503 - 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. Typically offered Fall.

504 - Real Analysis 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, and the Arzela-Ascoli theorem. Typically offered Fall.

510 - Vector Calculus Calculus of functions of several variables and of vector fields in orthogonal coordinate systems. Optimization problems, implicit function theorem, Green's theorem, Stokes' theorem, divergence theorems. Applications to engineering and the physical sciences. Not open to students with credit in MA 362 or 410. Typically offered Fall Spring Summer.

511 - Linear Algebra With Applications Real and complex vector spaces; linear transformations; Gram-Schmidt process and projections; least squares; QR and LU factorization; diagonalization, real and complex spectral theorem; Schur triangular form; Jordan canonical form; quadratic forms. Typically offered Summer.

514 - Numerical Analysis (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. Typically offered Fall Spring.

515 - Mathematics Of Finance An introduction to the mathematical tools and techniques of modern finance theory, in the context of Black-Scholes option pricing. Brownian motion and its stochastic calculus, Ito's formula, and Feynman-Kac formula. Pricing and hedging of claims on Black-Scholes assets. Incomplete markets. Path-dependent options. Stochastic portfolio optimization. Typically offered Spring.

516 - Advanced Probability And Options With Numerical Methods Stochastic interest rate models. American options from the probabilistic and PDE points of view. Numerical methods for European and American options, including binomial, trinomial, and Monte-Carlo methods. Typically offered Fall.

518 - Advanced Discrete Mathematics 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. Typically offered Spring.

519 - Introduction To Probability (STAT 519) 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. Typically offered Spring Fall.

520 - Boundary Value Problems Of Differential Equations Separation of variables; Fourier series; boundary value problems; Fourier transforms; Bessel functions; Legendre polynomials. Typically offered Fall Spring Summer.

521 - Introduction To Optimization Problems 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. Typically offered Fall.

523 - Introduction To Partial Differential Equations First order quasi-linear equations and their applications to physical and social sciences; the Cauchy-Kovalevsky theorem; characteristics, classification and canonical forms of linear equations; equations of mathematical physics; study of Laplace, wave and heat equations; methods of solution. Typically offered Fall Spring Summer.

525 - Introduction To Complex Analysis 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. Not open to students with credit in MA 425. Typically offered Fall Spring Summer.

527 - Advanced Mathematics For Engineers And Physicists I Courses 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. MA 511 is recommended. Typically offered Fall.

528 - Advanced Mathematics For Engineers And Physicists II 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. MA 510 is recommended. Typically offered Spring.

530 - Functions Of A Complex Variable I 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. More mathematically rigorous than MA 525. Typically offered Fall Spring.

531 - Functions Of A Complex Variable II Advanced topics. Typically offered Fall Spring Summer.

532 - Elements Of Stochastic Processes (STAT 532) 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. Typically offered Spring.

534 - Advanced Analysis For Engineers And Scientists An introduction to nomed linear spaces; Hilbert spaces; linear operations; spectral theory; selected applications. Typically offered Fall.

538 - Probability Theory I (STAT 538) Mathematically rigorous, measure-theoretic introduction to probability spaces, random variables, independence, weak and strong laws of large numbers, conditional expectations, and martingales. Typically offered Spring.

539 - Probability Theory II (STAT 539) Convergence of probability laws; characteristic functions; convergence to the normal law; infinitely divisible and stable laws; Brownian motion and the invariance principle. Typically offered Fall.

540 - Analysis I Metric spaces, compactness and connectedness, sequences and series, continuity and uniform continuity, differentiability, Taylor's Theorem, Riemann-Stieltjes integrals. Typically offered Fall Spring Summer.

541 - Analysis II Sequences and series of functions, uniform convergence, equicontinuous families, the Stone-Weierstrass Theorem, Fourier series, introduction to Lebesgue measure and integration. Typically offered Fall Spring Summer.

542 - Theory Of Distributions And Applications Definition and basic properties of distributions; convolution and Fourier transforms; applications to partial differential equations; Sobolev spaces. Typically offered Fall.

543 - Introduction To The Theory Of Ordinary Differential Equations Existence and uniqueness theorems for ordinary and functional differential equations; linear theory; self-adjoint problems; nonlinear and perturbation theory. Typically offered Fall Spring Summer.

544 - Real Analysis And Measure Theory Metric space topology; continuity, convergence; equicontinuity; compactness; bounded variation, Helly selection theorem; Riemann-Stieltjes integral; Lebesgue measure; abstract measure spaces; LP-spaces; Holder and Minkowski inequalities; Riesz-Fischer theorem. Typically offered Fall Spring.

545 - Functions Of Several Variables And Related Topics Differentation of functions; Besicovitch covering theorem; differentation of one measure with respect to another; Hardy-Littlewood maximal function; functions of several variables; Sobolev spaces. Typically offered Spring.

