501-502. Real and Complex Analysis (3 each) Cauchy's theorem, Laurent expansion, residue calculus, harmonic functions, conformal mapping, measure and integration, convergence theorems, Lp spaces, Hilbert spaces. Fourier analysis, Fubini's theorem, Hahn-Banach theorem, open mapping theorem, uniform boundedness principle, dual spaces, selected topics from functional analysis. Prerequisite: MATH 404 (for MATH 501 only); MATH 501 (for MATH 502 only).
503. Functional Analysis (3) Topological vector spaces, completeness, convexity, duality, Banach algebras, bounded operators on Hilbert space, the spectral theorem, unbounded operators, applications. Prerequisite: MATH 502.
504. Analysis in Euclidean Space (3) The Fourier transform in L1 and L2 and applications, interpolation of operators, Riesz and Marcinkiewicz theorems, singular integral operators. Prerequisite: MATH 502.
505. Mathematical Fluid Mechanics (3) Kinematics, balance laws, constitutive equations. Ideal fluids, viscous flows, boundary layers, lubrication. Gas dynamics. Prerequisite: MATH 402 or 404.
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.
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 502.
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.
509. Linear Analysis and Applications I (3) Vector spaces, linear transformations, integration, Fourier and Laplace transforms, distributions, differential operators. Prerequisite: MATH 401 or 411 or 412.
510. Linear Analysis and Applications II (3) Integral equations, compact operators, variational methods, partial differential equations. Prerequisite: MATH 509.
511. Ordinary Differential Equations I (3) Existence and uniqueness, linear systems, series methods, Poincare-Bendixson theory, stability. Prerequisite: MATH 411 or 412.
512. Ordinary Differential Equations II (3) Floquet theory, regular and singular boundary value problems, Green's functions, eigenfunction expansions. Prerequisite: MATH 511.
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 412.
514. Partial Differential Equations II (3) Sobolev spaces and elliptic boundary value problems, Schauder estimates, quasilinear symmetric hyperbolic systems, conservation laws. Prerequisite: MATH 502 and 513.
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 or 411 or 412.
516. (STAT 516) Stochastic Processes (3) Markov Chains; generating functions; limit theorems; continuous time and renewal processes; martingales, submartingales and supermartingales; diffusion processes; applications. Prerequisite: MATH (STAT) 416.
517. (STAT 517) Probability Theory (3) Measure theoretic foundations of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics. Prerequisite: MATH 502.
518. (STAT 518) Probability Theory (3) Measure theoretic foundations of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics. Prerequisite: MATH 517.
519. (STAT 519) Topics in Stochastic Processes (3) Selected topics in Stochastic processes, including Markov and Wiener processes; stochastic integrals, optimization and control; optimal filtering. Prerequisite: MATH (STAT) 516,517.
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.
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.
522. Complex Analysis: Theory and Applications II (3) Factorization theorems, prime number theorem, Mittag-Leffler theorem, Nevanlinna theory, Riemann surfaces, Hartog's theorems, holomorphic mappings and automorphisms of bounded domains. Prerequisite: MATH 521.
525. Theory of Functions of Several Complex Variables (3) Fundamental properties of holomorphic functions, reproducing kernels, integral representations, domain of holomorphy and pseudoconvexity, Weierstrass preparation theorem, complex manifolds. Prerequisite: MATH 502.
527. Geometry and Topology I (3) Topological spaces and continuous mappings, connectedness, compactness and separation, fundamental groups, Jordan curve theorem, singular homology, Brouwer Fixed Point Theorem. Prerequisite: MATH 429.
528. Geometry and Topology II (3) Manifolds, differentiable structures, implicit function theorem, vector fields and differential equations, differential forms, Poincare Lemma, integration, Stokes theorem, deRham's Theorem. Prerequisite: MATH 527.
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.
530. Differential Geometry (3) Distributions and Frobenius theorem, curvature of curves and surfaces, Riemannian geometry, connections, curvature, Gauss-Bonnet Theorem, geodesics and completeness. Prerequisite: MATH 528.
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.
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.
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.
535-536. Algebra (3 each) Permutation groups, Sylow theorems, Jordan-Hölder theorem, polynomial rings, unique factorization domains, algebraic and transcendental field extensions, Galois theory. Prerequisite: MATH 435 and a course in linear algebra (for MATH 535 only); MATH 535 (for MATH 536 only).
