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.
The Division of Applied Mathematics is an established leader in research on mathematical modeling for physical and physiological, multi-scale fluid mechanics. In general, fluid mechanics is an enabling science that describes dynamics over a wide spectrum of scales, ranging from the global scales of climate dynamics to the transport of suspended proteins through nano-pores. The focus of research in the Division has evolved as new challenges have emerged. This goes far beyond what has been seen as traditional fluid dynamics in the past and requires a broad scientific knowledge of biological and physio-chemical processes. The main activity of the fluids group is the theoretical description and numerical simulation of complex fluids, self-organization in active suspensions and biological processes relating to blood flow in the arterial tree, brain aneurysms, diseases of blood cells and bacterial locomotion. There exist ongoing interests in multi-scale phenomena in turbulence, flow-structure interactions and multiphase flows. The fluids group maintains strong connections with other faculty in Engineering and Physics and coordinates research seminars and graduate teaching. Collaborations with groups in biological and biomedical research are rapidly expanding.
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.
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.
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.
Vision research at Brown includes over 30 faculty from 10 different departments. What sets the Brown vision community apart is the unusually strong interaction between departments, and especially between faculty members in more quantitative disciplines (e.g. applied math, computer science, engineering, physics) and faculty in more biological or behavior-oriented disciplines (e.g. cognitive science, neuroscience, psychology). Our goal is to nurture multidisciplinary and translational research. Examples include theoretical studies of vision and visual plasticity in concert with experimental tests; biologically-inspired vision models implemented in artificial systems; and models of visual-cortical processing to address "high-level" visual deficits in developmental disorders such as Autism Spectrum Disorder.