546 - Introduction To Functional Analysis Fundamentals of functional analysis. Banach spaces, Hahn-Banach theorem. Principle of uniform boundedness. Closed graph and open mapping theorems. Applications. Hilbert spaces. Orthonormal sets. Spectral theorem for Hermitian operators and compact operators. Typically offered Fall.

547 - Analysis For Teachers I Inequalities, sequences, functions, limits. Application to such basic concepts as length and area and their implications for the teacher of mathematics. Open on to students in the M.A.T. program. Typically offered Fall Spring Summer.

548 - Analysis For Teachers II Elementary functions and basic theorems of calculus. Typically offered Fall Spring Summer.

550 - Algebra For Teachers I Definitions and elementary properties of groups, rings, integral domains, fields, vector spaces, and matrices, with major emphasis on the rings of integers, rational numbers, complex numbers, and polynomials. Open only to students in the M.A.T. program. Typically offered Fall Spring Summer.

551 - Algebra For Teachers II A continuation of MA 550. Open only to students in the M.A.T. program. Typically offered Fall Spring Summer.

553 - Introduction To Abstract Algebra Group theory: Sylow theorems, Jordan Hlder 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. Typically offered Fall.

554 - Linear Algebra 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. Typically offered Fall Spring.

555 - Algebraic Coding Theory This course studies error-correcting codes in depth, with an emphasis on their mathematical properties. Included will be discussions of: Hamming codes, Golay codes, BCH codes, cyclic codes, quadratic residue codes, as well as polynomials over finite fields and weight distributions. Typically offered Fall Spring Summer.

556 - Introduction To The Theory Of Numbers Divisibility, congruences, quadratic residues, Diophantine equations, the sequence of primes. Typically offered Fall Spring Summer.

557 - Abstract Algebra I Review of fundamental structures of algebra (groups, rings, fields, modules, algebras); Jordan-Holder 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. Typically offered Fall.

558 - Abstract Algebra II A continuation of MA 557. Typically offered Spring.

560 - Fundamental Concepts Of Geometry Foundations of Euclidean geometry, including a critique of Euclid's "Elements" and a detailed study of an axiom system such as that of Hilbert. Independence of the parallel axiom and introduction to non-Euclidean geometry. Typically offered Fall Spring Summer.

561 - Projective Geometry Ideal elements, duality, harmonic sets, projective metric, theory of conics, involution, imaginary elements. Typically offered Fall Spring Summer.

562 - Introduction To Differential Geometry And Topology 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 Poincare theorems on vector fields. Typically offered Fall.

563 - Advanced Geometry Topics in Euclidean and non-Euclidean geometry. Typically offered Fall Spring Summer.

571 - Elementary Topology Fundamentals of point set topology with a brief introduction to the fundamental group and related topics, topological and metric spaces, compactness, connectedness, separation properties, local compactness, introduction to function spaces, basic notions involving deformations of continuous paths. Typically offered Fall.

572 - Introduction In Algebraic Topology Singular homology theory; Eilenberg-Steenrod asioms; simplicial and cell complexes; elementary homotopy theory; Lefschetz fixed point theorem. Typically offered Spring.

575 - Graph Theory Introduction to graph theory with applications. Typically offered Summer Fall Spring.

580 - History Of Mathematics The origins of mathematical ideas and their evolution over time, from early number systems and the evolution of algebra, geometry, and calculus to twentieth-century results in the foundations of mathematics. Connections between mathematics and society, including the role of applications in the development of mathematical concepts. Typically offered Summer Fall Spring.

581 - Introduction To Logic For Teachers Sentential and general theory of inference and nature of proof: elementary axiom systems. Typically offered Fall Spring Summer.

583 - History Of Elementary Mathematics A survey of elementary mathematics before calculus. An effort will be made to link the history of mathematics to that of other sciences and to the social history of the relevant periods. Some acquaintance with ancient or medieval history of Europe is desirable. Typically offered Fall Spring Summer.

584 - Algebraic Number Theory 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. Typically offered Fall Spring.

585 - Mathematical Logic I Propositional and predicate calculus; the Goedel completeness and compactness theorem, primitive recursive and recursive functions; the Goedel incompleteness theorem; Tarski's theorem; Church's theorem; recursive undecidability; special topics such as nonstandard analysis. Typically offered Fall.

586 - Mathematical Logic II Topics from completeness and compactness theorems; Lowenheim-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. Typically offered Spring.

587 - General Set Theory 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. Typically offered Fall.

Repeatable for Additional Credit: Yes - May be repeated an unlimited number of times

611 - Methods Of Applied Mathematics I Banach and Hilbert spaces; linear operators; spectral theory of compact linear operators; applications to linear integral equations and to regular Sturm-Liouville problems for ordinary differential equations. Prerequisite: MA 511, 544. Typically offered Spring.

