University Bulletin

Graduate Degree Programs

MATH 401
**Introduction to Analysis I ** (3) Review of calculus, properties of real numbers, infinite series, uniform convergence, power series. Students who have passed Math. 403 may not schedule this course.

Effective: Fall 1983

Prerequisite:
MATH 230 orMATH 231

MATH 403
**Classical Analysis I ** (3) Topology of Rn, compactness, continuity of functions, uniform convergence, Arzela-Ascoli theorem in the plane, Stone-Wierstrass theorem.

Effective: Spring 1996

Prerequisite:
MATH 312

MATH 403H
**Honors Classical Analysis I ** (3) Development of a thorough understanding and technical mastery of foundations of classical analysis in the framework of metric spaces.

Effective: Spring 2010

Prerequisite:
MATH 311M, MATH 312H

MATH 404
**Classical Analysis II ** (3) Differentiation of functions from Rn to Rm, implicit function theorem, Riemann integration, Fubini's theorem, Fourier analysis.

Effective: Fall 1985

Prerequisite:
MATH 403

MATH 405
**Advanced Calculus for Engineers and Scientists I ** (3) Vector calculus, linear algebra, ordinary and partial differential equatinos. Students who have passed MATH 411 or 412 may not take this course for credit.

Effective: Spring 1994

Prerequisite:
MATH 231;MATH 250 orMATH 251

MATH 406
**Advanced Calculus for Engineers and Scientists II ** (3) Complex analytic functions, sequences and series, residues, Fourier and Laplace transforms. Students who have passed MATH 421 may not take this course for credit.

Effective: Spring 1994

Prerequisite:
MATH 405

MATH 408
**Advanced Calculus ** (3) Differential and integral calculus of functions of several variables, line and surface integrals, infinite series, series of functions, power series.

Effective: Spring 2007

Prerequisite:
MATH 141

MATH 410
**Complex Analysis for Mathematics and Engineering ** (3) Complex analytic functions; Cauchy-Riemann equations; complex contour integrals; Cauchy's integral formula; Taylor and Laurent series; residue theory; applications in engineering.

Effective: Summer 2014

Prerequisite:
MATH 230 orMATH 232

MATH 411
**Ordinary Differential Equations ** (3) Linear ordinary differential equations; existence and uniqueness questions; series solutions; special functions; eigenvalue problems; Laplace transforms; additional topics and applications.

Effective: Fall 1983

Prerequisite:
MATH 230 orMATH 231;MATH 250 orMATH 251

MATH 412
**Fourier Series and Partial Differential Equations ** (3) Orthogonal systems and Fourier series; derivation and classification of partial differential equations; eigenvalue function method and its applications; additional topics.

Effective: Spring 2009

Prerequisite:
MATH 230;MATH 250 orMATH 251

MATH 414
(STAT 414)
**Introduction to Probability Theory ** (3) Probability spaces, discrete and continuous random variables, transformations, expectations, generating functions, conditional distributions, law of large numbers, central limit theorems. Students may take only one course from MATH(STAT) 414 and 418 for credit.

Effective: Fall 2001

Prerequisite:
MATH 230 orMATH 231

MATH 415
(STAT 415)
**Introduction to Mathematical Statistics ** (3) A theoretical treatment of statistical inference, including sufficiency, estimation, testing, regression, analysis of variance, and chi-square tests.

Effective: Fall 1989

Prerequisite:
MATH 414

MATH 416
(STAT 416)
**Stochastic Modeling ** (3) Review of distribution models, probability generating functions, transforms, convolutions, Markov chains, equilibrium distributions, Poisson process, birth and death processes, estimation.

Effective: Spring 1984

Prerequisite:
MATH 318 orMATH 414;MATH 230

MATH 417
**Qualitative Theory of Differential Equations ** (3) Linear differential equations, stability of stationary solutions, ordinary bifurcation, exchange of stability, Hopf bifurcation, stability of periodic solutions, applications.

Effective: Spring 2009

Prerequisite:
MATH 220;MATH 250 orMATH 251

MATH 418
(STAT 418)
**Introduction to Probability and Stochastic Processes for Engineering ** (3) Introduction to probability axioms, combinatorics, random variables, limit laws, and stochastic processes. Students may take only one course from MATH(STAT) 414 and 418 for credit.

