IX. Graduate Math Courses

The department offers a wide range of graduate courses in a large variety of areas of mathematics. Following is a sample. In addition to the regular course offerings, numerous advanced topics and seminars are given each year. For a complete list and course descriptions please contact the department.

General Courses

503-Abstract Algebra.

Prerequisite : two upper-division mathematics courses, one on linear algebra and one on abstract algebra.

Group theory: definitions, examples, subgroups, quotient groups, homomorphisms, and isomorphism theorems. Ring theory: definitions, examples, homomorphisms, ideals, quotient rings, fraction fields, polynomial rings, Euclidean domains, and unique factorization domains. Field theory: algebraic field extensions, straightedge and compass constructions.

504-Real Analysis.

Prerequisite : two upper-division mathematics or engineering courses.

Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions, the Riemann- Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass Theorem, equicontinuity, the Arzela-Ascoli Theorem.

510-Vector Calculus.

Prerequisite : MA 262 or 272. Not open to students with credit in MA 362 or 410.

Calculus of functions of several variables and of vector fields in orthogonal coordinate systems; optimization problems; the implicit function theorem; Green's, Stokes', and the divergence theorems; applications to engineering and the physical sciences.

511-Linear Algebra with Applications.

Prerequisite : MA 262

Real and complex vector spaces; linear transformations; Gram-Schmidt process and projections; unitary and orthogonal diagonalization; Jordan canonical form; quadratic forms.

514-Numerical Analysis.

Prerequisite: Authorized equivalent courses or consent of instructor may be used in satisfying course pre-and co-requisites. (CS 514)

Iterative methods for solving nonlinear; linear difference equations, applications to solution of polynomial equations; differentiation and integration formulas; numerical solution of ordinary differential equations; roundoff error bounds.

515-Mathematics of Finance.

Prerequisite: MA/STAT 519 (or equivalent) or concurrent enrollment, MA 261 (or equivalent) and MA 262 or MA266 (or equivalent); or consent of instructor.

This is an introduction to the mathematical tools and techniques of modern finance theory. The market model will be restricted to the Black-Scholes world. Basic mathematical descriptions of financial instruments, such as stock prices, contingent claims and option prices will be given. Arbitrage, market completeness and hedging strategies will also be given. Some necessary background in stochastic calculus such as stochastic integrals, stochastic differential equations and their relations with partial differential equations will be provided.

516-Advanced Probability and Options with Numerical Methods.

Prerequisite: MA 515 or consent of instructor.

Stochastic interest rate models. American options from the probabilistic and partial differential equations point of view. Numerical methods for European and American options including binomial, trinomial and Monte-Carlo methods.

518-Advanced Discrete Mathematics.

Prerequisite: MA 262 or equivalent or consent of instructor.

The course covers mathematics useful in analyzing computer algorithms. Topics include recurrence relations, evaluation of sums, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods.

519-Intro to Probability (STAT 519).

Prerequisite: MA 510; or corequisite: MA 341 or 440.

Algebra of sets, sample spaces, combinatorial problems, independence, random variables, distribution functions, moment generating functions, special continuous and discrete distributions, distribution of a function of a random variable, limit theorems.

520-Boundary Value Problems of Differential Equations.

Prerequisite: MA 303 or 304, or equivalent.

Separation of variables; Fourier series; boundary value problems; Fourier transforms; Bessel functions; Legendre polynomials.

521-Intro to Optimization Problems.

Prerequisite: MA 362, 410, or 510, and 351 or 511.

Necessary and sufficient conditions for local extrema in programming problems and in the calculus of variations. Control problems; statement of maximum principles and applications. Discrete control problems.

523-Intro to Partial Differential Equations.

Prerequisite: MA 266 or 366, MA 440 and MA 362 410 or 510.

First order quasi-linear equations and their application to physical and social sciences; the Cauchy- Kovalevsky theorem; characteristics, classification, and canonical form of linear equations; equations of mathematical physics; study of the Laplace, wave and heat equations; methods of solution.

524-Finite Element Method for Partial Differential Equations.

Prerequisite: MA 362, 351, 523 or equivalent, or consent of instructor.

Mathematical aspects of the finite element method applied to elliptic, parabolic and hyperbolic partial differential equations. Topics in approximation theory in two dimensions and the numerical solution of sparse linear systems. Other topics at the discretion of the instructor. [At the present time, new courses on this topic are taught instead of 524: 598C in Fall and 598D in Spring.]

525-Intro to Complex Analysis.

Prerequisite: MA 362, 410, or 510.

Complex numbers and complex-valued functions of one complex variable; differentiation and contour integration; Cauchy's theorem; Taylor and Laurent series; residues; conformal mapping; applications.

