Research and Development

I am interested in projects  related to Elliptic Partial Differential Equations, Geometric Analysis, and Applied Mathematics with applications to Machine Learning Theory, Control Systems, Diffusion Models, and Biological Signal Processing. In particular I enjoy understanding mathematical questions related to existence, uniqueness, regularity estimates, asymptotic estimates, compactness, and perturbation theory.  Applications to Biological Signal processing focus on designing and engineering data systems used in procedure for patients with Cardiac Arrhythmias.

Patents

System and method for electro-anatomical mapping (EAM) based on a surface mesh

System and method for generating and electro-anatomical map with a user control mechanism to dynamically adjust a color scale associated said map

Research Papers

Danielli, D., Jain, R., 2020. Regularity results for a penalized boundary obstacle problem. arXiv:2003.00995

Jain, R., 2017. Uniform Estimates for the Penalized Boundary Obstacle Problem. arXiv:1510.06324.

Jain, R., 2017. The Fully Nonlinear Stochastic Impulse Control Problem. arXiv:1510.06394

Jain, R., 2016. The Classical Stochastic Impulse Control Problem. arXiv:1510.06317.

Carlson, W., Jain R., Train A., 2015. Order of Magnitude of Fourier Coefficients of Almost Periodic Functions. arXiv:1510.06313 

Survey Papers

Jain, R., Jo, J.,  2017. The Calabi Conjecture. arXiv:1703.06945.

Jain, R., 2017. Supersymmetric Schrodinger Operators with Applications to Morse Theory. arXiv:1703.06943.

Selected Presentations

CMS Summer Sectional Meeting, June 2nd, 2018 University of New Brunswick at Fredericton:

Presentation Slides

AMS Spring Sectional Meeting, April 15th, 2018 Portland State University:

Presentation Slides

PDE Seminar, November 28th, 2016 Bringham Young University:

Presentation Slides

AMS Fall Sectional Meeting, October 15th, 2016 St. Paul University:

Presentation Slides

International Workshop on Elliptic and Kinetic Equations, July 10th, 2015 IMPA:

Presentation Video

Introductory School on Free Boundary Problems and Related Topics, January 2015 Isaac Newton Institute for Mathematical Sciences:

Poster Presentation 

PhD Thesis

UT Austin 2016 

Outreach

Organizer for Math Research Conferences: 

McGill University Geometric  Analysis Workshop

PhD Candidacy Exam Committee Member: 

Viscosity Solutions, Convergent Schemes for PDEs and PDE Characterization of Convexity

Referee for Mathematical Journal Articles: 

AMS Special Session on Nonlocal Operators

Discrete Dynamical Systems A

Organizer for Graduate Math Research Seminars:

Graduate PDE Seminar

Junior Analysis Seminar

Undergraduate Research

Economic Development Seminar Paper: Analyzing Poverty alleviation in the Rural Sector