## Research

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.

## Teaching

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.

## Consulting

In the past I’ve done mathematical consulting for Vert Nova, Last Mile Skills, and Electronic Theatre Controls