# Graduate 500-Level Math Courses

The listing below describes all graduate mathematics courses in the University Park catalog. Not all courses are offered each year.

Some 400-level courses may also count for graduate credit.

MATH 501 REAL ANALYSIS (ANALYSIS A) (3) Lebesgue measure theory. Measurable sets and measurable functions. Lebesgue integration, convergence theorems. Lp spaces. Decomposition and differentiation of measures. Convolutions. The Fourier transform. Prerequisite: MATH 404

MATH 502 COMPLEX ANALYSIS (ANALYSIS B) (3) Complex numbers. Holomorphic functions. Cauchy's theorem. Meromorphic functions, Laurent expansions, residue calculus, conformal maps, topology of the plane. Prerequisite: MATH 404

MATH 503 FUNCTIONAL ANALYSIS (ANALYSIS C) (3) Banach and Hilbert spaces, dual spaces, linear operators, distributions, weak derivatives, Sobolev spaces, applications to linear differential equations. Prerequisite: MATH 501

MATH 504 ANALYSIS IN EUCLIDEAN SPACE (3) The Fourier transform in L1 and L2 and applications, interpolation of operators, Riesz and Marcinkiewics theorems, singular integral operators. Prerequisite: MATH 502

MATH 505 MATHEMATICAL FLUID MECHANICS (3) Kinematics, balance laws, constitutive equations; ideal fluids, viscous flows, boundary layers, lubrication; gas dynamics. Prerequisite: MATH 402 or MATH 404

MATH 506 ERGODIC THEORY (3) Measure-preserving transformations and flows, ergodic theorems, ergodicity, mixing, weak mixing, spectral invariants, measurable partitions, entropy, ornstein isomorphism theory. Prerequisite: MATH 502

MATH 507 DYNAMICAL SYSTEMS I (3) Fundamental concepts; extensive survey of examples; equivalence and classification of dynamical systems, principal classes of asymptotic invariants, circle maps. Prerequisite: MATH 501

MATH 508 DYNAMICAL SYSTEMS II (3) Hyperbolic theory; stable manifolds, hyperbolic sets, attractors, Anosov systems, shadowing, structural stability, entropy, pressure, Lyapunov characteristic exponents and non-uniform hyperbolicity. Prerequisite: MATH 507

MATH 511 ORDINARY DIFFERENTIAL EQUATIONS I (3) Existence and uniqueness, linear systems, series methods, Poincare-Bendixson theory, stability. Prerequisite: MATH 411 OR MATH 412

MATH 512 ORDINARY DIFFERENTIAL EQUATIONS II (3) Floquet theory, regular and singular boundary value problems, Green's functions, eigenfunction expansions. Prerequisite: MATH 511

MATH 513 PARTIAL DIFFERENTIAL EQUATIONS I (3) First order equations, the Cauchy problem, Cauchy-Kowalevski theorem, Laplace equation, wave equation, heat equation. Prerequisite: MATH 411 or MATH 412

MATH 514 PARTIAL DIFFERENTIAL EQUATIONS II (3) Sobolev spaces and Elliptic boundary value problems, Schauder estimates. Quasilinear symmetric hyperbolic systems, conservation laws. Prerequisite: MATH 502, MATH 513

MATH 515 CLASSICAL MECHANICS AND VARIATIONAL METHODS (3) Introduction to the calculus of variations, variational formulation of Lagrangian mechanics, symmetry in mechanical systems, Legendre transformation, Hamiltonian mechanics, completely integrable systems. Prerequisite: MATH 401, MATH 411, OR MATH 412

MATH 516 STOCHASTIC PROCESSES (3) Markov chains; generating functions; limit theorems; continuous time and renewal processes; martingales, submartingales, and supermartingales; diffusion processes; applications. Prerequisite: MATH 416

MATH 517(STAT) PROBABILITY THEORY (3) Measure theoretic foundation of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics. Prerequisite: MATH 403