615 - Numerical Methods For Partial Differential Equations I (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. Prerequisite: MA 514, 523. Typically offered Spring.

620 - Mathematical Theory Of Optimal Control Existence theorems; the maximum principle; relationship to the calculus of variations; linear systems with quadratic criteria; applications. Offered in alternate years. Prerequisite: MA 544. Typically offered Spring.

626 - Mathematical Formulation Of Physical Problems I Topics from classical and relativistic dynamics; continuum and fluid mechanics; electromagnetics; statistical mechanics; quantum theory; diffusion processes. Typically offered Fall.

631 - Several Complex Variables Power series, holomorphic functions, representation by integrals, extension of functions, holomorphically convex domains. Local theory of analytic sets (Weierstrass preparation theorem and consequences). Functions and sets in the projective space Pn (theorems of Weierstrass and Chow and their extensions). Prerequisite: MA 530. Typically offered Fall Spring Summer.

637 - Stochastic Integration 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. Ito's change of variables formula and its applications. Stochastic differential equations and their properties (existence and uniqueness, Markov properties, flows). Related topics. Prerequisite: MA 539 or STAT 539. Typically offered Spring.

638 - Stochastic Processes I (STAT 638) Advanced topics in probability theory which may include stationary processes, independent increment processes, Gaussian processes; martingales, Markov processes, ergodic theory. Prerequisite: MA 539. Typically offered Spring.

639 - Stochastic Process II (STAT 639) Continuation of MA 638. Typically offered Fall.

642 - Methods Of Linear And Nonlinear Partial Differential Equations I Second order elliptic equations including maximum principles, Harnack inequality, Schauder estimates, and Sobolev estimates. Applications of linear theory to nonlinear equations. Prerequisite: MA 523. Typically offered Fall Spring Summer.

643 - Methods Of Partial Differerntial Equations II Continuation of MA 642. Topics to be covered are Lp theory for solutions of elliptic equations, including Moser's estimates, Aleksandrov maximum principle, and the Calderon-Zygmund theory. Introduction to evolution problems for parabolic and hyperbolic equations, including Galerkin approximation and semigroup methods. Applications to nonlinear problems. Prerequisite: MA 642. Typically offered Spring.

644 - Calculus Of Variations Direct methods; necessary and sufficient conditions for lower semicontinuity of multiple integrals; existence theorems and connections with optimal control theory. Prerequisite: MA 544. Typically offered Fall Spring Summer.

646 - Banach Algebras And C*-Algebras 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. Prerequisite: MA 546. Typically offered Summer Fall Spring.

647 - Linear Partial Differential Equations I 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. Prerequisite: MA 542, 546. Typically offered Fall Spring Summer.

648 - Linear Partial Differential Equations II Continuation of MA 647. Specialized topics in partial differential equations, varied from time to time. Prerequisite: MA 647. Typically offered Fall Spring Summer.

650 - Commutative Algebra The study of those rings of importance in algebraic and analytic geometry and algebraic number theory. Prerequisite: MA 558. Typically offered Fall Spring Summer.

651 - Theory Of Rings And Algebras Advanced topics in associative ring theory. Prerequisite: MA 558. Typically offered Fall Spring Summer.

661 - Modern Differential Geometry Topics chosen by the instructor. Prerequisite: MA 544, 554. Typically offered Fall Spring Summer.

663 - Algebraic Curves And Functions I Algebraic functions of one variable from the geometric, algebraic, or function-theoretic points of view. Riemann-Roch theorem, differentials. Prerequisite: MA 558. Typically offered Fall Spring Summer.

664 - Algebraic Curves And Functions II Continuation of MA 663. Topics chosen by the instructor. Prerequisite: MA 663. Typically offered Fall Spring Summer.

665 - Algebraic Geometry Topics of current interest will be chosen by the instructor. Prerequisite: MA 650 or 663. Typically offered Fall Spring Summer.

672 - Algebraic Topology I A continuation of MA 572: cohomology; homotopy groups; fibrations; further topics. Prerequisite: MA 572. Typically offered Fall Spring Summer.

673 - Algebraic Topology II A sequel to MA 672 covering further advanced topics in algebraic and differential topology such as K-theory and characteristic classes. Prerequisite: MA 672. Typically offered Fall Spring Summer.

684 - Class Field Theory Ideles, adeles, L-functions, Artin symbol, reciprocity, local and global class fields, Kronecker-Weber Theorem. Prerequisite: MA 584. Typically offered Fall Spring.

687 - Axiomatic Set Theory Topics will include Godel's results on the consistency of the axiom of choice and the continuum hypothesis, and Cohen's results on the independence of these axioms. Prerequisite: MA 587. Typically offered Fall Spring Summer.