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Degrees & Accolades:

Ph.D. (Harvard University)
M.Sc. (McGill University)
B.Eng. (McGill University)

Research Profile:

I am interested in the broad area of sub-Riemannian geometry, especially in the interplay between the analytic aspects (hypoelliptic heat kernels and their short-time asymptotics) and the geometric ones (Carnot-Caratheodory distance, cut/conjugate loci, etc.). Their study often involves the use of stochastic analysis and Malliavin calculus techniques. Within geometric control theory, I am especially interested in problems that are of a topological nature, such as uncovering topological obstructions to the realization of a particular control objective. I am also interested in anything mathematical related to image processing, such as the study of partial differential equations in imaging, as well as inverse imaging problems.

I am continuously seeking highly motivated and well-prepared M.Sc. and Ph.D. students with an interest in pure and applied mathematics, especially in the general areas of sub-Riemannian geometry, stochastic analysis, and nonlinear and geometric control theory.

Research Areas:

Problem 1: A fundamental problem in sub-Riemannian geometry is that of the regularity of length-minimizing curves, the study of which is made challenging by the possible existence of “abnormal” extremals. It has been shown recently that sub-Riemannian length-minimizing curves cannot have corner-type singularities. The aim of this research is to identify additional types of singularities that sub-Riemannian length-minimizing curves cannot exhibit.

Problem 2: It is known that a suitably defined random walk on a sub-Riemannian manifold converges in law to a “horizontal” Brownian motion, directly tied to the sub-Laplacian of the underlying geometry. The aim of this research is to identify the geometric information that can be recovered from the properties of such processes, in particular, the structure of the cut and conjugate loci.

Problem 3: The KLS conjecture states that for any logconcave distribution in Rd , the Cheeger isoperimetric coefficient is equal to that achieved by half-spaces up to a universal constant factor. Recent approaches based on stochastic analysis tools, and, in particular, Eldan’s stochastic localization, have led to lower bounds on the Cheeger isoperimetric coefficient approaching the conjectured lower bound. The aim of this research is to investigate possible extensions to the stochastic localization method that would help improve those bounds.