Discrete Convexity, Curvature, and Implications

一个多世纪以来，连续域的凸性一直被认为是一个强大的性质和重要的工具。由于最近在最优输运、偏微分方程、几何和泛函分析等主题上的几个独立的发展，凸性和曲率的相互作用进一步丰富了对概率和分析中几个主题的理解。虽然在离散环境中，凸性通过重要的子模块化概念得到了发展，但曲率仍然是难以捉摸的。在过去的五年里，研究者和不同的合作者在马尔可夫链和图中引入了所谓的里奇曲率的离散概念，并得出了几个有趣的结果。然而，在离散空间的凸性和曲率方面，仍有许多有待理解和发展的地方。首先在数学和计算机科学中确定合适的应用将是发展任何新理论的重要指导原则。在离散概率中，Ahlswede-Daykin四函数定理有许多应用（通过FKG不等式和其他结果），类似地，Prekopa-Leindler不等式有助于加深对几何和函数不等式的理解，同时将（等效的）Brunn-Minkowski不等式扩展到概率、分析和偏微分方程的分支。该项目的一个主要技术目标是利用基于传输的最优耦合，将这两个开创性的结果联系起来，并推导出每个不等式的精炼版本。其他方面包括在体积增长和/或离散曲率的额外假设下，推导出拉普拉斯函数的第k个特征值与高阶（多部）Cheeger等周常数相关的高阶Buser不等式。本项目还涉及若干具体问题，供博士后和学生共同研究。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。

Convexity in continuous domains has been known to be a powerful property and an important tool for more than a century. Thanks to several independent recent developments in the topics of optimal transport, partial differential equations, geometric and functional analysis, the interplay of convexity and curvature has further enriched the understanding of several topics in probability and analysis. While convexity has been developed through the important notion of submodularity in discrete settings, curvature remains elusive. In the past five years, the investigator and various collaborators have introduced discrete notions of the so-called Ricci curvature in Markov chains and graphs and derived several interesting consequences. However, much remains to be understood and developed in terms of convexity and curvature in discrete spaces. Identifying suitable applications in mathematics and computer science first will be an important guiding principle in developing any new theory. In discrete probability, the Ahlswede-Daykin Four Functions Theorem has had numerous applications (by way of the FKG inequality and other consequences) and similarly the Prekopa-Leindler inequality has been instrumental in deepening the understanding of the geometric and functional inequalities, while expanding the applications of the (equivalent) Brunn-Minkowski inequality to branches of probability, analysis and PDEs. A main technical objective of this project is to relate these two seminal results, using optimal transport-based couplings, and derive refined versions of each inequality. Other aspects include deriving higher-order Buser inequalities relating the k-th eigenvalue of the Laplacian to the higher-order (multipartite) Cheeger isoperimetric constant, under additional assumptions on volume growth and/or discrete curvature. The project also involves several specific problems for investigation with postdoctoral fellows and students.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.