Overview
Division of Applied Mathematics Spans a Wide Range of Research Areas
The Division of Applied Mathematics at Brown University 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. Ongoing research in each of these 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.
Interdisciplinary Research and Study Opportunities are Available through Collaborative Projects and Courses
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, including joint appointments, courses, and research projects. There are also collaborative research projects and joint courses with faculty in the Departments of Computer Science, Biostatistics, Economics, Geological Sciences, and Neuroscience, as well as with faculty in the Medical School.
Some of the areas in which there are cooperative, interdisciplinary 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.
A Variety of Major Research Areas are Unified by a Common Philosophy and Approach to Research
Each research group in the Division of Applied Mathematics engages in close collaboration, in accordance with the Brown tradition. This tradition is responsible for a unifying philosophy and approach to mathematical research, though research techniques can vary from case to case.
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. Research in stochastic control and optimization includes virtually all of the problem areas of current interest. There is also a major program in numerical methods for all of the problem types related to stochastic control. Research in applied probability and statistics emphasizes image processing, computer and biological vision, and related complex problems in statistical inference.
Brown Emphasizes Applying Theoretical Studies to Real Data in Fields from Medical Imaging to Geosciences
Brown's graduate programs in applied mathematics emphasize the application of 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. For example, researchers collaborate with biologic and geologic investigators at Brown and beyond to examine statistical interference in high dimensions, with applications in molecular biology and the geosciences. 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 research in the division.
All theoretical studies in the Division of Applied Mathematics are motivated by their potential applications.
Brown's Division of Applied Mathematics Engages Actively in Fluid Mechanics Research
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 selfassembly of micro and nanoscale particles in suspension, active suspensions of microswimmers, and the flow properties of such systems.
Other aspects include uncertainty quantification in computational mechanics, noisy flow systems, and low-dimensional modeling. Both continuum and atomistic simulation methods are employed with high-performance computing required in most applications.
Innovative Research in Scientific Computing Contributes to a Multidisciplinary Atmosphere
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.
