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带到另一个领域，以开辟新的研究途径，推动知识的边界，并提供新的观点。许多被研究的问题都是出于在非正则空间(如分形图或粗糙度量空间)中发展理论的需要。其中一些问题的动机是物理、工程和数学金融。首席研究人员还将继续他的协同活动，包括写博客，组织会议，关于最近进展的讲座和迷你课程，以及指导研究生。研究活动的第一个也是最重要的部分围绕着对空间中曲率界限的理解，这些概念到目前为止都是难以捉摸的。在次黎曼流形类中，主要研究者和他的合作者感兴趣的是发展关于常曲率模型空间的比较几何。这包括次拉普拉斯比较定理、度量压缩性质和直径估计的研究。在Dirichlet空间类中，主要研究者和他的合作者对等周不等式和有限周长有界变差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，它允许定义关于任何Holder正则路径的积分。首席研究员和他的合作者对由一般高斯粗略路径驱动的微分方程解的几个性质感兴趣。该奖项反映了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。

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.