Research Interests

My current mathematical interests are in the intersection of geometry, stochastic analysis and PDEs. Recently I've beeen working on using coupling of diffusion processes to prove functional inequalities. Specifically, I've worked on an efficient non-Markovian successful coupling for Brownian motion on the Heisenberg group. We've used this coupling to yield local gradient estimates for Harmonic functions on the Heisenberg group. My research also consists of studying Kolmogorov Diffusions (Brownian motion with integrated brownian motion). We use a synchronous coupling technique to prove gradient bounds for the heat semigroup of a class of Kolmogorov diffusions where the Brownian motion lives on a Riemannian manifold.

I have also been involved in supervising an REU research project in the summer of 2017. We studied the Law of Large Numbers (LLN) and and Central Limit Theorems (CLT) for products of random matrices. The limit of the multiplicative LLN is called the Lyapunov exponent. We perturbed the random matrices with a parameter and looked to find the dependence of the the Lyapunov exponent on this parameter. We also studied the variance related to the multiplicative CLT. We proved and conjecture asymptotics of various parameter dependent plots.

My undergraduate research was in Coarse Geometry. Coarse Geometry is the study of metric spaces from a large scale perspective rather than the small scale perspective as done in topology. Coarse Geometry was found useful in obtaining partial results to the Novikov Conjecture and the Baum–Connes Conjecture.

Click here for my Research Statement.

Papers

Fabrice Baudoin, Maria Gordina, Phanuel Mariano, Gradient bounds for type Kolmogorov diffusions.(2017) (in preparation)

Phanuel Mariano and Hugo Panzo. Explicit variance in the CLT for products of random matrices related to random Hill's equations. (2017) (in preparation)

R. Majumdar, P. Mariano, H. Panzo, L. Peng and A. Sisti. Multiplicative LLN and CLT and their Applications (2017) (in preparation)

Sayan Banerjee, Maria Gordina, Phanuel Mariano, Coupling in the Heisenberg group and its applications to gradient estimates. Annals of Probability, to appear (2016) arXiv

Phanuel Mariano, On the Coarse Geometry of Lp, Rose-Hulman Undergraduate Math Journal Vol. 14 , No. 2 (2013)

Talks