Mathematical Sciences

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Head: Donald Drew

Chair of the Graduate Committee: Ashwani Kapila

Departmental Home Page: http://www.math.rpi.edu/index.html

Through the centuries, mathematics has been a central feature of our intellectual and technological development. Today its role in the physical sciences and engineering is well established. Its role in the life and social sciences, medicine, management, and the arts is undergoing remarkable growth—a virtual mathematization of the culture. The Department of Mathematical Sciences is directly engaged in this process through its educational and research programs. Our focus is the study and development of mathematical and computational methods and their application to problems of contemporary significance to our society.

The Department of Mathematical Sciences provides an in-depth education in both the foundations of mathematical thought as well in the applications of mathematics to real-life phenomena. For this reason, we offer a baccalaureate degree with a specialization in mathematics, applied mathematics, mathematics of computation, or operations research. The department’s programs are also designed to provide a broad spectrum of opportunities for students. This flexibility allows students and advisers to tailor programs to individual objectives and talents. As a result, the curricula are equally advantageous for individuals who will seek immediate employment upon graduation, for those who plan graduate-level education in the mathematical sciences, and for those who will apply their education to pursuits outside the mathematical arena. Our graduates have entered careers in law, medicine, engineering, management, and psychology, as well as in pure and applied mathematics, computer science, and operations research.

At the graduate level, Rensselaer is especially well-known as a center for advanced study and research in applied mathematics. The department’s M.S. and Ph.D. programs emphasize:

• Methods of applied mathematics, including ordinary and partial differential equations, approximation theory, asymptotic analysis, functional analysis, and numerical analysis;

• Applications in the physical sciences, biological sciences, and engineering;

• Scientific computing;
• Mathematical programming, including nonlinear, combinatorial, and multiple objective optimization and their applications.

At the highest level, continual interplay between the construction of the mathematical model and the solution of the resulting mathematical problem characterizes applied mathematics. The ideal applied mathematician, therefore, must be knowledgeable both in mathematics and in at least one field in which problem areas are found. A sound knowledge of the application area assists in constructing suitable models, and a high level of mathematical judgment and expertise may be required to solve the resulting mathematical problems.

Research Innovations and Initiatives

Faculty research activities in the Department of Mathematical Sciences center on applied mathematics, analysis, scientific computing, mathematical programming, and operations research. The faculty’s interest in applied research often leads to a synthesis of techniques from two or more research areas. Further, the formulation, solution, and interpretation of a problem often contain ideas that can be applied to problems in other areas. Focusing different research areas on real problems and the diversity of applications of real problem solutions creates an atmosphere of interaction and cooperation within the department and the university, as well as with other major research institutions.

Numerical Analysis and Scientific Computation
Investigations range from the study of fundamental problems in linear algebra to the development and analysis of numerical schemes for solving particular physical or life science problems. Research activities include the numerical solution of optimization problems, inverse eigenvalue problems, and free boundary problems; finite difference and finite element methods for stiff initial and boundary-value problems; and methods of resolving problems involving composite materials. Applications of these studies include reacting flows, shockwave propagation, semiconductor performance, biomathematics, acoustic signal propagation, and incompressible flow in various geometries.

Inverse Problems
This research involves the recovery of internal biological, mechanical, electrical, or magnetic properties of a system from boundary, spectral, or scattering data. The physical system is modeled by a partial differential equation or an ordinary differential equation with specific unknown terms representing, for example, stiffness in an elastic system or electric permittivity in an electromagnetic system. The goal of this work is to find the unknown properties from indirect measurements. Rensselaer has established a center for Inverse Problems at RPI. Current research applies functional analysis, perturbation theory, numerical analysis, and optimization to determine optimal datasets, to study the nonlinear dependence of the unknown physical quantities on the available data, and to obtain approximations of the nonlinear operators that will yield efficient reconstruction algorithms. There is a significant role for modeling, analysis, scientific computation, and algorithm development to obtain solutions to these problems.

Dynamical Systems
This research concentrates on the theory of dynamical systems and its applications in physics and engineering. Dynamical systems arise as mathematical models in various applications such as mechanics, optics, electric circuits, solid-state physics, fluid dynamics, optimal control, neural science and other fields. This research aims to discover and explain new and important phenomena found in experimental and numerical studies. Often involved is modeling a real-life problem by a dynamical system and then applying the ideas and methods of the theory to explain and predict complex behavior. Theoretical research is conducted in chaotic dynamics, Hamiltonian systems (KAM theory and applications, theoretical mechanics), bifurcation theory, and related fields. Mathematical methods used come from analysis, topology, differential geometry, combinatorics, and other fields. Computation may be used as an experimental tool.

