The availability of a masters program in applied and computational mathematics will benefit the on-campus and system-wide graduate programs in Engineering, at SMAST, in Biomedical Engineering, and across scientific disciplines. This program will provide courses in state-of-the-art techniques in scientific computing and in applied mathematics, which will be open to students in graduate programs in the sciences. The program will also attract talented students for joint masters programs and for five-year masters programs.

Statement of Purpose and Goals:

This program is designed for students interested in studying and developing computational methods for a broad range of applications. The program will provide both a strong foundation in computational and applied mathematics, and a deep understanding of the development, analysis and implementation of computational methods. The term computational mathematics is used in its broadest sense, to include computational tools used in mathematics and the physical and life sciences. The Masters in Computational Mathematics will prepare students for a career in industry, for a doctoral program in mathematics, applied mathematics, and various applied sciences and computing. It will benefit the graduate programs in the college of Engineering, at SMAST, and the new Umass system PhD program in biomedical engineering by providing courses in state-of-the-art scientific computing and applied mathematics for the doctoral students on campus. The proposed program aims to partner with the available masters and doctoral programs to provide interdisciplinary and collaborative opportunities for students pursuing a joint masters program. We draw upon the diverse strengths of the department to build a program which will appeal to students with a broad range of backgrounds and interests.

Advantages to other programs in UMD:

§ The availability of a wide variety of new courses (see list of courses, below) in computational and applied mathematics will benefit the current graduate programs in the College of Engineering and the UMass system PhD program in Biomedical Engineering. In addition to courses in numerical optimization, numerical solution of PDEs, numerical linear algebra, statistics and mathematical methods for the applied sciences, the program will have a "topics" course which will feature current topics of interest, including topics which are requested by the other graduate programs.
§ The option of a joint masters program will allow students who are in an existing doctoral or masters program on campus to take additional courses and obtain a joint masters' degree. A joint masters' in applied and computational mathematics and an applied science will help students in finding jobs, in applying to graduate programs (both within and outside of the Umass system), and in preparing for a career in all the applied sciences. This option will be attractive to students and will help us draw a talented pool of students into the different graduate programs at UMD.
§ Undergraduate students from mathematics and the physical and life sciences would benefit from a program that allows them to continue their education and finish an advanced degree efficiently in a streamlined curriculum. This option will be open to students from engineering or applied sciences who wish to specialize in computational methods in a joint track, and who wish to pursue a five-year masters program.
§ A multi-track system will also accommodate part-time students who work in local industries and government laboratories, and secondary school teachers who wish to continue their education and complete a master's degree program while working. The multi-track system would be also flexible enough to accommodate students in the existing Math Ed graduate program who are interested in applied mathematics.

Courses offered:

MTH 550: ANALYSIS I: Complex Analysis and Introduction to Real Analysis

MTH 551: ANALYSIS II: Real and Functional Analysis

MTH 555 and 556 Statistics I and II

MTH 560 and 561: Mathematical Methods: Theory of distributions, Cylindrical and Spherical coordinates, boundary value problems, Sturm-Liouville problem, Green's function, Variational methods, Perturbation Methods, Perturbation of eigenvalue problems.

MTH 571 Partial Differential Equations: An introduction to partial differential equations. Topics include: the classification of partial differential equations, the heat equation, the potential equation, separation of variables, Fourier series, the wave equation, and Sturm-Liouville eigenvalue problems.

MTH 572 Numerical Methods for Partial Differential Equations: Numerical methods for solving parabolic, hyperbolic, and elliptic partial differential equations. The course will emphasize the concepts of consistency, convergence and stability. Topics include: implicit and explicit methods, truncation error, Von Neumann stability analysis, and the Lax equivalence theorem.

MTH 573 Numerical Linear Algebra: An introduction to numerical linear algebra. Numerical linear algebra is fundamental to all areas of computational mathematics. This course will cover direct numerical method for solving linear systems and linear least squares problems, stability and conditioning, computational methods for finding eigenvalues and eigenvectors, and iterative methods for both linear systems and eigenvalue problems.

MTH 574 Numerical Optimization: An introduction to constrained and unconstrained optimization. Numerical optimization is an essential tool in a wide variety of applications. The course covers fundamental topics in unconstrained optimization and also methods for solving linear and nonlinear constrained optimization problems.

MTH 582 Advanced Numerical Solution Partial Differential Equations: Numerical methods for solving hyperbolic partial differential equations. The course includes Algorithm development, analysis, implementation and application issues. Methods covered include finite difference, finite element and spectral methods for the solution of time-dependent partial differential equations. We include approximation and stability theory for Fourier spectral methods. This course covers both theoretical and practical aspects as well as an introduction to time stepping methods.

MTH 583 Topics in computational Mathematics: Will include courses in computational fluid dynamics, systems biology, dynamical systems, computational biology and numerical relativity.

MTH 585 Graduate faculty seminars:

MTH 600 Analysis of non-linear dynamical systems: A computational oriented introduction to the analysis of data arising from nonlinear dynamical systems. These dynamical systems include such diverse things as fluid flow, bacterial growth, predator-prey dynamics, traffic flow, growth of tumors, nonlinear circuits, chatter in machine tools, and neural activity. Specific topics to be addressed include the behavior of solutions of finite difference equations, and differential equations, cellular automata, phase space reconstruction for time series, fractal dimension of chaotic attractors, calculation of Lyapunov exponents, control and chaos, synchronization of chaotic systems, and Hamiltonian chaos.