# Research Areas

### Climate

Mathematical climate research seeks to develop models of the climate system, including interactions between the atmosphere, ocean, and ecosystems, and methods for interpretation of complex environmental data. This field is inherently interdisciplinary and leverages mathematical research in disciplines such as scientific computing, dynamical systems, and probability in collaboration with environmental scientists. Problems range from fundamental questions about the physics of the climate system to applied questions such as the climate impacts in a given region or the deployment of a particular climate solution.

- Mara Freilich (starting in July 2023): network analysis, stochastic models, fluid dynamics, oceanography, ecology

### Dynamical Systems and Partial Differential Equations

Research in this area focuses on nonlinear differential equations and dynamical systems that arise in the physical, social, and life sciences. Among the equations considered are finite-dimensional dynamical systems, reaction-diffusion systems, hyperbolic conservation laws, max-plus operators and differential delay equations. Questions that are addressed for these systems include the existence and stability of nonlinear waves and patterns, kinetic theory, phase transitions, domain coarsening, and statistical theories of turbulence, to name but a few. Even though the techniques can vary widely from case to case, a unifying philosophy is the combination of applications and theory that is in the great Brown tradition in this area of mathematics, which is being fostered by close collaboration among the members of the group.

- Hongjie Dong: Linear and nonlinear elliptic and parabolic PDEs, fluid equations
- Yan Guo: Partial differential equations, kinetic theory and fluids
- John Mallet-Paret: Dynamical systems; differential–delay, lattice, and reaction–diffusion equations
- Govind Menon: Kinetics of phase transitions and models of domain coarsening, integrable systems, random matrix theory, statistical theories of turbulence
- Bjorn Sandstede: Applied dynamical systems, nonlinear waves and patterns, data science, computational biology

Dynamical Systems and Partial Differential Equations

### Pattern Theory, Statistics, and Computational Molecular Biology

Research in pattern theory seeks to develop models of complex systems and statistical methods and algorithms for the interpretation of high-dimensional data. Pattern theory research is also typically interdisciplinary; it includes collaborations with computer scientists, engineers, cognitive and neural scientists, and molecular biologists. Most of pattern theory research relies on tools from mathematical analysis, probability theory, applied and theoretical statistics, and stochastic processes. Recent applications include the development of models for computation and representation in primate visual pathways, as well as the development of statistical methods for Bayesian non-parametrics, network analysis, the interpretation of multi-electrode neurophysiological recordings, image processing and image analysis, and the analyses of genome-wide expression data and cellular regulatory pathways.

- Elie Bienenstock: theoretical neuroscience, computational vision, computational linguistics
- Stuart Geman: compositionality, statistical analysis of neurophysiological data, neural representation and neural modeling, statistical analysis of natural images, timing and rare events in the markets
- Basilis Gidas: Bayesian statistics, computer vision, speech recognition, computational molecular biology
- Matthew Harrison: statistics, machine learning, applications in neuroscience, neural engineering, ecology

### Probability and Stochastic Processes

The Division has long been a leader in stochastic systems theory and its applications, as well as at the forefront of current developments in probability theory, random processes and related computational methods. Research in probability theory and stochastic processes include stochastic partial differential equations, nonlinear filtering, measure-valued processes, deterministic and stochastic control theory, probabilistic approaches to partial differential equations, stability and the qualitative theory of stochastic dynamical systems, the theory of large deviations. Our research endeavors also include Monte Carlo simulation, Gibbs measures and phase transitions, as well as stochastic networks. There also exists a major program in numerical methods for a variety of stochastic dynamical systems, including Markov chain approximations and spectral methods.

- Hongjie Dong: stochastic processes, stochastic control theory, probabilistic approaches of PDEs
- Paul Dupuis: applied probability, control theory, large deviation, numerical methods, Monte Carlo
- Oanh Nguyen: probability theory, stochastic processes, stochastic networks, phase transition, random polynomials
- Kavita Ramanan: probability theory and stochastic processes, reflected diffusions, Gibbs measures and phase transitions, large deviations, measure-valued processes, stochastic networks
- Hui Wang: stochastic optimization, large deviations, stochastic networks, fast simulation

### Scientific Computation and Numerical Analysis

This research area is inherently multidisciplinary. It has undergone phenomenal growth in response to the successes of modern computational methods in increasing the understanding of fundamental problems in science and engineering. The Division’s program in scientific computation and numerical analysis has kept pace with these developments and relates to most of the other research activities in the Division. Special emphasis has been given to newly developed, high-order techniques for the solution of the linear and nonlinear partial differential equations that arise in control theory and fluid dynamics. Numerical methods for the discontinuous problems that arise in shock wave propagation and for stochastic PDEs and uncertainty modeling are being studied. Emphasis is also being placed on the solution of large-scale linear systems and on the use of parallel processors in linear and nonlinear problems.

- Mark Ainsworth: Finite element methods: adaptivity, a-posteriori error control, Implementation using Bernstein- Bézier techniques, Dissipative and dispersive properties.
- Jerome Darbon: Efficient algorithms for variational/Bayesian estimations and connections with Hamilton-Jacobi PDEs, Combinatorial optimization (especially network flows and graph-based algorithms), Stochastic sampling algorithms (especially perfect samplers), Algorithm/architecture co-design including low level implementation, Applications to denoising, geomorphology, remote sensing, biological, medical, historical, radar and inverse problems in imaging sciences
- Johnny Guzmán: Discontinuous Galerkin methods, Mixed methods/compatible discretizations, Local error analysis
- George Em Karniadakis: Stochastic PDEs and stochastic multi-scale modeling, Fractional PDEs, Spectral element methods, Parallel computing
- Brendan Keith: PDE-Constrained Optimization, Stochastic and Fractional PDEs, Scientific Machine Learning, Uncertainty Quantification, Finite Element Analysis
- Chi-Wang Shu: High order methods for hyperbolic and convection dominated PDEs, Computational fluid dynamics

### Research Groups

### CRUNCH Group

Research conducted by the CRUNCH Group focuses on the development of stochastic multiscale methods for physical and biological applications, specifically numerical algorithms, visualization methods and parallel software for continuum and atomistic simulations in biophysics, fluid and solid mechanics, biomedical modeling and related applications. The main approach to numerical discretization is based on spectral/hp element methods, on multi-element polynomial chaos, and on stochastic molecular dynamics (DPD). The group is directed by Prof. George Em Karniadakis. We invite you to visit both our DPD Club and Crunch FPDE Club websites.