MATH 518(STAT) PROBABILITY THEORY (3) Measure theoretic foundation of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics. Prerequisite: STAT 517

MATH 519(STAT) TOPICS IN STOCHASTIC PROCESSES (3) Selected topics in stochastic processes, including Markov and Wiener processes; stochastic integrals, optimization, and control; optimal filtering. Prerequisite: STAT 516, STAT 517

MATH 520 INTRODUCTION TO OPERATOR ALGEBRAS (3) Basic properties of C*-algebras, classification of von Neumann algebras into types, functionals and representations, tensor products, automorphisms, crossed products. Prerequisite: MATH 503

MATH 521 COMPLEX ANALYSIS: THEORY AND APPLICATIONS I (3) Conformal mappings, Schwarz-Cristoffel transformations, Dirichlet and Neumann problems, electrostatics and fluid flow, transform methods, asymptotic methods, Runge approximation theorems. Prerequisite: MATH 502

MATH 523 NUMERICAL ANALYSIS I (3) Approximation and interpolation, numerical quadrature, direct methods of numerical linear algebra, numerical solutions of nonlinear systems and optimization. Prerequisite: MATH 456

MATH 524 NUMERICAL LINEAR ALGEBRA (ALGEBRA C) (3) Matrix decompositions. Direct methods of numerical linear algebra. Eigenvalue computations. Iterative methods. Prerequisite: MATH 535

MATH 527 METRIC AND TOPOLOGICAL SPACES (3) Metric spaces, continuous maps, compactness, connectedness, and completeness. Topological spaces, products, quotients, homotopy, fundamental group, simple applications. Prerequisite: MATH 429

MATH 528 DIFFERENTIABLE MANIFOLDS (3) Smooth manifolds, smooth maps, Sards' theorem. The tangent bundle, vector fields, differential forms, integration on manifolds, de Rham cohomology; simple applications. Lie groups, smooth actions, quotient spaces, examples. Prerequisite: MATH 527

MATH 529 ALGEBRAIC TOPOLOGY (3) Manifolds, Poincare duality, vector bundles, Thom isomorphism, characteristic classes, classifying spaces for vector bundles, discussion of bordism, as time allows. Prerequisite: MATH 528

MATH 530 DIFFERENTIAL GEOMETRY (3) Distributions and Frobenius theorem, curvature of curves and surfaces, Riemannian geometry, connections, curvature, Gauss-Bonnet theorem, geodesic and completeness. Prerequisite: MATH 528

MATH 531 DIFFERENTIAL TOPOLOGY (3) DeRham's theorem, geometry of smooth mappings, critical values, Sard's theorem, Morse functions, degree of mappings, smooth fiber bundles. Prerequisite: MATH 528

MATH 533 LIE THEORY I (3) Lie groups, lie algebras, exponential mappings, subgroups, subalgebras, simply connected groups, adjoint representation, semisimple groups, infinitesimal theory, Cartan's criterion. Prerequisite: MATH 528

MATH 534 LIE THEORY II (3) Representations of compact lie groups and semisimple lie algebras, characters, orthogonality, Peter-Weyl theorem, Cartan-Weyl highest weight theory. Prerequisite: MATH 533

MATH 535 LINEAR ALGEBRA (ALGEBRA A) (3) Vector spaces. Linear transformations. Bilinear forms. Canonical forms for linear transformations. Multilinear algebra. Prerequisite: MATH 435 and a course in linear algebra

MATH 536 ABSTRACT ALGEBRA (ALGEBRA B) (3) Permutation groups, Sylow theorems. Rings, ideals, unique factorization domains, finitely generated modules. Fields, algebraic and transcendental field extensions, Galois theory. Prerequisite: MATH 435

MATH 537 FIELD THEORY (3) Finite and infinite algebraic extensions; cyclotomic fields; transcendental extensions; bases of transcendence, Luroth's theorem, ordered fields, valuations; formally real fields. Prerequisite: MATH 536

