Free Probability, Transport, and Applications

最优运输是研究如何在将大量物体从一个地方移动到另一个地方的同时将成本降到最低。例如，1781年，法国数学家蒙格考虑了以最佳方式移动成堆的泥土来填充洞穴的问题。这个领域后来发展成为一个繁荣的数学领域，与概率论、随机分析、微分方程式和许多其他领域有联系。目前的项目旨在根据沃库列斯库的自由概率理论开发最优运输技术。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。该奖项支持提倡者在这一领域的研究，并通过研究生的培训为美国劳动力的发展做出贡献。该项目涉及PDE方法的紧急使用，以产生算子代数和自由概率论的结果。这些方法使我们能够构造自由迁移映射，证明算子代数的同构和分解结果；它们给出了对Voulescu的自由熵理论和自由随机微积分的新见解；加强了随机矩阵理论和自由概率理论之间的联系。我们提出了一个混合的问题，一些来自现有的研究方向，一些来自探索新的调查路线。这个奖项反映了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。

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.