Geometric and Functional Inequalities in Sub-Riemannian and Non-Smooth Dirichlet Spaces and Analysis of Random Rough Paths

该研究项目涉及数学的几个领域：分析、几何和概率论。首席研究员对这些领域之间的相互作用特别感兴趣，并将工具从一个领域引入另一个领域，以开辟新的研究路径，推动知识的边界，并提供新的观点。许多研究问题的动机是需要在非规则空间（如分形或粗糙度量空间）中发展理论。其中一些问题是由物理学、工程学和数学金融学引起的。首席研究员还将继续他的协同活动，包括写博客、组织会议、就最新进展举办讲座和迷你课程，以及指导研究生。研究活动的第一个也是最重要的部分是围绕对空间曲率边界的理解展开的，迄今为止，这些概念都是难以捉摸的。在次黎曼流形的类别中，主要研究者和他的合作者对开发一个关于常曲率模型空间的比较几何感兴趣。这包括研究次拉普拉斯比较定理，测量收缩性质和直径估计。在Dirichlet空间中，主要研究者和他的合作者对等周不等式和有限周长和有界变分函数集的相关研究感兴趣。研究项目的第二部分涉及随机粗糙路径理论中的问题。粗糙路径理论是一种新的理论，它允许定义关于任意Holder规则路径的积分。首席研究员和他的合作者对一般高斯粗糙路径驱动的微分方程解的几个性质感兴趣。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。

The research project lies at the intersection of several areas of mathematics: analysis, geometry and probability theory. The principal investigator is particularly interested in the interplay between those fields and bringing tools from one field to another in order to open new paths of research, push the boundary of knowledge, and offer new point of views. Many of the investigated problems are motivated by the need to develop theories in spaces which are not regular (like fractals or rough metric spaces). Some of those problems are motivated by physics, engineering and mathematical finance. The principal investigator will also continue his synergistic activities including blogging, organizing conferences, giving lectures and mini-courses on recent advances, and mentoring graduate students.A first and most important part of the research activity evolves around the understanding of curvature bounds in spaces for which such notions have so far been elusive. In the class of sub-Riemannian manifolds, the principal investigator and his collaborators are interested in developing a comparison geometry with respect to constant curvature model spaces. This includes the study of sub-Laplacian comparison theorems, measure contraction properties and diameter estimates. In the class of Dirichlet spaces, the principal investigator and his collaborators are interested in isoperimetric inequalities and the related study of sets of finite perimeter and bounded variation functions. A second part of the research project deals with problems in the theory of random rough paths. Rough paths theory is a recent theory that allows to define integrals with respect to any Holder regular path. The principal investigator and his collaborators are interested in several properties of the solutions of differential equations driven by general Gaussian rough paths.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.