527-Advanced Mathematics for Engineers and Physicists I.

Prerequisite: MA 262; MA 511 is recommended. MA 527 and 528 constitute a two-semester sequence covering a broad range of subjects useful in early graduate engineering courses.

Topics in MA 527 include linear algebra, systems of ordinary differential equations, Laplace transforms., Fourier series and transforms, and partial differential equations.

528-Advanced Mathematics for Engineers and Physicists II.

Prerequisite: MA 262; MA 510 is recommended. MA 527 and 528 constitute a two-semester sequence covering a broad range of subjects useful in early graduate engineering courses.

Topics in MA 528 include divergence theorem, Stokes' theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory. 530-Functions of a Complex Variable I. Prerequisite or corequisite : MA 544. (More mathematically rigorous than MA 525). Complex numbers and complex-valued functions of one complex variable; differentiation and contour integration; Cauchy's theorem; Taylor and Laurent series; residues; conformal mapping; special topics. 531-Functions of a Complex Variables II. Prerequisite : MA 530. Advanced topics.

532-Elements of Stochastic Processes (STAT 532).

Prerequisite: MA 519.

A basic course in stochastic models, including discrete and continuous time Markov chains and Brownian motion, as well as an introduction to topics such as Gaussian processes, queues, epidemic models, branching processes, renewal processes, replacement, and reliability problems.

533-Fractals and Chaos with Applications in the Earth Sciences.

Prerequisite: MA 262 or 265/266 or 351/366.

An introduction to the theory and phenomenology of nonlinear dynamics, chaos, self-similarity, and fractal geometry, for advanced undergraduate and beginning graduate students. Includes applications of this theory to geophysical problems.

538-Probability Theory I (STAT 538).

Prerequisite : MA 544.

Mathematically rigorous, measure-theoretic introduction to probability spaces, radom variables, independence, weak and strong laws of large numbers, conditional expectations and martingales.

539-Probability Theory II (STAT 539).

Prerequisite : MA 530 and 538.

Convergence of probability laws, characteristic functions; convergence to the normal law; infinitely divisible and stable laws; Brownian motion and the invariance principle. 542-Theory of Distributions and Applications. Prerequisite : MA 510 and 525 or equivalent. Definition and basic properties of distributions: convolution and Fourier transforms; applications to partial differential equations; Sobolev spaces. 543-Intro to the Theory of Ordinary Differential Equations. Prerequisite : MA 361. Existence and uniqueness theorems for ordinary and functional differential equations; linear theory; self- adjoint problems; nonlinear and perturbation theory.

544-Real Analysis and Measure Theory.

Prerequisite : MA 442 or 504.

Metric space topology; continuity, convergence; equicontinuity; compactness; bounded variation, Helly selection theorem; Riemann-Stieltjes integral; Lebesgue measure; abstract measure spaces; Lp-spaces; H¬older and Minkowski inequalities; Riesz-Fischer theorem.

545-Functions of Several Variables and Related Topics.

Prerequisite: MA 544.

Differentiation of functions; Besicovitch covering theorem; differentiation of one measure with respect to another; Hardy-Littlewood maximal function; functions of several variables; Sobolev spaces.

546-Intro to Functional Analysis.

Prerequisite : MA 544.

Fundamentals of functional analysis; Banach spaces. Hahn-Banach theorem; principle of uniform boundedness; closed graph and open mapping theorem; applications; Hilbert spaces; orthonormal sets; spectral theorem for Hermitian operators and for compact operators. 553-Intro to Abstract Algebra. Prerequisite : MA 453. Group theory: Sylow theorems, Jordan-H¬older theorem, solvable groups. Ring theory unique factorization in polynomial rings and principal ideal domains. Field theory: ruler and compass constructions, roots of unity, finite fields, Galois theory, solvability of equations by radicals.

554-Linear Algebra.

Prerequisite: MA 350 or equivalent.

Review of basics: vector spaces; dimension; linear maps; matrices; determinants; linear equations. Bilinear forms; inner product spaces; spectral theory; eigenvalues. Modules over a principal ideal domain; finitely generated abelian groups; Jordan and rational canonical forms for a linear transformation.

557-Abstract Algebra I.

Prerequisite: MA 454.

Review of fundamental structures of algebra (groups, rings, fields, modules, algebras); Jordan-H¬older and Sylow theorems; Galois theory; bilinear forms; modules over principal ideal domains; Artinian rings and semisimple modules; Polynomial and power series rings; Noetherian rings and modules; localization; integral dependence; rudiments of algebraic geometry and algebraic number theory; ramification theory.

