Chair   Mark H. Holmes
Chair of Graduate Committee   William Siegmann
Department Home Page   http://www.math.rpi.edu/index.html

Through the centuries, mathematics has been a central feature of man’s intellectual and technological development. Today its role in 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 our culture.

The Department of Mathematical Sciences is directly engaged in this process on a broad front in its educational and research programs. These emphasize analysis, applied mathematics, scientific computing, and mathematical programming. They focus on the study and development of mathematical and computational methods and their application to problems of contemporary significance to our society.

Mathematical Sciences at Rensselaer is interpreted in the broadest sense with emphasis on both the foundations of mathematical thought and the application of mathematics to the physical sciences, the life sciences, business, and engineering. The applied mathematics emphasis at the undergraduate level is very real, with the opportunity for students to take courses and become involved in projects that will make meaningful employment available upon graduation. One area of concentration is operations research where, like other branches of applied mathematics, emphasis is placed on modeling real phenomena and testing and analyzing its models for useful conclusions.

The Department of Mathematical Sciences provides an in depth education in mathematics, applied mathematics, mathematics of computation, and operations research leading to a baccalaureate degree with a specialization in one of these areas.

The department’s B.S. programs are designed to provide a broad spectrum of opportunities for students. A wide choice is built in so that students and advisers may tailor programs to individual objectives and talents. This makes the curricula equally advantageous for individuals who will seek immediate employment upon graduation, for those who plan graduate education in the mathematical sciences, and for those who will apply their education to pursuits outside the mathematical arena. Mathematical sciences graduates have gone on to careers in law, medicine, engineering, management, and psychology, as well as in pure and applied mathematics, computer science, and operations research.

Applied Mathematics   Rensselaer is especially well-known as a center for advanced study and research in applied mathematics. In the department’s M.S. and Ph.D. programs emphasis is given to:

• 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, particularly computational methods in fluid dynamics and heterogeneous materials;
• mathematical programming, including nonlinear, combinatorial, and multiple objective optimization and their applications.

At the highest level, applied mathematics is characterized by continual interplay between the construction of the mathematical model and the solution of the resulting mathematical problem. 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 area of application is required in order to construct suitable models, and a high level of mathematical judgment and expertise may be required to solve the resulting mathematical problems.

Areas of Advanced Research and Study

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 naturally leads to interaction among mathematics, computer science, and operations research, since many applied problems require the synthesis of techniques from two or more research areas. Further, the formulation, solution, and interpretation of a problem for an application in a field in the physical sciences often contain ideas that can be applied to problems in other areas such as the physical, biological, and social sciences. This focusing of different research areas on real problems and the diversity of applications of solutions of real problems 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   Numerical analysis and scientific computing are active fields of research at Rensselaer. Investigations range from the study of fundamental problems in linear algebra to the development and analysis of numerical schemes for solving particular physical 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 for solving problems involving composite materials. Applications of these studies include reacting flows, shock wave propagation, semiconductor performance, biomathematics, acoustic signal propagation, and incompressible flow in various geometries.

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

Dynamical Systems   This research is concentrated 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, and other fields. The aim of this research is to discover and to explain new and important phenomena found in experimental and numerical studies. This research often involves modeling a real-life problem by a dynamical system and then applying the ideas and methods of the theory to explain and predict the behavior that seems complex or mysterious. 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) to study 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 and provide 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 is concerned with finding optimal or very good solutions for a broad range of problems, with areas of application 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. Specific 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, genetic algorithms and tabu search, mathematical programming approaches to problems in artificial intelligence such as machine learning, neural networks, support vector machines, pattern recognition, and planning, combining operations research and intelligence problem solving methods, scalability of these methods to very 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 are in cardiac imaging, 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; 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.

Functional Analysis   Research is done in conjunction with particular applied problems and problems are both linear and nonlinear. Areas of application include inverse and ill posed problems, numerical methods for large-scale problems, approximation problems, image restoration and fixed-point methods in complementarity problems. Nonlinear functional analysis is applied to inverse and ill posed problems to find optimal data sets, to establish existence, uniqueness and stability results, to find accurate finite dimensional approximations infinite dimensional problems, and to use optimization techniques to develop numerical algorithms to construct solutions. Studies involve parameter dependent operators in Hilbert and Banach spaces where the parameters are the unknown solutions of nonlinear functional equations.

Fluid Mechanics   Methods of applied mathematics are being used to study how fluids with different characteristics behave in various geometric configurations. 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, the flow of water past a ship, and the theory of flow in a gas centrifuge. Transonic flow problems and perturbations of exactly sonic flow and of shock-free flow are being analyzed. Studies also include the evolution of non-Newtonian (e.g., polymer) fluid flow. The mathematics involved includes bifurcation theory for eigenvalue problems and asymptotic, perturbation, and numerical techniques.

