# 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.*