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.