Wave Propagation
These studies focus on the behavior of acoustic wave propagation. A major area of interest is underwater sound transmissions. Mathematical models are being developed and analyzed to describe the influences of ocean environmental features (such as internal waves and sediment variations) on the study of the propagation of signals in both frequency and time domains, and to improve the accuracy of known numerical methods. Improved numerical and asymptotic methods are derived and tested, providing new ways to extract information from complex propagation environments. Stochastic propagation effects are modeled and analyzed, and results are used to explain variability observed by ocean scientists. Results are extended and applied to acoustic propagation environments ranging from the atmospheres of Jupiter and the Earth to the upper layer of the Earth’s crust.

Mathematical Programming and Operations Research
Mathematical programming endeavors to find optimal solutions for a broad range of problems including medical, financial, scientific, and engineering problems. Research is conducted on the development, evaluation, and comparison of serial and parallel algorithms for a variety of mathematical programming problems. Current research topics include interior point methods for linear, integer, and nonlinear programming; branch-and-bound and branch-and-cut approaches to integer programming problems; column generation methods; financial optimization; and genetic algorithms and tabu search. Also under investigation are mathematical programming approaches to problems in artificial intelligence such as machine learning, neural networks, support vector machines, pattern recognition, and planning. This research also considers combining operations research and artificial intelligence problem-solving methods, scalability of these methods to large problems in data mining, mathematical programming approaches to other areas in computer science such as database query optimization, and stochastic programming.

Biomathematics
Mathematical biology is a very active area of applied mathematical research. This is an interdisciplinary endeavor, with a strong interaction with biological and biomedical scientists. Projects of current interest include cardiac imaging and the use of computer graphics to construct pictures of the heart, mechanoreception, mathematical modeling of biological systems that transform mechanical stimuli (e.g., sound, touch, etc.) into ionic or neural signals and molecular systems in cells. Also being studied are nonlinear ionic diffusion in polyelectrolytic gels and the mechanics of multiphasic tissues like cartilage and the cornea. Numerical analysis, asymptotics, and functional analysis are used to investigate mathematically posed problems resulting from the models.

Fluid Mechanics
Methods of applied mathematics are being used to study how fluids behave under a wide spectrum of conditions. The physical problems usually lead to partial differential equations, which may be linear or nonlinear. Current problems deal with fluid mechanics in engineering systems, the flow and stability of two-phase mixtures, the transition from laminar to turbulent flow in boundary layers, fluid mechanical models of atmospheric events and the theory of flow in a gas centrifuge. Studies also include the evolution of non-Newtonian (e.g., polymer) fluid flow.

Combustion Theory
Investigations include mathematical modeling of combustion and flame propagation phenomena, and analysis of the resulting systems of nonlinear ordinary and partial differential equations. Topics of current interest are bifurcation and stability of reactive systems, evolution and interaction of waves in reactive gases, combustion and vortex breakdown in swirling flows, and transition from deflagration to detonation in granular explosives.

Applied Geometry
Included are problems dealing with surface design, curve design, robot path planning, packing, tiling, computational geometry, and artificial intelligence as it applies to geometry. Students take advantage of related courses in electrical engineering, mechanical engineering, computer science, and mathematics.

Approximation Theory
This branch of mathematics strives to understand the fundamental limits in optimally representing different signal types. “Signals” here may mean a database of digital audio signal, a collection of digital mammograms, solutions of a class of integral equations, or triangulated compact surfaces acquired by a 3-D scanner. These signals are typically modeled mathematically based on their intrinsic smoothness or oscillatory characteristics. Current research effort involves the design and analysis of various multiresolution techniques that have provable optimality properties for these models. Such optimal representations are invariably the key ingredients to successful data compression, estimation, and computer-aided geometric design. Exploited tools range from mathematical analysis (e.g., Littlewood-Paley theory) to fast numerical algorithms, to information theory, to algebraic and differential geometry, and to spline and subdivision theory.

Complex Systems
This includes an investigation into nonlinear phenomena that arise in such diverse areas as semiconductor laser theory, nonlinear and fiber optics, surface water waves, acoustic waves and gas lasers. Although these topics are seemingly disconnected and have different physical characteristics, they all can be viewed as complex systems composed out of interacting particles or waves. There is a general theoretical framework for their description called weak turbulence theory. The research in this area involves development of weak turbulence theory and how to use this theory to study complex systems.

