Graduate Program

RESEARCH AREAS

The Division of Applied Mathematics is devoted to training and research in a broad spectrum of applied mathematics. It explores the connections between mathematics and its applications at both the research and the educational levels. The principal areas of research activities are ordinary, functional, and partial differential equations; stochastic control theory; applied probability, statistics and stochastic systems theory; neuroscience and computational biology; numerical analysis and scientific computation; and the mechanics of solids and fluids.

The effort in virtually all the research areas ranges from applied and algorithmic problems to the study of fundamental mathematical questions. This breadth is one of the great strengths of the program.

Scholarship and research in the Division of Applied Mathematics are augmented by many points of contact with the Department of Mathematics and the Division of Engineering. There are joint appointments, joint courses, and joint research projects. There are also joint research projects with faculty in the Departments of Computer Science, Economics, Geological Sciences and Neurosciences and with faculty in the Medical School. Some of the areas in which there are cooperative research projects with other departments include partial differential equations, fluid and solid mechanics, robotics and computer vision, scientific and parallel computing, stochastic systems theory and medical statistics.

Research in the areas of differential equations and dynamical systems focuses on the qualitative properties of solutions of the nonlinear differential equations that arise in the physical sciences, biological sciences and economics. A great variety of differential equations is being studied: ordinary, functional (with delays), and partial, including fluid equations, hyperbolic conservation laws, kinetic equations, reaction-diffusion systems and relativistic wave equations. Even though the techniques can vary widely from case to case, a unifying philosophy in the approach has been generated by the great Brown tradition in this area of mathematics and is being fostered by close collaboration among the members of the group.

Research in stochastic control and optimization includes virtually all of the problem areas of current interest: optimization methods, stability and the qualitative theory of stochastic dynamical systems, small-noise problems, singular perturbations, approximation methods, stochastic networks and data processing systems, methods of large deviations, applications to queuing networks, applications to financial and economic models, nonlinear filtering, recursive stochastic algorithms and their applications in communication and adaptive control theory, asymptotic methods, singular control problems, problems with wide-band noise, control under partial information and heavy traffic approximations. There is a major program in numerical methods for all of the problem types.

Research in applied probability and statistics emphasizes image processing, computer and biological vision, and related complex problems in statistical inference. Algorithms are developed, based on mathematical models, for restoration, enhancement, reconstruction, and high-level analysis of digital images. The mathematical models, predominantly probabilistic and pattern-theoretic in nature, enable the use of classical statistical principles as a foundation. Considerable attention is given to the applications of the methods to real data, in areas such as medical imaging, industrial automation, and design of intelligent systems for object detection and recognition. Closely related work is concerned with other complex signal processing and inference problems including, for example, speech recognition. Other statistically oriented work concerns the analysis of data from problems in medicine and health care.

Research focuses on statistical inference in high dimensions with applications primarily in molecular biology but also in the geosciences. The development of novel inference principles, methods, and algorithms appropriate to the special characteristics of high-dimensional discrete spaces are the main activities. Challenging high dimensional statistical inference problems emerging from the sequencing of genomes and from numerous high throughput data acquisition technologies of the post genome era provide the focus of the largest component of our research. In addition, sequences of data emerging form paleoclimatology studies of the earth’s history are another important application area. Bayesian statistical approaches dominated our work, and we generally employ either direct sampling strategies using recursive equations to obtain key normalizing constants, or Markov chain Monte Carlo (MCMC) algorithms. All of our work involves real applications in collaboration with biologic and geologic investigators at Brown and beyond, and all of our theoretical studies have been motivated by these applications.

Research in fluid mechanics focuses on problems in complex fluids, bioflows, microflows, and oceanography. Specific applications include multiscale modeling of the human arterial tree, aneurisms, and blood rheology. Some further topics of interest are the dynamic self-assembly of micro and nanoscale particles in suspension, active suspensions of micro-swimmers, and the flow properties of such systems. Other aspects include uncertainty quantification in computational mechanics, noisy flow systems, and low-dimensional modeling. Both continuum as well as atomistic simulation methods are employed with high-performance computing required in most applications.

Scientific computing, as a science and as a method of research, is inherently multidisciplinary. It has undergone phenomenal growth in response to the successes of modern computational methods in increasing 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 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.

A large number of regular seminars cover the principal areas of scientific interest in the Division. Some involve speakers from all over the world, and others are used to augment the formal courses by providing expositions of special topics in a short series of lectures. The research seminars are an integral part of the graduate program.

The Division typically has a large number of postdoctoral and faculty visitors who actively contribute to research programs and graduate education. It is not uncommon for the visiting researchers to outnumber the regular faculty.

The Division of Applied Mathematics also cooperates actively with Computation and Mathematics of Mind (CMM), The Center for Computational Molecular Biology (CCMB), Institute for Brain and Neural Systems, the Center for Biophysical and Biomedical Engineering, the Center for Gerontology and Health Care Research and the Center for Statistical Science, the latter two being operated by the Brown University School of Medicine. These affiliations reflect growing interest in the Division with the applications of mathematics to the nonphysical sciences.