Mathematical Sciences

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Head: Peter Kramer

Associate Head: Bruce Piper

Chair of the Graduate Committee: John Mitchell

Departmental Home Page:  https://science.rpi.edu/mathematical-sciences/

Through the centuries, mathematics has been a central feature of 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. Its focus is the study and development of mathematical and computational methods and their application to problems of contemporary significance to 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, it offers a baccalaureate degree in mathematics with capstones in (pure) mathematics, applied mathematics, mathematics of computation, and mathematics of 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. Department 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;
• computational mathematics, including the development, analysis, and implementation of numerical methods for mathematical models that arise in applications;                                                                                                   
• 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

The research activities of the faculty in the Department of Mathematical Sciences center on applied mathematics, analysis, scientific computing, mathematical programming, and optimization. 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.

Mathematical Biology
Biological sciences have undergone a great expansion, beginning in about the middle of the last century, and introduced a wealth of new areas studying systems ranging in size from the molecular to that of ecosystems. The tools of investigation in many areas of modern biology have grown to be increasingly quantitative and reliant on other sciences, particularly mathematics. The biosciences have thus become a rich source of mathematical problems, inspiring advances in modeling, analysis, and computational methods. Biomathematicians create these advances, as well as aid biological and medical scientists with the quantitative and predictive aspects of their discovery process.

Faculty in the department work in a wide variety of areas in biomathematics, including neuroscience, DNA and RNA modeling, cellular systems and transport, sensory systems, disease modeling and diagnoses, organ imaging, and tissue mechanics. They have been using and advancing all three types of the above mathematical problem areas associated with biological or medical problems. Many of the research projects in this area involve close collaboration with researchers in the relevant biological or medical area.

Operations Research and Data Science
A central focus in operations research and data science is the study of real-world problems with the aim of making better decisions. Analytical techniques, such as mathematical programming, machine learning, data mining, probability and statistics, and mathematical modeling and simulation, are used to formulate and solve mathematical models and optimization problems. A common feature of the research is the collection and use of data, ranging from big data (massive collections of data) and network domain data (e.g. street maps) to abstract knowledge and assumptions about how processes work. Applications of operations research and data science abound in many areas of engineering, business, science and medicine, leading to collaborative interdisciplinary research that is both interesting and challenging.

The research of math faculty is driven by many compelling real-world problems. Current work includes developing effective emergency responses to natural disasters, tracking infectious diseases (such as tuberculosis) so that they may be controlled effectively, and applying advanced data analysis for more reliable and efficient health-care solutions. Faculty are also using data analysis and modeling to increase the effectiveness of tracking motions in videos, to improve energy sustainability by increasing wind turbine output, and to analyze medical surfaces. Methods of nonlinear bi-level programming, optimization, probabilistic, and computational differential geometry have been developed for general data analysis that are applicable to wide range of problems in data science.

Geophysical and Environmental Modeling
Understanding how human activities impact the environment and ecosystems involves a web of interconnecting biological, chemical, and physical components. In addition to the disciplinary expertise required for each of these elements, mathematics plays a strong role in effectively analyzing and computing how the key drivers and parameters influence the various metrics of health of the ecosystem and environment. In particular, mathematical techniques and considerations are applied to derive quantitative representations of the complex models, to develop effective computational schemes which can accurately handle the multiple scales and equation types in the models, and to integrate historical and observational data into models which can help predict the effects of changes in policy and/or climate.

Particular aspects of geophysical and environmental modeling with which mathematics faculty have been engaged include the analysis and simulation of turbulent wave interactions in the ocean, effective representations of chemical transport by flow structures in the ocean, the study of interacting vortex dynamics in planetary atmospheres, and spatio-temporal models for plant-herbivore interactions. The recently launched Jefferson Project at Lake George promises to deliver an unprecedented amount of data and information regarding the physical, chemical, and ecological state of the lake and new collaboration opportunities for faculty.

Inverse Problems and Imaging
For inverse problems, the sought after solution is indirectly related to the measurable data, and it is either impossible to obtain the data more directly or it is not desirable to do so. An important aspect of the problem is that the mathematical model of the process that produces the data is fundamentally used in the algorithm that produces the image. Challenges include: modeling of the physical problem, creating new mathematics for analysis of the model, identifying appropriate (often large) and/or rich data sets, and working with scientific computations and visualization aids. Participants are from the Schools of Science and Engineering and share common mathematical interests and tools. Common societal impacts are to human health and safety.

