Research Interests
Deep Learning theory, Optimization. Random matrices, Integrable Systems, Analysis of non-linear (possibly integrable) PDEs. Riemann-Hilbert problems.
My Math research is focused on the use of Riemann-Hilbert techniques to study physically meaningful quantities arising in the field of Random Matrix Theory or Integrable PDEs, in particular their asymptotic behaviour in certain limit or critical regime and their connection with integrable systems (Lax pair and Painlevé equations).
I currently have a keen interest in applications of Random Matrix Theory and Integrable Systems to Generalization analysis and Optimization (Machine Learning).
Research seminar notes
Here are some notes from past seminars and lectures given at CSU and Tulane University:
Determinantal Point Processes, notes from the lecture given at the Special Functions course (MATH 7720) at Tulane University in October 2019.
Random Matrices, notes from the lecture given at the Special Functions course (MATH 7720) at Tulane University in October 2019.
Orthogonal Polynomials and Riemann Hilbert Problem, notes from the lecture given at the Special Functions course (MATH 7720) at Tulane University in October 2019.
The Vortex Filament Equation and the Nonlinear Schrödinger equation, notes from the lecture given at the PDELab seminar at Colorado State University in November 2018.
Variational formulation of a PDE, notes from the lecture given at the PDELab seminar at Colorado State University in October 2017.
Conference and tutorial videos
You might have seen me here:
Smallest singular value distribution and large gap asymptotics for products of random matrices, workshop "Six vertex models, dimers, shapes and all that", Simons Center for Geometry and Physics, Stony Brook University (NY), 2016.
"Integrable" gap probabilities for the Generalized Bessel process, conference "Painlevé Equations and Discrete Dynamics", Banff International Research Station (BIRS), Banff (AB), 2016.
A note on the condition numbers of first order optimization, summer workshop "Statistical Physics and Machine Learning", École de Physique des Houches, Les Houches (France), 2020.
Random Matrix Theory for uninitiated (and ML applications), Montréal Machine Learning and Optimization (MTL MLOpt), 2020.
A study on condition numbers for first-order optimization, East Coast Optimization Meeting, George Mason University, Fairfax (VA), 2021.
Rigorous Asymptotics of a KdV Soliton Gas, New horizons in dispersive hydrodynamics, Isaac Newton Institute, Cambridge (UK), 2021.
Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes, Connections And Introductory Workshop: Universality And Integrability In Random Matrix Theory And Interacting Particle Systems, MSRI, Berkeley (CA), 2021.