558-Abstract Algebra II.

Prerequisite: MA 557.

A continuation of MA 557. This course is usually an introduction to commutative algebra. Noetherian rings and modules, localization, integral dependence, Going Up and Going Down Theorems, Hilbert's Basis Theorem and Nullstellensatz, Noether Normalization Theorem, and Primary Decomposition.

560-Fundamental Concepts of Geometry.

Prerequisite : MA 261.

Foundations of Euclidean geometry, including a critique of Euclid's ÒElementsÓ and a detailed study of an axiom system such as that of Hilbert. Independence of the parallel axiom and introduction to non-Euclidean geometry.

562-Intro to Differential Geometry and Topology.

Prerequisite: MA 351 and 442.

Smooth manifolds; tangent vectors; inverse and implicit function theorems; submanifolds; vector fields; integral curves; differential forms; the exterior derivative; DeRham cohomology groups; surfaces in E³; Gaussian curvature; two-dimensional Riemannian geometry; Gauss-Bonnet and Poincar«e theorems on vector fields.

571-Elementary Topology.

Prerequisite: MA 440 or 504.

Fundamentals of point set topology with an introduction to the fundamental group and related topics: topological spaces, product topology, quotient topology, compactness and connectedness, function spaces, homotopy of paths and the fundamental group, covering spaces, Seifert-van Kampen theorem.

572-Intro to Algebraic Topology.

Prerequisite: MA 571.

Singular homology theory; Eilenberg-Steenrod axioms; simplicial and cell complexes; elementary homotopy theory; Lefschetz fixed point theorem.

575-Linear Graph Theory.

Prerequisite: MA 351 or equivalent.

Introduction to graph theory with applications.

584-Algebraic Number Theory.

Prerequisite: MA 553, 554. Authorized equivalent courses or consent of instructor may be used in satisfying course pre-and co-requisites.

Dedekind domains, norm, discriminant, different, finiteness of class number, Dirichlet unit theorem, quadratic and cyclotomic extensions, quadratic reciprocity, decomposition and inertia groups, completions and local fields.

585-Mathematical Logic I.

Prerequisite: MA 385 or 453.

Propositional and predicate calculus; the G¬odel completeness and compactness theorems; primitive recursive and recursive functions; the G¬odel incompleteness theorem; Tarski's theorem; Church's theorem; rescursive undecidability, special topics such as nonstandard analysis.

586-Mathematical Logic II.

Prerequisite: MA 585.

Topics from completeness and compactness theorems; L¬owenheim-Skolem theorems; omitting types and interpolation theorems; homogeneous and saturated models; elimination of quantifiers; Boolean algebras; complete, model complete and decidable theories; ultraproducts; nonstandard analysis.

587-General Set Theory.

Prerequisite: MA 387 or 441 or 453.

Set algebra; functions and relations; ordering relations; transfinite induction; cardinal and ordinal numbers; the axiom of choice; maximal principles; the continuum hypothesis; the axiom of constructibility; applications to algebra, analysis, and topology.

598-Topics in Mathematics

Supervised reading courses as well as dual-level special topics courses are given under this number.

611-Methods of Applied Mathematics I.

Prerequisite: MA 511 or equivalent and MA 544.

Banach and Hilbert spaces; linear operators; spectral theory of compact linear operators; applications to linear integral equations and to regular Sturm-Louville problems for ordinary differential equations.

615-Numerical Methods For Partial Differential Equations I.

Prerequisite: MA 514, 523. Authorized equivalent courses or consent of instructor may be used in satisfying course pre-and co-equisites. (CS 615)

Finite element method for elliptic partial differential equations; weak formulation; finite-dimensional approximations; error bounds; algorithmic issues; solving sparse linear systems; finite element method for parabolic partial differential equations; backward difference and Crank-Nicholson time-stepping; introduction to finite difference methods for elliptic, parabolic, and hyperbolic equations; stability, consistency, and convergence; discrete maximum principles.

620-Mathematical Theory of Optimal Control.

Prerequisite: MA 544

Existence theorems; the maximum principle; relationship to the calculus of variations; linear systems with quadratic criteria; applications.

626-Mathematical Formulation of Physical Problems I

Topics from classical and relativistic dynamics; continuum and ßuid mechanics; electromagnetics; statistical mechanics; quantum theory; diffusion processes.

631-Several Complex Variables.

Prerequisite: MA 530.

Power series, holomorphic functions, representation by integrals, extension of functions holomorphically to convex domains. Local theory of analytic sets (Weierstrass preparation theorem and consequences). Functions and sets in the projective space P ⁿ (theorems of Weierstrass and Chow and their extensions).

