# 500-Level Math Courses

The listing below describes all graduate mathematics courses in the Penn State Graduate Bulletin. Not all courses are offered each year.

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

MATH 501 **REAL ANALYSIS (ANALYSIS A)** (3) This course develops Lebesgue measure and integration theory. This is a centerpiece of modern analysis, providing a key tool in many areas of pure and applied mathematics. The course covers the following topics: 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) This course is devoted to the analysis of differentiable functions of a complex variable. This is a central topic in pure mathematics, as well as a vital computational tool. The course covers the following topics: 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) This course develops the theory needed to treat linear integral and differential equations, within the framework of infinite-dimensional linear algebra. Applications to some classical equations are presented. The course covers the following topics: 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 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: MATH 517(STAT)

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 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 method of numerical linear algebra. Eigenvalue computations. Iterative methods. Prerequisite: MATH 535

MATH 527 **TOPOLOGY** (3) Fundamental group, homology theories, index theory, CW complexes, and examples of calculations. Prerequisite: MATH 429

MATH 528 **DIFFERENTIABLE MANIFOLDS** (3) Smooth manifolds, smooth maps, Sard's theorem. The tangent bundle, vector fields, differential forms, integration on manifolds. Foliations. 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 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. Inner products and quadratic forms. Theory of endomorphisms of a finite-dimensional vector space. Orthogonal bases, spectral theorem and applications. Prerequisite: MATH 435 and a course in linear algebra

MATH 536 **ABSTRACT ALGEBRA (ALGEBRA B)** (3) Groups. Sylow's 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 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 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 **ALGEBRAIC NUMBER THEORY I** (3) Algebraic number fields, algebraic integers, Dedekind domains, factoring primes in number fields, ideal class group, Dirichlet's unit theorem, introduction to p-adic fields. Prerequisite: MATH 465, MATH 536.

MATH 568 **ANALYTIC NUMBER THEORY I** (3) Arithmetical functions. Chebyshev's inequalities for the prime counting function and Merten's theorem. Dirichlet series and the prime number theorem. Primes in arithemtic progressions. Further topics will be chosen from the following. The approximation of real numbers by rationals, criteria for irrationality. Uniform distribution. The theory of exponential sums. The Selberg sieve. Applications. Prerequisite: MATH 421

MATH 569 **ALGEBRAIC NUMBER THEORY II** (3) Galois theory of prime ideals, Frobenius automorphisms, cyclotomic fields, class field theory, local fields, ideles and adeles. Prerequisite: MATH 567

MATH 570 **ELLIPTIC CURVES** (3) The arithmetic of elliptic curves, number of rational points over finite fields, analytic uniformization over complex numbers, good and bad reduction,

Mordell-Weil theorem, Siegel's theorem on finiteness of integral points. Prerequisite: MATH 421, MATH 569.

MATH 571 **ANALYTIC NUMBER THEORY II** (3) The large sieve. The Selberg sieve. The Bombieri-Vinogradov theorem that the generalised Riemann hypothesis holds on average. Further topics will be chosen from the following. The Zhang-Maynard-Tao theorems on bounded gaps in the primes. The Green-Tao theorem on primes in arithmetic progression. The Hardy-Littlewood method, including the Vinogradov three primes theorem and applications to Waring's problem. The Vinogradov mean value theorem and applications, including to the zero-free region of the Riemann zeta function. Prerequisite: MATH 421

MATH 572 **MODULAR FORMS** (3) A basic introduction to modular functions, modular forms, modular groups, and Hecke operators. More advanced topics might include Eichler-Selberg trace formula, connection to Galois representations, Langlands program, applications to theory of partitions and class field theory. Prerequisite: MATH 521, MATH 536.

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 578 **THEORY AND APPLICATION OF WAVELETS** (3) Theory and physical interpretation of continuous and discrete wavelet transforms for applications in different engineering disciplines. Prerequisite: 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) A concise introduction to von Neumann algebra theory, beginning with the basic definitions and proceeding through modular theory. The currently important subjects of index theory and free probability theory will be introduced. Prerequisite: MATH 503

MATH 585 **TOPICS IN MATHEMATICAL MODELING** (3) Introduction to mathematical modeling, covering the basic modeling and common mathematical techniques for problems from physical, biological and social sciences. Prerequisite: MATH 403 , MATH 411 , and MATH 412

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 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) Formal courses given on a topical or special interest subject which may be offered infrequently; several different topics may be taught in one year or term. Syllabus

MATH 601 **Ph.D. DISSERTATION FULL-TIME**

MATH 610 **THESIS RESEARCH OFF CAMPUS**

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