MATH 538 COMMUTATIVE ALGEBRA (3) Topics selected from Noetherian rings and modules, primary decompositions, Dedekind domains and ideal theory, other special types of commutative rings or fields. Prerequisite: MATH 536

MATH 542 GROUP THEORY I (3) Topics selected by instructor from abelian, solvable, and nilpotent groups; finite presentations; free products; group extensions; group representations. Prerequisite: MATH 535

MATH 547 ALGEBRAIC GEOMETRY I (3) Affine and projective algebraic varieties; Zariski topology; Hilbert Nullstellensatz; regular functions and maps; birationality; smooth varieties normalization; dimension. Prerequisite: MATH 536

MATH 548 ALGEBRAIC GEOMETRY II (3) Topics may include algebraic curves, Riemann-Roch theorem, linear systems and divisors, intersectino theory, schemes, sheaf cohomology, algebraic groups. Prerequisite: MATH 547

MATH 550 (CSE) NUMERICAL LINEAR ALGEBRA (3) Solution of linear systems, sparse matrix techniques, linear least squares, singular value decomposition, numerical computation of eigenvalues and eigenvectors. Prerequisite: MATH 441 or MATH 456

MATH 551 (CSE) NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (3) Methods for initial value and boundary value problems; convergence and stability analysis, automatic error control, stiff systems, boundary value problems. Prerequisite: MATH 451 OR MATH 456

MATH 552 (CSE) NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS (3) Finite difference methods for elliptic, parabolic, and hyperbolic differential equations; solutions techniques for discretized systems; finite element methods for elliptic problems. Prerequisite: MATH 402 OR MATH 404; MATH 451 OR MATH 456

MATH 553 (CSE) INTRODUCTION TO APPROXIMATION THEORY (3) Interpolation; remainder theory; approximation of functions; error analysis; orthogonal polynomials; approximation of linear functionals; functional analysis applied to numerical analysis. Prerequisite: MATH 401, 3 credits of computer science and engineering

MATH 554 APPROXIMATION THEORY (3) Approximation in normed spaces; existence, uniqueness, characterization, computation of best approximations; error bounds; degree of approximation; approximation of linear functionals. Prerequisite: MATH 451 OR MATH 456; MATH 501

MATH 555 (CSE) NUMERICAL OPTIMIZATION TECHNIQUES (3) unconstrained and constrained optimization methods, linear and quadratic programming, software issues, ellipsoid and Karmarkar's algorithm, global optimization, parallelism in optimization. Prerequisite: MATH 456

MATH 556 (CSE) FINITE ELEMENT METHODS (3) Sobolev spaces, variational formulations of boundary value problems; piecewise polynomial approximation theory, convergence and stability, special methods and applications. Prerequisite: MATH 502, MATH 552

MATH 557 MATHEMATICAL LOGIC (3) The predicate calculus; completeness and compactness; Godel's first and second incompleteness theorems; introduction to model theory; introduction to proof theory. Prerequisite: MATH 435 OR MATH 457

MATH 558 FOUNDATIONS OF MATHEMATICS I (3) Decidability of the real numbers; computability; undecidability of the natural numbers; models of set theory; axiom of choice; continuum hypothesis. Prerequisite: any 400 level math course

MATH 559 RECURSION THEORY I (3) Recursive functions; degrees of unsolvability; hyperarithmetic theory; applications to Borel combinatorics. Computational complexity. Combinatory logic and the Lambda calculus. Prerequisite: MATH 459, MATH 557, OR MATH 558

MATH 561 SET THEORY I (3) Models of set theory. Inner models, forcing, large cardinals, determinacy. Descriptive set theory. Applications to analysis. Prerequisite: MATH 557 OR MATH 558

MATH 565 FOUNDATIONS OF MATHEMATICS II (3) Subsystems of second order arithmetic; set existence axioms; reverse mathematics; foundations of analysis and algebra. Prerequisite: MATH 557, MATH 558