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 studies of 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. A research level seminar is regularly attended by graduate students and Rensselaer faculty. Discussions center on industrially motivated problems at the interface between mathematics and computer science and engineering.

Semiconductor Device Equations   Semiconductor devices are the heart and soul of the microelectronics revolution. Formulation of the physics of operation of these devices results in a set of coupled and strongly nonlinear partial differential equations. Faculty are currently active in the investigation of analytical and numerical methods for modeling diverse semiconductor device problems. Areas of specialization include prediction of fundamental limits on metal-oxidesemiconductor field-effect transistor (MOSFET) dimensions and environmental radiation effects.

Mathematical Finance   Mathematical techniques are applied to the study of a variety of investment problems such as the selection of an optimal portfolio and securities, the effect of transaction costs on rebalancing schemes, and the equivalent worth pricing of a derivative. Of particular interest is the needs of the long run investor.

Approximation Theory   This branch of mathematics strives for the understanding of the fundamental limits in optimally representing different kinds of signals. “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 multiresoluton 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. We exploit tools ranging 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.

Faculty

Professors

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.
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).
Handelman, G.H.   Ph.D. (Brown University); applied mathematics, elasticity, wave propagation, mathematical biology (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.
Luchins, E.H.   Ph.D. (University of Oregon); algebra, mathematics of psychology, mathematics education, number theory, history of mathematics, women in mathematics (emeritus).
McLaughlin, H.W.   Ph.D. (University of Maryland); applied geometry.
McLaughlin, J.R.   Ph.D. (University of California, Riverside); inverse vibration and inverse scattering problems, wave propagation, analysis, applied mathematics.
Roytburd, V.   Ph.D. (University of California, Berkeley); applied mathematics, combustion theory.
Rubenfeld, L.A.   Ph.D. (New York University); applied mathematics, mathematics and science education.
Siegmann, W.L.   Ph.D. (Massachusetts Institute of Technology) applied mathematics, wave propagation.

Associate Professors

Bennett, K.P.   Ph.D. (University of Wisconsin); mathematical programming, operations research, machine learning, data mining, artificial intelligence.
Kovacic, G.   Ph.D. (California Institute of Technology); applied mathematics, nonlinear dynamics, nonlinear optics.
Lim, C.C.   Ph.D. (Brown University); mathematical modeling, vortex dynamics, applications of graph theory.
Mitchell, J.E.   Ph.D. (Cornell University); mathematical programming, integer programming, interior point methods, column generation methods, financial optimization, stochastic programming.
Piper, B.R.   Ph.D. (University of Utah); computer-aided geometric design, numerical analysis, computer graphics.
Schwendeman, D.W.   Ph.D. (California Institute of Technology); applied mathematics, scientific computing.

Assistant Professors

Kramer, Peter R.   Ph.D. (Princeton University); turbulent diffusion, stochastic processes.
Lvov, Y.   Ph.D. (University of Arizona); mathematical physics and nonlinear phenomena.
Nolan, Clifford J.   Ph.D. (Rice University); medical and seismic imaging using microlocal analysis.
Yu, T.P.-Y.   Ph.D. (Stanford University); wavelets and applications in signal and image reconstruction.

Clinical Assistant Professors

Blackford, J.T.   Ph.D. (Ohio State University); algebraic coding theory, combinatories, codes over rings.
Kiehl, M.   Ph.D. (Rensselaer Polytechnic Institute); biomathematics.
Schmidt, D. A.   Ph.D. (Rensselaer Polytechnic Institute); graph theory, qualitative matrix analysis, mathematics education.

Joint Appointments with Computer Science

Professors

Flaherty, J.E.   Ph.D. (Polytechnic Institute of Brooklyn); scientific computation, numerical analysis, applied mathematics.
Rogers, E.H.   Ph.D. (Carnegie Institute of Technology); VLSI architecture, computer applications.

Undergraduate Curricula

Mathematics has always been the cornerstone of scientific development. At Rensselaer, our 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 mathematics program at the undergraduate level is designed to educate students in a variety of mathematical areas. The flexibility in this program, with its numerous options, permits selection of courses ranging from pure theory, with the idea of building a foundation for further studies at the advanced level, to applied subjects, with mathematical modeling and the solution of real world problems. In particular, Rensselaer’s Department of Mathematical Sciences is one of the few in this country with a strong faculty orientation toward applications of mathematics. This emphasis is reflected in the many undergraduate courses dealing with areas of mathematical applications and in the applied flavor with which department faculty typically teach undergraduate courses.

Four curricula leading to a Bachelor of Science in Mathematics have been designed to aid students in constructing programs that reflect their interests and career objectives.

• Mathematics (a traditional program emphasizing the elements of pure and applied mathematics).
• Applied Mathematics (a program 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 (a program emphasizing the use of mathematics in developing and studying analytical models of discrete systems, especially those that arise in management, engineering, and social sciences).

