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.
PapersFabrice 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)
- UVA Probability Seminar. University of Virginia. Couplings for hypoelliptic diffusions and applications to gradient estimates. Invited talk. Fall 2017
- Lehigh Probability & Statistics Seminar. Lehigh University. Coupling on the Heisenberg group and its applications to gradient estimates. Invited talk. Spring 2017
- UConn Sigma Seminar. Solving differential equations with probability. Spring 2017
- Seminar in Stochastic Processes. University of Virginia. Functional inequalities of hypoelliptic operators using coupling. Poster. Spring 2017
- Fifteenth Northeast Probability Seminar. Coupling on the Heisenberg group and its applications to gradient estimates. Fall 2016
- UConn Analysis Learning Seminar. Probabilistic Techniques in Analysis. Fall 2016
- UConn General Exam. Gradient Estimates on Manifolds Using Coupling for Diffusion Processes. Spring 2016
- UConn Sigma Seminar. Coarse Geometry. Fall 2014
- UConn Math Club Talk. The volume of the unit ball in n dimensions. Spring 2014
- JMM – AMS Special Session. On the Coarse Geometry of Lp: A Course Equivalence. Winter 2013