MATH 567 NUMBER THEORY I (3) Congruences, quadratic residues, arithmetic functions, partitions, classical multiplicative ideal theory, valuations and p-adic numbers; primes in arithmetic progression, distribution of primes. Prerequisite: MATH 421

MATH 568 NUMBER THEORY II (3) Congruences, quadratic residues, arithmetic functions, partitions, classical multiplicative ideal theory, valuations and p-adic numbers; primes in arithmetic progression, distribution of primes. Prerequisite: MATH 421

MATH 569 ALGEBRAIC NUMBER THEORY I (3) Dedekind rings; cyclotomic and Kummer extensions; valuations; ramification, decomposition, inertial groups; Galois extensions; locally compact groups of number theory. Prerequisite: MATH 536, MATH 568

MATH 570 ALGEBRAIC NUMBER THEORY II (3) Topics chosen from class field theroy; integral quadratic forms; algebraic and arithmetic groups; algebraic function of one variable. Prerequisite: MATH 569

MATH 571 ANALYTIC NUMBER THEORY I (3) Improvements of the prime number theorem, L-functions and class numbers, asymptotic and arithmetic properties of coefficients of modular forms. Prerequisite: MATH 421

MATH 572 ANALYTIC NUMBER THEORY II (3) Distribution of primes, analytic number theory in algebraic number fields, transcendental numbers, advanced theory of partitions. Prerequisite: MATH 571

MATH 574 TOPICS IN LOGIC AND FOUNDATIONS (3 - 6) Topics in mathematical logic and the foundations of mathematics. Prerequisite: MATH 558

MATH 577 (M E) STOCHASTIC SYSTEMS FOR SCIENCE AND ENGINEERING (3) The course develops the theory of stochastic processes and linear and nonlinear stochastic differential equations for applications to science and engineering. Prerequisite: MATH 414 or MATH 418; M E 550 or MATH 501

MATH 580 INTRODUCTION TO APPLIED MATHEMATICS I (3) A graduate course of fundamental techniques including tensor, ordinary and partial differential equations, and linear transforms. Prerequisite: Basic knowledge of linear algebra, vector calculus and ODE, MATH 405

MATH 581 INTRODUCTION TO APPLIED MATHEMATICS II (3) A graduate course of fundamental techniques including Ordinary, Partial, and Stochastic Differential Equations, Wavelet Analysis, and Perturbation Theory. Prerequisite: MATH 580, or consent of instructor

MATH 582 INTRODUCTION TO C* ALGEBRA THEORY (3) Basic properties of C* algebras, representation theory, group C* algebras and crossed products, tensor products, nuclearity and exactness. Prerequisite: MATH 503

MATH 583 INTRODUCTION TO K-THEORY (3) K-theory groups for compact spaces and C*-algebras. Long exact sequences, Bott periodicity, index theory and the Pimsner-Voiculescu theorem. Prerequisite: MATH 503

MATH 584 INTRODUCTION TO VON NEUMANN ALGEBRAS (3) Comparison of projections, traces, tensor products, ITPFI factors and crossed products, the Jones index, modular theory, free probability. Prerequisite: MATH 503

MATH 588 (CSE) COMPLEXITY IN COMPUTER ALGEBRA (3) Complexity of integer multiplication, polynomial multiplication, fast Fourier transform, division, calculating the greatest common divisor of polynomials. Prerequisite: CSE 465

MATH 590 COLLOQUIUM (1 - 3) Continuing seminars which consist of a series of individual lectures by faculty, students, or outside speakers.

MATH 596 INDIVIDUAL STUDIES (1 - 9) Creative projects, including nonthesis research, which are supervised on an individual basis and which fall outside the scope of formal courses.

MATH 597 SPECIAL TOPICS (1 - 9)

MATH 598 SPECIAL TOPICS (1 - 9)

MATH 599 (IL) FOREIGN STUDIES (1 -12 per semester, maximum of 24)

NOTE: Courses in computer science and statistics are listed separately.