637-Stochastic Integration.

Prerequisite: MA/STAT 539 or equivalent, or consent of instructor.

Review of martingale theory, including the Martingale Convergence Theorem, Doob's Optional Sampling Theorem, Doob's maximal quadratic inequality. Brownian motion and related processes, with emphasis on properties relevant to stochastic integration (sample path properties, martingale properties, quadratic variation). Stochastic integration and its properties. Itöo change of variables formula and its applications. Stochastic differential equations and their properties (existence and uniqueness, Markov properties, ßows). Related topics.

638-Stochastic Processes I (STAT 638).

Prerequisite: MA 539.

Advanced topics in probability theory which may include stationary processes, independent increment processes, Gaussian processes; martingales, Markov processes, ergodic theory.

639-Stochastic Processes II (STAT 639)

Continuation of MA 638.

642-Methods of Linear and Nonlinear Partial Differential Equations I.

Prerequisite: MA 523 and 611.

Second order elliptic equations including maximum principles, Harnack inequality, Schauder estimates, and Sobolev estimates. Applications of linear theory to nonlinear equations.

643-Methods of Linear and Nonlinear Partial Differential Equations II.

Prerequisite: MA 642.

Continuation of MA 642. Topics to be covered are: L^p theory for solutions of elliptic equations including Moser's estimates, Aleksandrov maximum principle, Calderon-Zygmund theory. Introduction to evolution problems for parabolic and hyperbolic equations including Galerkin approximation and semigroup methods. Applications to nonlinear problems.

644-Calculus of Variations.

Prerequisite: MA 544.

Direct methods; necessary and sufficient conditions for lower semicontinuity of multiple integrals; existence theorems and connections with optimal control theory.

646-Banach Algebras and C*-algebras.

Prerequisite: MA 546 or equivalent.

Banach algebras, Gelfand theory, the commutative Gelfand-Naimark theorem and applications to normal operators. C*-algebras and representations, the noncommutative Gelfand-Naimark theorem, von Neumann algebras, and Murray-von Neumann equivalence. Some operator theory or other topics may be included as time permits.

647-Linear Partial Differential Equations I.

Prerequisite: MA 542 and 546.

Cauchy-Kovalevaka and Holmgren's theorems. Cauchy and mixed problems for hyperbolic systems. Mixed problems for parabolic equations. Boundary value problems of the Lopatinski type for elliptic equations. Construction of kernels, regularity of the solutions in the interior and up to the boundary.

648-Linear Partial Differential Equations II.

Prerequisite: MA 647.

Continuation of MA 647. Specialized topics in partial differential equations, varied from time to time.

650-Commutative Algebra.

Prerequisite: MA 558. The study of those rings of importance in algebraic and analytic geometry and algebraic number theory. Topics are at the discretion of the instructor. Possible topics include regular rings and Cohen-Macaulay rings, homological algebra, ßatness, Gorenstein rings, projective dimension.

651-Theory of Rings and Algebras.

Prerequisite: MA 558.

Advanced topics in associative ring theory.

661-Modern Differential Geometry.

Prerequisite: MA 544, 554.

Differential manifolds, tangent vectors, vector fields and differential forms, tensor fields, DeRham's theorems, imbedding theorems, Riemannian geometry, curvatures, harmonic integrals.

663-Algebraic Curves and Functions I.

Prerequisite: MA 558.

Algebraic functions of one variable from the geometric, algebraic, or function-theoretic points of view. Riemann-Roch theorem, differentials.

664-Algebraic Curves and Functions II.

Prerequisite: MA 663.

Continuation of MA 663. Topics chosen by the instructor.

665-Algebraic Geometry.

Prerequisite: MA 650 or 663.

Topics of current interest will be chosen by the instructor.

672-Algebraic Topology I.

Prerequisite: MA 572.

A continuation of MA 572: cohomology; homotopy groups; fibrations; further topics.

673-Algebraic Topology II.

Prerequisite: MA 672.

A sequel to MA 672 covering further advanced topics in algebraic and differential topology such as K-theory and characteristics classes.

684-Class Field Theory.

Prerequisite: MA 584. Authorized equivalent courses or consent of instructor may be used in satisfying course pre-and co-requisites.

Ideles, adeles, L-functions, Artin symbol, reciprocity, local and global class fields, Kronecker-Weber Theorem.

690-Topics in Algebra

691-Topics in Logic and Foundations

692-Topics in Applied Mathematics

693-Topics in Analysis

694-Topics in Differential Equations

696-Topics in Geometry

697-Topics in Topology

699-Research Ph.D. Thesis