The curricula serve as a framework on which each student, with the help of his or her adviser, can build a strong and individually meaningful program.

The four curricula have several features in common. First, they each contain nine free electives. These give students the freedom to design a program that is unique. They also give students an opportunity to concentrate on a subject in addition to mathematics, to obtain a broad-based education, or to complement their mathematics program. The second common feature is the H&SS requirement. They all require 24 credits. Finally, all four curricula require 124 credits for graduation.

Dual Majors   The requirements for a dual major are described elsewhere in this catalog. Interest in this program 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 for dual majors.

The First Two Years

The four curricula are the same for the first two years, so you do not need to make a decision at an early stage. Descriptions of each curricula, and the requirements for the junior and senior years, are given later. For the first two years the requirements for each are as follows:

Detailed and up-to-date descriptions of the mathematics courses, including special topics courses, can be found at the department’s web site at http://www.math.rpi.edu/index.html.

The seminar courses Art and Science of Math (MATH-1900, MATH-1910) in the first year are not required but are strongly recommended. This is a weekly seminar for math majors where interesting and challenging mathematical problems and ideas are discussed. Also, attention should be given to the survey course Fundamentals of Mathematics (MATH-2700). This is a second semester sophomore course that provides a good background for the junior and senior mathematics courses.

The science electives must be chosen so at least three different disciplines in the School of Science, outside of math, are represented in your total program. In particular, since CSCI-1100 is already required, the science electives must include two, or more, other disciplines. Note that math science includes all courses with prefixes MATH and MATP (and any course cross listed with a MATH or MATP course); computer science includes all courses with prefix CSCI (and any course cross listed with a CSCI course); courses with prefixes PHYS and ASTR are considered a single discipline.

Curriculum Requirements

Mathematics Curriculum   This curriculum provides a broad and basic education in mathematics. It is especially suited to those who intend to continue their graduate education in mathematics or in some other scientific and engineering field. There is considerable flexibility built in so that students and their advisers may tailor programs to individual objectives. Thus, with appropriate choices in the mathematical options provided, the curriculum is equally useful to those students who will seek immediate employment upon graduation.

The mathematics options listed above are any 4000, or higher, level course from the Department of Mathematical Sciences. Those planning on going on to graduate work should take Linear Algebra (MATH-4100).

Applied Mathematics Curriculum   What exactly do applied mathematicians do? Well, for one thing they study mathematical problems that arise in science, engineering, or management. The applied mathematics curriculum therefore stresses courses that involve the construction, analysis, and evaluation of mathematical models of real-world problems and those areas of mathematics that are most widely useful in solving them. The program allows for considerable flexibility, but it is recommended that a student acquire a solid background in the three principal areas of applied mathematics. These are modeling, analysis or solution methods, and numerical analysis.

The mathematics options listed above are any 4000, or higher, level course from the Department of Math Science. It is recommended that students take Physics I, II (PHYS-1100, PHYS-1200) and those who may continue on to graduate school should consider taking Mathematical Analysis II (MATH-4210) and Linear Algebra (MATH-4100).

Mathematics of Computation Curriculum   Computers and computational methods play an important role in all fields of science and engineering. The mathematics of computation curriculum focuses on the mathematical development, analysis, and application of numerical methods. Surrounding this main focus are courses that build mathematical expertise in analysis, modeling, and applications. There is also flexibility to pursue courses in computer science as well as courses in other fields of science and engineering.

The mathematics options listed above are any 4000, or higher, level course from the Department of Mathematical Sciences. The computation option is either Introduction to Numerical Methods for Differential Equations (MATH-4820) or Computational Optimization (MATP-4820). The CS options are any 2000 level, or higher, courses from Computer Science (i.e., courses with prefix CSCI and not cross listed with any math course).

It is recommended that students take Physics I, II (PHYS-1100, PHYS-1200) and Computer Science II (CSCI-1200). Those who may continue on to graduate school should consider taking Mathematical Analysis II (MATH-4210).

Mathematics of Operations Research Curriculum   The mathematics of operations research curriculum emphasizes the use of mathematics for developing and studying analytical models of systems. These models are used to help make better decisions in areas such as management, engineering, and the social sciences. In mathematical programming a problem is modeled as an objective function with constraints on the possible solutions, then the resulting model is optimized. Computer programs are used to solve the models. Algebra, analysis, and discrete mathematics all play a role in the analysis of the models and the development of computer algorithms to solve the models. Frequently the inputs and outcomes of the model are not known with certainty, thus probability and statistics are used. A list of suggested courses is given below.