Researchers in the mathematics department address some problems at the basic scientific level: for example finding properties of the earth’s substructure from seismic measurements or determining material properties of mechanical or biological systems. Other problems focus on direct applications: finding tumors in biological tissue, distinguishing abnormal from normal tissue, identifying fault locations in earthquake active regions, establishing the integrity of dikes, locating objects concealed by vegetation cover, and locating mines in the sea environment. This work can involve a significant amount of mathematical modeling of the application problem; noted in particular is that there is significant biomechanical modeling of tissue prior to addressing the inverse and imaging problem.

Electromagnetics, Optics, and Plasmas
Electromagnetism is a fundamental branch of physics and a key component of many natural and engineered systems. Since its inception, electromagnetism has been a rich source of fundamental mathematical problems, especially in the theory and numerical computation of solutions to partial differential equations. In the past several decades, much research effort has been focused on phenomena arising from the interaction of electromagnetic fields, such as light, and underlying optical media, such as plasmas, glass fibers, gasses or crystals composed of active atoms, or artificial composites containing materials with different response properties. Analytical, asymptotic, and computational methods developed by applied mathematicians have proven to be important for investigating and understanding these phenomena.

Researchers in the department are investigating fundamental problems in electromagnetic wave propagation using these three classes of techniques, while at the same time further developing the mathematical tools. They have contributed to fields ranging from exactly solvable nonlinear partial differential equations used in optical pulse propagation, modeling of electromagnetic responses of composite materials and ionized plasmas, to highly accurate computational algorithms. These algorithms are designed to preserve important features of the underlying physical systems, which, for example, enable accurate simulations of the problems and even of their long-time asymptotic behavior. These computational tools can be deployed on some of the largest computers in the world in order to help describe basic questions relating to electromagnetic phenomena.

High Performance Computing and Numerical Analysis
Research in high performance computing and numerical analysis involves the development of algorithms designed to compute solutions of difficult, often nonlinear, mathematical problems spanning a wide range of applications. An essential element of the research concerns the analysis of the algorithms, which seeks to uncover important properties of the methods, such as stability and convergence, so that the algorithms can be employed with a clear understanding of their behavior and accuracy. To obtain well-resolved solutions of problems in complex multidimensional configurations, high-performance computers, such as those available at Rensselaer’s CCI, are often needed, and an exciting aspect of the research involves implementing algorithms effectively and efficiently for such platforms.

Faculty in the department are active in this area of research. Recent work has led, for example, to the development of a wide class of stable and efficient partitioned algorithms for fluid-structure interaction problems, high-order accurate variational finite-difference methods, exactly divergence-free central discontinuous Galerkin finite-element methods, energy-conserving or asymptotically preserving high-order methods for kinetic equations, upwind methods for hyperbolic equations in second-order form, high-order conservative methods for Maxwell’s equations, and numerical methods in differential geometry.

Acoustics, Combustion, and Fluid-Structure Interactions
Research in acoustics, combustion, and fluid-structure interactions share a common theme of mathematical modeling and numerical simulation of problems in fluid and solid mechanics. Applications in these areas often involve wave propagation in complex constitutive materials, such as acoustic wave propagation in non-uniform media, detonation in heterogeneous explosives, or nonlinear deformation in elastic solids, among others. Mathematical models in these areas involve systems of partial differential equations, usually nonlinear and of hyperbolic type, together with matching conditions at interfaces in multi-material applications. For the latter applications, the dynamics of interfaces separating materials is an important feature of the problem, and developing stable and robust numerical tools for their accurate simulation is essential.

Researchers in mathematics are tackling interesting modeling issues related to long-range acoustic wave propagation in the ocean interacting with multicomponent sediments at the ocean floor, high-speed compressible flow in multiphase reactive materials, and compressible and incompressible fluids interacting with deformable structures, such as blood flow in veins and arties. A significant aspect of the active research concerns the development of numerical algorithms for the accurate solution of the model equations, and faculty in the math department are among the leaders in developing advanced simulation tools in these exciting areas of research.

High-Dimensional Stochastic Modeling, Analysis, and Simulation
A common situation in modern scientific research is that the number of factors affecting quantities of interest (such as the shape of a biomolecule, changes in regional climate, or the biodiversity of the ecosystem in a certain lake) is so large that a model comprising all of them would be analytically and/or computationally intractable. A typical way of modeling such systems is to include a manageable number of explicit dynamical variables, and represent the others by some suitable statistical or stochastic terms. Much of the recent mathematical research in such stochastic systems involves the appropriate formulation of such models when the number of retained degrees of freedom is still large, and the development of analytical methods and computational approaches to characterize how the key quantities of interest are impacted by the combination of the nonlinear dynamics of the explicit variables and their stochastic driving by unresolved variables.

