学领域，与概率论、随机分析、微分方程和许多其他领域都有联系。目前的项目旨在在Voiculescu的自由概率理论的背景下开发最佳的运输技术。该理论的一个显著特征是变量不再交换：它们乘法的顺序很重要。这就产生了一个极其丰富的理论，可以对经典对象进行自由的概率推广，比如偏微分方程、布朗运动等等。该奖项支持申请人在该领域的研究，并通过培养研究生为美国劳动力发展做出贡献。这个项目处理了PDE方法在算子代数和自由概率论中产生结果的紧急使用。这些方法允许我们构造自由传输映射，并证明算子代数的同构和分解结果；他们对沃库列斯库的自由熵理论和自由随机微积分有了新的认识；加强随机矩阵理论与自由概率论的联系。我们提出了一些混合问题，一些来自现有的研究方向，一些探索新的研究方向。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Optimal transport is a study of ways of moving large numbers of objects from one place to another while minimizing cost. For example, in 1781 French mathematician Monge considered the problem of moving piles of dirt to fill holes in an optimal way. This area has since developed into a thriving mathematical field with connections to probability theory, stochastic analysis, differential equations, and many other fields. The current project aims to develop optimal transportation techniques in the context of Voiculescu’s free probability theory. 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. This award supports the proposer's research in this area and contributes to US workforce development through the training of graduate students.This project deals with the emergent use of PDE methods to produce results in operator algebras and free probability theory. These methods allow us to construct free transport maps and prove isomorphism and decomposition results for operator algebras; they give new insights into Voiculescu's theory of free entropy and free stochastic calculus; and strengthen the connection between random matrix theory and free probability theory. We propose a mixture of problems, some coming from existing research directions and some exploring new lines of inquiry.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.