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About Us

 

The Division of Applied Mathematics is one of the most prominent departments at Brown, and one of the oldest and strongest of its type in the country. The Division had its origin in a program of Advanced Instruction and Research in Mechanics, established in 1941 on the recommendation of a committee of the National Research Council. This early program focused on solid and fluid mechanics, electromagnetic theory, mathematical methods in applied physics, numerical analysis and probability theory—the principal interests of the faculty for many years. 

Since then the interests of the faculty have extended, as the Division has maintained a leading role in the development of applied mathematics. In 1964, for example, the Center for Dynamical Systems was established to coordinate the research of a large group of people working in ordinary and partial differential equations and their applications. More recently, strong programs of research in scientific computing and in applied probability and statistics have been established.

The Division is devoted to training and research in a broad spectrum of applied mathematics. It explores the connections between mathematics and its applications at both research and 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, materials science and fluids. The effort in virtually all research ranges from applied and algorithmic problems to the study of fundamental mathematical questions. This breadth is one of the great strengths of the program.

The Division emphasizes applied mathematics as a unifying theme. To facilitate cooperation among faculty and students, some research programs are partly organized around interdepartmental research centers. The centers facilitate funding and cooperative research and help to maintain at the highest level the research and educational atmosphere of the Division. 

Along with Engineering and Mathematics at Brown, the Division participates in the Lefschetz Center for Dynamical Systems. Research in the center is focused on modern approaches to dynamical systems, partial differential equations (particularly nonlinear wave propagation and conservation laws) and stochastic control. 

In addition, the Division is one of five participating units in the Center for Computational Molecular Biology. Plus the Division cooperates actively with the Center for Neural Sciences, the Center for Biophysical and Biomedical Engineering, the Center for Gerontology and Health Care Research and the Center for Statistical Science, the latter two part of Brown Medical School. 

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 visiting researchers to outnumber regular faculty.

The Division and its research groups maintain significant high-performance computing and data storage resources in close association with the Center for Computation and Visualization. Current high-performance computing facilities dedicated to the Division include three Linux clusters of 80, 64 and 36 CPUs, a parallel file service system with 50 terabytes of disk storage, and a tape archival/backup system with a 100-terabyte-tape library. Desktop research systems include Linux workstations, Sun Solaris workstations and SGI IRIX workstations. Other resources available in the Center for Computation and Visualization include an IBM R/S6000 SP system with 140 CPUs, an IBM Linux cluster with 100 CPUs, an AMD Linux cluster with 110 CPUs, and a Virtual Reality visualization laboratory with a 4-wall Cave display.

  

Broad Research Areas

  

Differential Equations and Dynamical Systems

Research in this area 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 are studied: ordinary, functional (with delays), and partial, including Schroedinger equations, hyperbolic conservation laws, Hamilton-Jacobi equations, reaction-diffusion systems, relativistic wave equations, kinetic equations and hierarchies. 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.

 

Stochastic Control and Optimization

 

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

 

Applied Probability and Statistics

Image processing, computer vision, and related complex problems are formulated as 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 for the methods. 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, in collaboration with the Department of Neuroscience, is concerned with modeling of the primate vision system, and the statistical analysis and interpretation of microelectrode recordings from a variety of animal preparations. Other statistically oriented work concerns the analysis of data from problems in medicine and health care.

 

Fluid Mechanics

Fluid mechanics encompasses a wide range of physical phenomena from nano-scales to ocean dynamics. Research is presently focused on complex fluids, whose microscopic properties differ from simple fluids such as air or water. This includes work on polymeric flows and two-phase flows and extends to bio-fluids. One such project is on the multiscale modeling of blood platelet phenomena, where the chemical activation of platelets and the formation of clots at the micro-scale are integrated through to flow of blood in the complete human arterial tree. Other projects cover the nano-scale properties of fluid-surface and fluid-particle interactions, and dynamic self-assembly of particles in suspension. There is a continuing focus on turbulent flows and methods for combining flow measurements with simulations.

 

Scientific Computation

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.

  

Computational Molecular Biology 

 

Research in this area emphasizes the development of modern computational and statistical methods for studying fundamental problems in molecular biology. The major focus of current research is on global gene regulation and regulatory networks including understanding and control of DNA transcription, regulatory events effecting messenger RNA, and RNA structure. Numerous genome scale data resources are employed in these analyses including microarray data, ChIp-chip data, and multiple genome sequences.  Our main theoretical interest concerns statistical inference in discrete high dimensional spaces with application to genomics.

 

Vision Research

 

The Center for Vision Research, part of Brown’s interdisciplinary Brain Science Program, promotes and facilitates research on biological vision, computational aspects of machine vision, visual disorders, and the brain mechanisms underlying vision. The CVR provides in-depth training in vision research to postdoctoral fellows, medical residents, graduate students, and undergraduates and serves as a unifying organization spanning traditional departments, as well helping to bridge the gap between basic research and clinical practice. Vision Research at Brown includes over 30 faculty from 10 departments. What sets the Brown vision community apart is the unusually strong interactions 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.