Effective: Fall 2011

Prerequisite:
MATH 230 orMATH 231

MATH 419
(PHYS 419)
**Theoretical Mechanics ** (3) Principles of Newtonian, Lagrangian, and Hamiltonian mechanics of particles with applications to vibrations, rotations, orbital motion, and collisions.

Effective: Spring 2007

Prerequisite:
MATH 230 orMATH 231;MATH 250 orMATH 251;PHYS 212, PHYS 213 andPHYS 214

MATH 421
**Complex Analysis ** (3) Infinite sequences and series; algebra and geometry of complex numbers; analytic functions; integration; power series; residue calculus; conformal mapping, applications.

Effective: Summer 1993

Prerequisite:
MATH 230, MATH 232 orMATH 405;MATH 401 orMATH 403

MATH 425
**Introduction to Operations Research ** (3) Nature of operations research, problem formulation, model construction, deriving solution from models, allocation problems, general linear allocation problem, inventory problems.

Effective: Spring 2012

Prerequisite:
MATH 141 andMATH 220

MATH 426
**Introduction to Modern Geometry ** (3) Plane and space curves; space surfaces; curvature; intrinsic geometry of surfaces; Gauss-Bonnet theorem; covariant differentiation; tensor analysis.

Effective: Spring 1994

Prerequisite:
MATH 401 orMATH 403

MATH 427
**Foundations of Geometry ** (3) Euclidean and various non-Euclidean geometries and their development from postulate systems. Students who have passed MATH 427 may not schedule MATH 471.

Effective: Spring 1994

Prerequisite:
MATH 230 orMATH 231

MATH 428
**Geometry for Teachers ** (1) Research in mathematics education using ideas from Euclidean and non-Euclidean geometry. Students who have passed MATH 471 may not schedule MATH 427.

Effective: Spring 2007

Prerequisite:
MATH 311W . Prerequisite or concurrent:MATH 427

MATH 429
**Introduction to Topology ** (3) Metric spaces, topological spaces, separation axioms, product spaces, identificaiton spaces, compactness, connectedness, fundamental group.

Effective: Spring 1994

Prerequisite:
MATH 311W

MATH 430
**Linear Algebra and Discrete Models I ** (3) Vector spaces, linear transformations, matrices determinants, characteristic values and vectors, systems of linear equations, applications to discrete models.

Effective: Spring 2010

Prerequisite:
MATH 220

MATH 431
**Linear Algebra and Discrete Models II ** (3) Vector spaces and linear transformations, matrices, determinants, characteristics values and vectors, systems of linear equations, applications to discrete models.

Effective: Spring 2007

Prerequisite:
MATH 430

MATH 435
**Basic Abstract Algebra ** (3) Elementary theory of groups, rings, and fields. Students who have passed MATH 435 may not schedule MATH 470.

Effective: Spring 2010

Prerequisite:
MATH 311W orMATH 315

MATH 436
**Linear Algebra ** (3) Vector spaces and linear transformations, canonical forms of matrices, elementary divisors, invariant factors; applications. Students who have passed MATH 436 may not schedule MATH 441.

Effective: Fall 1983

Prerequisite:
MATH 311W

MATH 437
**Algebraic Geometry ** (3) Study of curves in the plane defined by polynomial equations p(x,y)= 0. Projective equivalence, singular points, classification of cubics.

Effective: Spring 2009

Prerequisite:
MATH 230 orMATH 231;MATH 311W

MATH 441
**Matrix Algebra ** (3) Determinants, matrices, linear equations, characteristic roots, quadratic forms, vector spaces. Students who have passed Math 436 may not schedule this course.

Effective: Fall 1985

Prerequisite:
MATH 220

MATH 444
**Mathematical Statistics and Applications I ** (3) Distributions of random variables, special distributions, limiting distributions, sampling, statistical inference, point and interval estimation, orthogonal polynomials, and least squares.

Effective: Spring 2007

Prerequisite:
MATH 141

MATH 446
**Introduction to Applied Statistics I ** (3) Descriptive statistics, probability theory, discrete and continuous probability distributions, statistical inferences for means and proportions.

Effective: Spring 2007

Prerequisite:
MATH 022 orMATH 040

MATH 447
**Introduction to Applied Statistics II ** (3) Regression, correlation, analysis of variance, contingency tables, nonparametric methods, time series, index numbers.

Effective: Spring 2007

MATH 449
**Applied Ordinary Differential Equations ** (3) Differential and difference equations and their application to biology, chemistry, and physics; techniques in dynamical systems theory.

