AMCS 602-603. Algebraic Techniques for Applied Mathematics and Computational Science. Staff. First semester: We begin with an introduction to group theory. The emphasis is on groups as symmetries and transformations of space. After an introduction to abstract groups and the basic facts about finite groups, we turn our attention to compact Lie groups and their representations. In the latter connection we explore the connections between orthogonal polynomials, classical transcendental functions and group representations. This unit is completed with a discussion of finite groups and their applications in coding theory. Second semester: We turn to linear algebra and the structural properties of linear systems of equations relevant to their numerical solution. In this context we introduce eigenvalues and the spectral theory of matrices. Methods appropriate to the numerical solution of very large systems are discussed. We then turn to the problem of solving systems of polynomial equations, introducing basic properties of rings, ideals and modules. This allows us to define Grobner bases and their use in the numerical solution of algebraic equations. The theoretical content of this course is illustrated and supplemented throughout the year with substantial computational examples and assignments.
AMCS/MATH 608 Analysis I. (Prereqs: 508-509, usually offered in the Fall) Complex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. Introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C0 and the Riesz-Markov theorem, Lp-spaces, Fubini Theorem.
AMCS/MATH 609 Analysis II. (Prereqs: 508-509 and 608, usually offered in the Fall) Real analysis: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L2-theory of the Fourier transform. Functional analysis: normed linear spaces, convexity, the Hahn-Banach theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, Lp-theory for the Fourier transform.
AMCS/MATH 610 Functional Analysis. (Prereq: 608 or 609, some elementary complex analysis is essential, usually offered in the Spring) Convexity and the Hahn Banach Theorem. Hilbert Spaces, Banach Spaces, and examples: Sobolev spaces, Holder spaces. The uniform boundedness principle, Baire category theorem, bounded operators, open mapping theorem, closed graph theorem and applications. The concepts of duality and dual spaces. The Riesz theory of compact operators and Fredholm theory. Functional calculus and elementary Spectral Theory. Interpolation theorems. Applications to partial differential equations and approximation theory.
Through the Math and Statistics departments we also offer a one year course on Probability and Stochastic processes:
STAT 530-531/MATH 546-547. Probability, and Stochastic Processes for Applied Mathematics and Computational Science. Staff. Probability and statistics are the language most appropriate to the discussion of experimental data. This course introduces and places on a firm foundation, methods and ideas arising in probability theory and the theory of stochastic processes, and their applications to problems of empirical science. First semester: Basic concepts of measure theory, as measurement theory, probability theory, random variables, expectation, and independence, the weak and strong laws of large numbers, the central and Poisson limit theorems. Second semester: Measure theory revisited, conditional expectations, Martingales, Brownian motion, diffusion processes and stochastic integration and differential equations. Applications to the analysis of partial differential equations.
The following courses are sometimes taken by students in the AMCS Masters program (generally 400-level MATH courses cannot be used by Math majors seeking a masters degree):
AMCS 510. (x-listed MATH 410) Complex Analysis. (C) Staff. Prerequisite(s): MATH 241 or permission of instructor. Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.
AMCS 520. (x-listed MATH 420) Ordinary Differential Equations. (C) Staff. Prerequisite(s): MATH 241 or permission of instructor. After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.
AMCS 525. (x-listed MATH 425) Partial Differential Equations. (A) Staff. Prerequisite(s): MATH 241 or permission of instructor. Knowledge of PHYS 150-151 will be helpful. Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.
AMCS 530. (x-listed MATH 430) Introduction to Probability. (M) Staff. Prerequisite(s): MATH 240.
Random variables, events, special distributions, expectations, independence, law of large numbers, introduction to the central limit theorem, and applications.
AMCS 532. (x-listed MATH 432) Game Theory. (C) Staff. A mathematical approach to game theory, with an emphasis on examples of actual games. Topics will include mathematical models of games, combinatorial games, two person (zero sum and general sum) games, non-cooperating games and equilibria.