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.
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.
539-540. Ring Theory (3 each) Selected topics including Nötherian and Artinian modules and rings, semisimple rings, Wedderburn theorems Jacobson radical and density theorem. Prerequisite: MATH 536 (for MATH 539 only); MATH 539 (for MATH 540 only).
542-543. Group Theory I and II (3 each) Topics selected by instructor from abelian, solvable, and nilpotent groups; finite presentations; free products; group extension; group representations. Prerequisite: MATH 535 (for MATH 542 only); MATH 542 (for MATH 543 only).
544. Applied Algebra (3) Basic algorithms of algebra, application to number theory, group theory, field theory, linear algebra and combinatorics. Prerequisite: MATH 435, 436 and ability to use a computer.
546. Semigroup Theory and Applications (3) Basic algebraic properties of semigroups, finite transformation semigroups, free semigroups, formal languages and combinatorics. Prerequisite: MATH 435, 535.
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.
548. Algebraic Geometry II (3) Topics may include: algebraic curves, Riemann-Roch theorem, linear systems and divisors, intersection theory, schemes, sheaf cohomology, algebraic groups. Prerequisite: MATH 547.
549. Mathematical Programming (3) Quadratic and convex programming, Integer and combinatorial programming, dynamic and stochastic programming. Prerequisite: MATH 484.
550. (CSE 550) 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 (CSE) 456 or MATH 441.
551. (CSE 551) Numerical Solutions 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 (CSE) 451 or 456, MATH 411.
552. (CSE 552) 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 404; MATH (CSE) 451 or 456.
553. (CSE 553) 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 and 3 credits of computer science and engineering.
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 (CSE) 451 or 456, MATH 501.
555. (CSE 555) 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 (CSE) 456.
556. (CSE 556) 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 and 552.
557. Mathematical Logic (3) The predicate calculus. Completeness and compactness. Gödel's first and second incompleteness theorems. Introduction to model theory. Introduction to proof theory. Prerequisite: MATH 435 or 457 or equivalent.
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 or equivalent.
559-560. Recursion Theory I, II (3 each) Recursive functions; degrees of unsolvability. Hyperarithmetic theory; applications to Borel combinatorics. Computational complexity. Combinatory logic and the lambda calculus. Prerequisite: MATH 459 or 557 or 558.
561-562. Set Theory I, II (3 each) Models of set theory. Inner models, forcing, large cardinals, determinacy. Descriptive set theory. Applications to analysis. Prerequisite: MATH 557 or 558.
563-564. Model Theory I, II (3 each) Interpolation and definability. Prime and saturated models. Stability. Additional topics. Applications to algebra. Prerequisite: MATH 557.
565. Foundations of Mathematics II (3) Subsystems of second order arithmetic. Set existence axioms. Reverse mathematics. Foundations of analysis and algebra. Prerequisite: MATH 557 and 558.
567-568. Number Theory I, 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 435 (for MATH 567 only); MATH 567 and prerequisite or concurrent: MATH 421 (for MATH 568 only).
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, 568.
570. Algebraic Number Theory II (3) Topics chosen from: class field theory; integral quadratic forms; algebraic and arithmetic groups; algebraic functions of one variable. Prerequisite: MATH 569.
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 568.
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.
574. Topics in Logic and Foundations (3-6; may be taken repeatedly) Topics in mathematical logic and the foundations of mathematics. Prerequisite: MATH 558.
577. (ME 577) Stochastic Systems for Science and Engineering (3) Develops the theory of stochastic processes and linear and non-linear stochastic differential equations for applications to science and engineering. Prerequisite: MATH(STAT) 414 or 418, and MATH 501 or ME 550.
588. (CSE 588) Complexity in Computer Algebra (3) Complexity of integer multiplication, fast Fourier transform, division, calculating the greatest common divisor of polynomials, computing determinant solving linear systems. Prerequisite: MATH (CSE) 465.
590. Colloquium (1-3)
596. Individual Studies (1-9)
597. Special Topics (1-9)
598. Special Topics (1-9)
599. Foreign Study (1-12)
602. Supervised Experience in College Teaching (1-3 per semester, maximum of 6)