Effective: Spring 2007

Prerequisite:
MATH 250 orMATH 251

MATH 450
**Mathematical Modeling ** (3) Constructing mathematical models of physical phenomena; topics include pendulum motion, polymer fluids, chemical reactions, waves, flight, and chaos.

Effective: Spring 2007

Prerequisite:
MATH 315 andMATH 430 orMATH 405 orMATH 412

MATH 451
(CMPSC 451)
**Numerical Computations ** (3) Algorithms for interpolation, approximation, integration, nonlinear equations, linear systems, fast FOURIER transform, and differential equations emphasizing computational properties and implementation. Students may take only one course for credit from MATH 451 and 455.

Effective: Spring 2008

Prerequisite:
3 credits of programming;MATH 230 orMATH 231

MATH 455
(CMPSC 455)
**Introduction to Numerical Analysis I ** (3) Floating point computation, numerical rootfinding, interpolation, numerical quadrature, direct methods for linear systems. Students may take only one course for credit from MATH 451 and MATH 455.

Effective: Spring 2008

Prerequisite:
CMPSC 201, CMPSC 202 orCMPSC 121;MATH 220;MATH 230 orMATH 231

MATH 456
(CMPSC 456)
**Introduction to Numerical Analysis II ** (3) Polynomial and piecewise polynomial approximation, matrix least squares problems, numerical solution of eigenvalue problems, numerical solution of ordinary differential equations.

Effective: Spring 2008

Prerequisite:
MATH 455

MATH 457
**Introduction to Mathematical Logic ** (3) Propositional logic, first-order predicate logic, axioms and rules of inference, structures, models, definability, completeness, compactness.

Effective: Summer 2011

Prerequisite:
MATH 311W ; 3 additional credits in philosophy

MATH 461
(PHYS 461)
**Theoretical Mechanics ** (3) Continuation of Math.(Phys.) 419. Theoretical treatment of dynamics of a rigid body, theory of elasticity, aggregates of particles, wave motion, mechanics of fluids.

Effective: Fall 1986

Prerequisite:
MATH 419

MATH 465
**Number Theory ** (3) Elements, divisibility of numbers, congruences, residues, and forms.

Effective: Spring 2009

Prerequisite:
MATH 311W

MATH 467
(CMPSC 467)
**Factorization and Primality Testing ** (3) Prime sieves, factoring, computer numeration systems, congruences, multiplicative functions, primitive roots, cryptography, quadratic residues. Students who have passed MATH 465 may not schedule this course.

Effective: Spring 1995

Prerequisite:
MATH 311W

MATH 468
**Mathematical Coding Theory ** (3) Shannon's theorem, block codes, linear codes, Hamming codes, Hadamard codes, Golay codes, Reed-Muller codes, bounds on codes, cyclic codes.

Effective: Fall 1983

Prerequisite:
MATH 311W ; advanced calculus

MATH 470
**Algebra for Teachers ** (3) An introduction to algebraic structures and to the axiomatic approach, including the elements of linear algebra. Designed for teachers and prospective teachers. Students who have passed Math 435 may not schedule this course.

Effective: Fall 1988

Prerequisite:
MATH 311W

MATH 471
**Geometry for Teachers ** (4) Problem solving oriented introduction to Euclidean and non-Euclidean geometries; construction problems and geometrical transformations via "Geometer's Sketchpad" software. Intended primarily for those seeking teacher certification in secondary mathematics. Students who have passed MATH 427 may not schedule this course.

Effective: Spring 1996

Prerequisite:
MATH 311W

MATH 475W
(US;IL)
**History of Mathematics ** (3) A global survey of the history of mathematics as viewed as a human response to cultural, political, economic, and societal pressures.

Effective: Spring 2012

Prerequisite:
MATH 315 orMATH 311W

MATH 479
(PHYS 479)
**Special and General Relativity ** (3) Mathematical description, physical concepts, and experimental tests of special and general relativity.

Effective: Spring 2007

Prerequisite:
PHYS 237, PHYS 400, PHYS 419;MATH 250 orMATH 251;MATH 230 orMATH 231

MATH 482
**Mathematical Methods of Operations Research ** (3) Survey of linear and nonlinear programming; mathematics of optimization; queues; simulation.