Bioinformatics
The massive volume of new data being produced by genome sequencing projects point to an increasing need for bioinformatics. This is a highly interdisciplinary field, involving faculty in mathematical sciences, biology, computer science, chemistry and several departments in the school of engineering. Rensselaer has established a joint bioinformatics center with the nearby Wadsworth Laboratories in the New York State Department of Public Health. Current activities at Rensselaer comprise the development and application of algorithms that aim to solve biological problems using DNA and amino acid sequence, structure, and related information. Some of the problems addressed are the search for patterns in biomolecular sequences that are functionally important, such as transcription binding sites; the prediction of structure or function from nucleic acid or protein sequence data; the development of methods and databases to classify large amounts of biological information, and the development of algorithms and software that are important for current biotechnology applications.

Undergraduate Programs

Mathematics has always been the cornerstone of scientific development. Rensselaer’s aim is to provide an education in mathematics, both as a subject in itself and as a discipline to aid in the development of other social and scientific fields. The undergraduate mathematics program educates students in a variety of mathematical areas. The flexibility in this program, with its numerous options, permits selection of courses ranging from pure theory (which builds a foundation for more advanced studies), to applied subjects focusing on mathematical modeling and the solution of real-world problems. In particular, Rensselaer’s Department of Mathematical Sciences is one of the few American programs with a strong faculty orientation toward mathematics applications. Reflecting this emphasis are the many undergraduate courses dealing with areas of mathematical applications and the applied flavor with which department faculty typically teach them.

Baccalaureate Programs

Four curricula leading to a B.S. in Mathematics have been designed to permit the construction of programs that reflect individual student interests and career objectives. These curricula include:

• Mathematics—a traditional program emphasizing the elements of pure and applied mathematics.

• Applied Mathematics—emphasizing both the modeling of physical phenomena and methods of analyzing the resulting mathematical problems.

• Mathematics of Computation—a program bridging mathematics and computer science, with emphasis on numerical methods for solution of problems in science and engineering.

• Mathematics of Operations Research—emphasizing the use of mathematics in developing and studying analytical models of discrete systems, especially those that arise in management, engineering, and social sciences.

These four curricula share several common features. First, they each contain eight free electives that permit students to design unique programs. These electives also allow students to concentrate on a subject in addition to mathematics, to obtain a broad-based education, or to complement their mathematics program. A second common feature is the Humanities and Social Sciences requirement of 24 credits. Finally, completion of all four curricula requires a total of 124 credits.

An immediate choice among these four curricula is not necessary, since for the first two years, all mathematics students follow the same basic curriculum. This initial two-year course of study is outlined below and is followed by sample junior/senior curricula for each of the department’s four undergraduate programs. In addition to the specific requirements in each track, it is strongly recommended that students planning to pursue graduate study in mathematics take the following courses:

MATH 4200 Mathematical Analysis I

MATH 4210 Mathematical Analysis II

MATH 4100 Linear Algebra

MATH 4300 Introduction to Complex Variables

Dual Major Programs

The requirements for a dual major are described in the section on Academic Information and Regulations. Interest in such programs is increasing, and recent combinations have included math and physics, math and computer science, and math and psychology. Typical schedules for such combinations can be found at the department’s Web site under dual majors.

Accelerated Programs

Qualified students may earn a B.S. and M.S. degree in the same or different areas in a shorter-than-usual time. They may do so through the use of advanced placement credit, by taking additional courses during the fall and spring semesters, and/or by taking summer courses.

For example, a student with advanced placement credit for Calculus I and II may earn the B.S. and M.S. degrees within four years by taking an additional course each regular fall and spring semester. Since a student may take up to 21 credit hours per semester at no additional charge, it may be possible to earn both degrees for the cost of a B.S. alone. As a second example, rather than taking more courses during the academic year, a student may earn two degrees in four years by taking eight courses distributed over three summers.

Such a joint degree program requires that the student apply to and be accepted by the Graduate School at an appropriate stage. A wide variety of joint degree programs can be arranged depending on the student’s background, interests, and desired rate of progress. The interested student should consult the faculty adviser to design an optimum program.

Graduate Programs

The Department of Mathematical Sciences offers programs leading to the M.S. and Ph.D. degrees. Each curriculum is highly flexible, and each student’s program of study is individually designed.

A departmental colloquium series, in which both mathematics faculty and guest lecturers present current research work, supplements course work. In addition, graduate students organize a weekly seminar, in which they present material from their research. Moreover, each semester, faculty and students organize informal seminars that explore topics of mutual interest. In a special course called Introduction to Research in Mathematics, each week a faculty member discusses his or her research program and describes current problems for graduate students to investigate. In addition, through formal course work and individual contact with the faculty, students become familiar with all departmental research activities. The department’s Web site also provides an overview of these research activities and lists the faculty working in each area.

