Free Probability, Transport, and Applications

免费概率、传输和应用

• 批准号: 2054450
• 负责人: Dimitri Shlyakhtenko
• 金额: $ 36万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054450&HistoricalAwards=false
• 关键词: Free; Probability; Transport; Applications

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.

最佳运输是对将大量物体从一个地方移动到另一个地方的方式的研究，同时最小化了成本。例如，在1781年，法国数学家蒙格（Monge）考虑了将灰尘钢琴移动以最佳方式填充孔的问题。此后，该领域已发展为蓬勃发展的数学领域，并与概率理论，随机分析，微分方程和许多其他领域有联系。当前的项目旨在在Voiculescu的自由概率理论的背景下开发最佳的运输技术。该理论的一个独特特征是变量不再命令：其乘法的顺序很重要。这导致了非常丰富的理论，它导致了经典对象（例如部分微分方程，布朗运动等）的自由概率概括。该奖项支持该提案在该领域的研究，并通过培训研究生为美国的劳动力发展做出了贡献。该项目介绍了PDE方法的紧急使用，以在操作员代数和自由概率理论中产生结果。这些方法使我们能够构建免费的传输图并证明对操作员代数的同构和分解结果；他们对Voiculescu的自由熵和游离随机演算的理论提供了新的见解。并加强随机矩阵理论与自由概率理论之间的联系。我们提出了各种问题的混合，其中一些来自现有的研究方向以及一些探索新的询问线。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为是通过评估而被视为珍贵的支持。

项目成果

Duality for Optimal Couplings in Free Probability

• DOI: 10.1007/s00220-022-04480-0
• 发表时间: 2021-05
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: W. Gangbo;David Jekel;Kyeongsik Nam;D. Shlyakhtenko

Fractional free convolution powers

分数自由卷积幂

• DOI: 10.1512/iumj.2022.71.9163
• 发表时间: 2022
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: Shlyakhtenko, Dimitri;Tao, Terence
• 通讯作者: Tao, Terence

An inequality for non-microstates free entropy dimension for crossed products by finite abelian groups

有限阿贝尔群交叉积的非微观状态自由熵维不等式

• DOI: 10.4171/lem/1056
• 发表时间: 2023
• 期刊: L’Enseignement Mathématique
• 作者: Shlyakhtenko, Dimitri

Tracial smooth functions of non-commuting variables and the free Wasserstein manifold

非交换变量的轨迹平滑函数和自由 Wasserstein 流形

• DOI: 10.4064/dm843-10-2021
• 发表时间: 2022
• 期刊: Dissertationes Mathematicae
• 影响因子: 1.8
• 作者: Jekel, David;Li, Wuchen;Shlyakhtenko, Dimitri
• 通讯作者: Shlyakhtenko, Dimitri

Von Neumann Algebras of Sofic Groups with β x(2) x1 = 0 Are Strongly 1-Bounded

β x(2) x1 = 0 的 Sofic 群的冯诺依曼代数是强 1 有界的

• 发表时间: 2021
• 期刊: Journal of operator theory
• 影响因子: 0.8
• 作者: D. Shlyakhtenko