Effective: Spring 2007

Prerequisite:
MATH 220, MATH 230, STAT 301

MATH 484
**Linear Programs and Related Problems ** (3) Introduction to theory and applications of linear programming; the simplex algorithm and newer methods of solution; duality theory.

Effective: Spring 1987

Prerequisite:
MATH 220;MATH 230 orMATH 231

MATH 485
**Graph Theory ** (3) Introduction to the theory and applications of graphs and directed graphs. Emphasis on the fundamental theorems and their proofs.

Effective: Spring 1987

Prerequisite:
MATH 311W

MATH 486
**Mathematical Theory of Games ** (3) Basic theorems, concepts, and methods in the mathematical study of games of strategy; determination of optimal play when possible.

Effective: Spring 2006

Prerequisite:
MATH 220

MATH 494
**Research Project ** (1-12) Supervised student activities on research projects identified on an individual or small-group basis.

Effective: Spring 1995

MATH 494H
**Research Project ** (1-12) Supervised student activities on research projects identified on an individual or small-group basis.

Effective: Fall 2007

MATH 495
**Internship ** (1-18) Supervised off-campus, nongroup instruction including field experiences, practica, or internships. Written and oral critique of activity required.

Effective: Spring 2007

Prerequisite:
prior approval of proposed assignment by instructor

MATH 496
**Independent Studies ** (1-18) Creative projects, including research and design, which are supervised on an individual basis and which fall outside the scope of formal courses.

Effective: Fall 1983

MATH 497
**Special Topics ** (1-9) Formal courses given infrequently to explore, in depth, a comparatively narow subject which may be topical or of special interest.

Effective: Fall 1983

MATH 498
**Special Topics ** (1-9) Formal courses given infrequently to explore, in depth, a comparatively narrow subject which may be topical or of special interest.

Effective: Fall 1992

MATH 499
(IL)
**Foreign Studies ** (1-12) Courses offered in foreign countries by individual or group instruction.

Effective: Summer 2005

MATH 501
**Real Analysis** (3) Legesgue measure theory. Measurable sets and measurable functions. Legesgue integration, convergence theorems. Lp spaces. Decomposition and differentiation of measures. Convolutions. The Fourier transform.

Effective: Spring 2013

Prerequisite:
MATH 404

MATH 502
**Complex Analysis** (3) Complex numbers. Holomorphic functions. Cauchy's theorem. Meromorphic functions. Laurent expansions, residue calculus. Conformal maps, topology of the plane.

Effective: Spring 2013

Prerequisite:
MATH 501

MATH 503
**Functional Analysis** (3) Banach spaces and Hilbert spaces. Dual spaces. Linear operators. Distributors, weak derivatives. Sovolev spaces. Applications to linear differential equations.

Effective: Spring 2013

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.

Effective: Spring 1992

Prerequisite:
MATH 502

MATH 505
**Mathematical Fluid Mechanics** (3) Kinematics, balance laws, constitutive equations; ideal fluids, viscous flows, boundary layers, lubrication; gas dynamics.

Effective: Spring 1992

Prerequisite:
MATH 402 orMATH 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.

Effective: Spring 1992

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.

Effective: Spring 1992

Prerequisite:
MATH 502

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.

Effective: Spring 1992

Prerequisite:
MATH 507

MATH 511
**Ordinary Differential Equations I** (3) Existence and uniqueness, linear systems, series methods, Poincare-Bendixson theory, stability.

Effective: Spring 1992

Prerequisite:
MATH 411 orMATH 412

MATH 513
**Partial Differential Equations I** (3) First order equations, the Cauchy problem, Cauchy-Kowalevski theorem, Laplace equation, wave equation, heat equation.

Effective: Spring 1992

Prerequisite:
MATH 411 orMATH 412

MATH 514
**Partial Differential Equations II** (3) Sobolev spaces and Elliptic boundary value problems, Schauder estimates. Quasilinear symmetric hyperbolic systems, conservation laws.

Effective: Spring 1992

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.

Effective: Spring 1992

Prerequisite:
MATH 401, MATH 411 orMATH 412

MATH 516
**Stochastic Processes** (3) Markov chains; generating functions; limit theorems; continuous time and renewal processes; martingales, submartingales, and supermartingales; diffusion processes; applications.

Effective: Summer 1995

Prerequisite:
MATH 416

MATH 517
(STAT 517)
**Probability Theory ** (3) Measure theoretic foundation of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics.