High-dimensional stochastic systems are being developed and explored by mathematics faculty in the context of microbiology, geophysics, optics, and epidemiology. Examples of recent and ongoing research includes the impact of network topology and statistics on the synchrony of a neuronal network, social influence dynamics on random networks, hydrodynamic fluctuations in suspensions of swimming microorganisms, propagation of light through a disordered active medium, the interaction of molecular motor proteins in intracellular transport, and turbulent dynamics of waves in the ocean.

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 the 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 programs in the U.S. 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.

Outcomes of the Undergraduate Curricula

Students who successfully complete this program will be able to demonstrate:

• skills in the use of basic elements of mathematics and mathematical modeling.
• an ability to write mathematical proofs and an appreciation of the rigorous and logical structure of mathematics.
• an ability and the confidence to self-learn new mathematical concepts.
• an ability to apply critical and logical thought and mathematical principles to diverse problems.
• specialized knowledge of advanced mathematics in one or more of the tracks of analysis and algebra, mathematics of computation, applied mathematics, or operations research.

Baccalaureate Programs

The curriculum leading to a B.S. in Mathematics has been designed to permit the construction of programs that reflect individual student interests and career objectives. These programs include capstones in the following areas:

• 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 capstones share several common features. First, they each contain 11 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, Arts, 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 capstones is not necessary, since for the first two years, all mathematics students follow the same basic curriculum. A template for the undergraduate curriculum is outlined in the Programs section of this catalog.

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 the two summers following the freshman and junior years.

Such a joint degree program requires that the student apply to and be accepted by the Office of Graduate Education 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 their 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

Banks, J.—Ph.D. (Rensselaer Polytechnic Institute); numerical methods for partial differential equations, fluid-structure interaction, computational fluid dynamics and solid mechanics, scientific computing, wave phenomenon, laser plasma interaction.

Bennett, K.P.—Ph.D. (University of Wisconsin); mathematical programming, operations research, machine learning, data mining, artificial intelligence, applications of these methods to problems in engineering, business, medicine, biology, chemistry, and public health.

Henshaw, W. D.—Ph.D. (California Institute of Technology); numerical methods for partial differential equations, multigrid methods, computational fluid mechanics, computational solid mechanics, fluid-structure interactions, computational electromagnetics, overset grids and grid generation, scientific computing.

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

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

Kramer, P.R.—Ph.D. (Princeton University); stochastic processes, statistical modeling, applications to cellular biology and environmental science.

Li, F.—Ph.D. (Brown University); numerical analysis and scientific computing; efficient and highly accurate structure-preserving numerical methods for simulating MHD, Maxwell’s equations in linear and nonlinear media, Vlasov-type models, radiation transfer etc; model order reduction techniques for transport dominated models.

Lim, C.C.—Ph.D. (Brown University); network science, nonequilibrium statistical physics, applied probability and stochastic process, vortex dynamics and quasi-2D fluid flows.

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

Mitchell, J.E.—Ph.D. (Cornell University); optimization, integer programming, conic optimization, mathematical programs with complimentary constraints, column generation methods, stochastic optimization, applications of optimization.

Schwendeman, D.W.—Ph.D. (California Institute of Technology); mathematical modeling and analysis; numerical methods for partial differential equations; computational methods for reactive and nonreactive gas dynamics, solid mechanics, fluid-structure interactions and conjugate heat transfer; computational electromagnetics; numerical methods using overset grids.

Associate Professors

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

Xu, Y. —Ph.D. (Rice University); continuous optimization, stochastic optimization, matrix and tensor optimization, distributed computing, and applications of optimization methods to machine learning, statistical learning, signal processing, and data mining.

Assistant Professors

Faulstich, F. —Ph.D. (University of Oslo); numerical methods for high-dimensional problems; numerical analysis; embedding and multi-scale methods; quantum computing; strongly-correlated quantum chemistry.

Senior Lecturers

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

Kucinski, G. —Ph.D. (Binghamton University); mathematics education.

Lecturers

Brown, E. —Ph.D. (Rensselaer Polytechnic Institute); mathematics education and underwater acoustics.

Galo da Silva, B. —Ph.D. (University of Rochester/University of São Paulo); harmonic analysis, geometric measure theory, analytic combinatorics; one-dimensional dynamics.

Kam, E. —Ph.D. (The Hong Kong Polytechnic University); computational aeroacoustics, finite difference lattice Boltzmann method, computational fluid dynamics.

Klyachko, K. —Ph.D. (University at Albany); mathematics education.

Stevenson, D. —Ph.D. (University at Albany); analytic function spaces; operator theory; mathematics education.

Traldi, E. —Ph.D. (Rutgers University); mathematics education.

*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.

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