Undergraduates with backgrounds in mathematics or any related major with significant mathematical content are admissible to the graduate program.

Course Descriptions

Courses directly related to all Mathematical Sciences curricula are described in the Course Description section of this catalog under the department code MATH or MATP.

Faculty*

Professors

Bennett, K.P.—Ph.D. (University of Wisconsin); mathematical programming, operations research, machine learning, data mining, artificial intelligence.

Boyce, W.E.—Ph.D. (Carnegie Institute of Technology); applied mathematics, mathematics education (emeritus).

Cheney, M.—Ph.D. (Indiana University); inverse problems, wave propagation, applications in engineering and biology, partial differential equations.

Drew, D.A.—Ph.D. (Rensselaer Polytechnic Institute); applied mathematics, fluid mechanics, mathematical biology.

Ecker, J.G.—Ph.D. (University of Michigan); mathematical programming, multiobjective programming, geometric programming, mathematical programming applications, ellipsoid algorithms.

Fleishman, B.A.—Ph.D. (New York University); nonlinear differential equations, mathematics education (emeritus).

Habetler, G.J.—Ph.D. (Carnegie Institute of Technology); functional analysis, numerical analysis (emeritus).

Herron, I.—Ph.D. (Johns Hopkins University); applied mathematics, fluid mechanics, hydrodynamics, stability.

Holmes, M.—Ph.D. (University of California, Los Angeles); perturbation methods, biomathematics, nonlinear continuum mechanics.

Isaacson, D.—Ph.D. (New York University); mathematical physics, biomedical applications.

Jacobson, M.J.—Ph.D. (Carnegie Institute of Technology); applied mathematics, acoustic and electromagnetic wave propagation (emeritus).

Kapila, A.—Ph.D. (Cornell University); applied mathematics, combustion, fluid mechanics.

Lim, C.C.—Ph.D. (Brown University); mathematical modeling, vortex dynamics, applications of graph theory.

McLaughlin, H.W.—Ph.D. (University of Maryland); applied geometry.

McLaughlin, J.R.—Ph.D. (University of California, Riverside); inverse bioelasticity problems, inverse vibration and inverse scattering problems, wave propagation, analysis, applied mathematics.

Mitchell, J.E.—Ph.D. (Cornell University); mathematical programming, integer programming, interior point methods, column generation methods, financial optimization, stochastic programming.

Roytburd, V.—Ph.D. (University of California, Berkeley); applied mathematics, combustion theory.

Rubenfeld, L.A.—Ph.D. (New York University); applied mathematics, mathematics, science education.

Siegmann, W.L.—Ph.D. (Massachusetts Institute of Technology); applied mathematics, wave propagation.

Schwendeman, D.W.—Ph.D. (California Institute of Technology); applied mathematics, scientific computing.

Zuker, M.—Ph.D. (Massachusetts Institute of Technology); bioinformatics.

Associate Professors

Kovacic, G.—Ph.D. (California Institute of Technology); applied mathematics, nonlinear dynamics, nonlinear optics.

Kramer, P.R.—Ph.D. (Princeton University); turbulent diffusion, stochastic processes.

Lvov, Y.—Ph.D. (University of Arizona); mathematical physics and nonlinear phenomena.

Piper, B.R.—Ph.D. (University of Utah); computer-aided geometric design, numerical analysis, computer graphics.

Assistant Professor

Li, F.—Ph.D. (Brown University); numerical analysis and scientific computing; finite element methods; discontinuous Galerkin methods; numerical methods for Maxwell equations, Maxwelleign-problems, conservation laws, Hamilton-Jacobi equations, magneto-hydrodynamics equations.

Clinical Associate Professor

Kiehl, M.—Ph.D. (Rensselaer Polytechnic Institute); biomathematics.

Clinical Assistant Professor

Schmidt, D.A.—Ph.D. (Rensselaer Polytechnic Institute); graph theory, qualitative matrix analysis, mathematics education.

Research Assistant Professor

Nolan, C.J.—Ph.D. (Rice University); medical and seismic imaging using microlocal analysis.

Joint Appointments with Computer Science—Professors

Rogers, E.H.—Ph.D. (Carnegie Institute of Technology); VLSI architecture, computer applications (emeritus).

 * Departmental faculty listings are accurate as of the date generated for inclusion in this catalog. For the most up-to-date listing of faculty positions, including end-of-year promotions, please refer to the Faculty Roster section of this catalog, which is current as of the May 2008 Board of Trustees meeting.

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