The OR option is either Probability Theory and Applications (MATP-4600) or Computational Optimization (MATP-4820). For the OR curriculum the mathematics options are any 4000, or higher, level course from the Department of Math Science (MATH-xxxx or MATP-xxxx) plus up to two 4000 or higher level courses from decision science (DSES-xxxx) or computer science (CSCI-xxxx). In other words, of the four mathematics options a minimum of two must have a prefix of MATH or MATP.

Four-Year Mathematical Sciences B.S.-M.S. Degrees   A qualified student may earn a B.S. degree and an M.S. Degree in the same or different areas in a shorter-than-usual time. This may be done by using advanced placement credit, by taking additional courses during the fall and spring semesters and/or by taking courses during summers.

For example, a student with advanced placement credit for Calculus I and II may earn the B.S. And M.S. degrees in four years by taking one more course than the number the typical program requires in each of the regular fall and spring semesters. Since a student may take up to 21 credit hours per semester at no additional charge, he or she may be able to earn both degrees for the cost of a B.S. alone. As a second example, the same student may earn two degrees in four years by taking no more courses than the number the typical program requires during the academic years but by taking eight courses distributed over three summers.

Of course, 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 is asked to consult his or her faculty adviser to design an optimum program.

Minor in Mathematics   The Department of Mathematical Sciences provides a minor program in mathematics. Students not majoring in mathematics may receive a minor in math by taking four courses at the 4000 level or above from the MATH and MATP course groups. These courses should form a coherent program and have the prior approval of the chairman of the Department of Mathematical Sciences.

Graduate Programs

The Department of Mathematical Sciences offers programs leading to the M.S. and Ph.D. degrees.

Each curriculum is highly flexible, and in effect each student’s program of study is individually designed.

Course work is supplemented by the departmental colloquium series, in which both the mathematics faculty and guest lecturers present current research work. There is also a weekly seminar organized by the graduate students in which they present material from their research. Moreover, a number of informal seminars are organized each semester by faculty and students to explore topics of mutual interest. Through formal course work, these additional activities, and individual contact with the faculty, the student becomes familiar with the research activities within the department. There is an overview of the research activities on the Department’s web pages listing the research areas and the faculty working in each area.

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

Graduate Degree Requirements

Master of Science in Applied Mathematics   The emphasis of this program is on mathematics and how it can be employed to study problems in science, engineering, or management. It stresses the construction, analysis, and evaluation of mathematical models of real-world problems, and those areas of mathematics that are most widely useful in solving them. The requirements for the degree are such that students can prepare themselves for entry into the Ph.D. program in applied mathematics or for employment in business, industry, or government.

The student must meet the degree requirements of the Graduate School and follow a plan of study acceptable to the Department of Mathematical Sciences and the Graduate School. Each student’s program of study must include: (a) at least 4 four-credit courses at the graduate (6000) level, of which at least two must be in math MATH-6xxx or MATP-6xxx: (b) at least 4 four-credit courses with prefix MATH or MATP: (c) at least one three- or four-credit course at the 4000 or 6000 level outside the Department (i.e., with a prefix other than MATH or MATP and not cross listed with any Department course), selected in consultation with your math adviser: (d) participation in a professional experience such as an applied mathematics master’s project, the Applied Mathematics Industry Workshop, or a graduate cooperative education internship. A Department faculty member must approve plans for this experience in advance and will certify its satisfactory completion. This experience may, but need not, be arranged for up to six academic credits.

Master of Science in Mathematics   The student must meet the degree requirements of the Graduate School and follow a plan of study acceptable to the Department of Mathematical Sciences and the Graduate School. The plan of study should represent a reasonably broad program in mathematics and must contain (a) at least 4 four-credit courses at the graduate (6000) level, of which at least two must have numbers in the range MATH-6000 to MATH-6390 (b) at least 4 four-credit courses with prefix MATH or MATP, (c) participation in a professional experience such as a mathematics master’s project or a graduate cooperative education internship. A Department faculty member must approve plans for this experience in advance and will certify its satisfactory completion. This experience may, but need not, be arranged for up to six academic credits.

Doctor of Philosophy   A student working for the doctorate must demonstrate high achievement both in scholarship and in independent research. All programs must be in accord with the general rules of the Graduate School.

The Ph.D. degree can be obtained by following a program of study in mathematics or in applied mathematics. In either case, the student’s program of study must include (a) at least 6 four-credit (non-thesis) graduate mathematics courses (i.e., those with numbers MATH-6xxx or MATP-6xxx), (b) at least one three- or four-credit course at the graduate (6000) level, outside the Department (i.e., with a prefix other than MATH or MATP and not cross listed with any Department course), selected in consultation with your math adviser, (c) at most 30 thesis/research credits. All doctoral students must pass a written preliminary exam as well as an oral qualifying examination, and complete an oral candidacy presentation. Descriptions of these requirements can be found on the department’s web pages for the Ph.D. program.

Any deviations from these requirements must have the approval of the department’s Graduate Committee.