My research interests lie in developing mathematical analysis to study a broad range of applied problems. I use techniques such as the calculus of variations, PDE, non-linear, linear and non-smooth analysis, and probability theory to study challenging applied problems. Application areas that I have studied include:
- Phase transitions in material science (e.g. Cahn-Hilliard and coagulation-fragmentation equations)
- Decision making with limited information (e.g. reinforcement learning, decentralized optimization/control)
- Asymptotic consistency for statistical estimation (e.g. regularized risk minimization, extreme value processes)
- Fluid dynamics (e.g. non-uniqueness in Euler’s equation, Reynolds-averaged Navier-Stokes equations)
Publications on these topics includes:
- Second Order Γ-Limit for the Cahn-Hilliard Functional. G. Leoni, R. Murray. ARMA. 2016. Published, Preprint
- Algebraic Decay to Equilibrium for the Becker-Doring Equations. R. Murray, B. Pego. SIMA. 2016. Published, Preprint.
- Slow Motion for the Nonlocal Allen Cahn Equation in n-dimensions. R. Murray, M. Rinaldi. Calc Var PDE 2016. Published, Preprint
- Cutoff Estimates for the Linearized Becker-Doring Equations. R. Murray, B. Pego. Comm. Math. Sci. 2017. Published, Preprint.
- Local minimizers and slow motion for the mass preserving Allen–Cahn equation in higher dimensions. G. Leoni, R. Murray. To Appear in Proceedings of the AMS. Preprint
- On Best-Response Dynamics in Potential Games. B. Swenson, R. Murray, S. Kar. SIAM Control, 2018. Published. Preprint.
- Regular Potential Games. B. Swenson, R. Murray, S. Kar. Submitted January 2018. Preprint,
- A new analytical approach to consistency and overfitting in regularized empirical risk minimization. N. Garcia Trillos, R. Murray. EJAM, 2017. Published Preprint
- Revisiting Normalized Gradient Descent: Evasion of Saddle Points. R. Murray, B. Swenson, S. Kar. Transactions in automatic control, 2018. Preprint.
- A model for system uncertainty in reinforcement learning. R. Murray, M. Palladino. Systems and Control Letters 2019. Preprint.
- Wright-Fisher dynamics with periodic fitness fluctuations. R. Murray, G. Young. Journal of theoretical biology 2019.
- Modelling uncertainty in reinforcement learning. R. Murray, M. Palladino, IEEE CDC 2019.
- A maximum principle argument for the uniform convergence of graph Laplacian regressors. N. Garcia Trillos, R. Murray. Submitted. Preprint 2019.
- From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds. N. Garcia Trillos, R. Murray, M. Thorpe. Preprint 2020.
- On Self-similar Solutions to the Incompressible Euler Equations. A. Bressan, R. Murray. JDE, 2020.
- Distributed Stochastic Gradient Descent and Convergence to Local Minima. B. Swenson, R. Murray, S. Kar, V. Poor. Preprint 2020.
My teaching at NCSU
- Math 715 (Measure theory) Spring 2020
- Math 305 (Linear algebra) Fall 2019
My teaching at PSU
- Math 484 (Linear programming) Spring 2019
- Math 405 (Advanced engineering math) Fall 2018
- Math 230 (Calc in 3D) Fall 2017
- Math 141 (Integral calculus) Spring 2017
- Math 232 (Integral vector calculus) Fall 2016
Teaching outside of university
- Two summers I taught an extended summer school on data science, based on parts of BYU’s applied math curriculum.
In the past I’ve done mathematical consulting for Vert Nova, Last Mile Skills, and Electronic Theatre Controls