Effective: Summer 2000

Prerequisite:
MATH 403

MATH 518
(STAT 518)
**Probability Theory** (3) Measure theoretic foundation of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics.

Effective: Fall 1983

Prerequisite:
STAT 517

MATH 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.

Effective: Fall 1984

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.

Effective: Summer 2002

Prerequisite:
MATH 456

MATH 524
**Numerical Linear Algebra ** (3) Matrix decompositions. Direct method of numerical linear algebra. Eigenvalue computations. Iterative methods.

Effective: Spring 2013

Prerequisite:
MATH 535

MATH 527
**Metric and Topological Spaces** (3) Metric spaces, continuous maps, compactness, connectedness, and completeness. Topolocial spaces, products, quotients, homotopy, fundamental group, simple applications.

Effective: Spring 2013

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.

Effective: Spring 2013

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.

Effective: Spring 1992

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.

Effective: Spring 1992

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.

Effective: Spring 1992

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.

Effective: Spring 1992

Prerequisite:
MATH 533

MATH 535
**Linear Algebra** (3) Vector spaces. Linear transformations. Bilinear forms. Canonical forms for linear transformations. Multilinear algebra.

Effective: Spring 2013

Prerequisite:
MATH 436

MATH 536
**Abstract Algebra** (3) Groups. Sylow's theorems. Rings. Ideals, unique factorization domains. Finitely generated modules. Fields. Algebraic and transcendental field extensions, Galois theory.

Effective: Spring 2013

Prerequisite:
MATH 535

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.

Effective: Fall 1983

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.

Effective: Winter 1978

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.

Effective: Fall 1983

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.

Effective: Spring 1992

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.

Effective: Spring 1992

Prerequisite:
MATH 547

MATH 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.

Effective: Summer 1996

Prerequisite:
MATH 441 orMATH 456

MATH 551
(CSE 551)
**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.

Effective: Summer 1996

Prerequisite:
MATH 451 orMATH 456

MATH 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.

Effective: Summer 1996

Prerequisite:
MATH 402 orMATH 404;MATH 451 orMATH 456

MATH 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.

Effective: Summer 1996

Prerequisite:
MATH 401 3 credits of computer science and engineering

MATH 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.

Effective: Summer 1996

Prerequisite:
MATH 456

MATH 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.

Effective: Summer 1996

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.

Effective: Spring 1992

Prerequisite:
MATH 435 orMATH 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.

Effective: Spring 1992

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.

Effective: Summer 2011

Prerequisite:
MATH 557 orMATH 558

MATH 561
**Set Theory I** (3) Models of set theory. Inner models, forcing, large cardinals, determinacy. Descriptive set theory. Applications to analysis.

Effective: Spring 1992

Prerequisite:
MATH 557 orMATH 558

MATH 565
**Foundations of Mathematics II** (3) Subsystems of second order arithmetic; set existence axioms; reverse mathematics; foundations of analysis and algebra.

Effective: Spring 1992

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.

Effective: Spring 1994

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.

Effective: Spring 1994

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.

Effective: Spring 1992

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.

Effective: Spring 1992

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.

Effective: Spring 1994

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.

Effective: Spring 1992

Prerequisite:
MATH 571

MATH 574
**Topics in Logic and Foundations** (3-6 per semester) Topics in mathematical logic and the foundations of mathematics.

Effective: Spring 1992

Prerequisite:
MATH 558

MATH 577
(M E 577)
**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.

Effective: Summer 1998

Prerequisite:
MATH 414 orMATH 418;M E 550 orMATH 501

MATH 578
(M E 578)
**Theory & Applications of Wavelets ** (3) Theory and physical interpretation of continuous and discrete wavelet transforms for applications in different disciplines.

Effective: Spring 2012

Prerequisite:
MATH 501 orM E 550

MATH 580
**Introduction to Applied Mathematics I** (3) A graduate course of fundamental techniques including tensor, ordinary and partial differential equations, and linear transforms.

Effective: Fall 2003

Prerequisite:
Basic knowledge of linear algebra vector calculus and ODEMATH 405

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.

Effective: Summer 2006

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.

Effective: Summer 2006

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.

Effective: Summer 2006

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.

Effective: Summer 2012

Prerequisite:
MATH 403, MATH 411 andMATH 412

MATH 588
(CSE 588)
**Complexity in Computer Algebra** (3) Complexity of integer multiplication, polynomial multiplication, fast Fourier transform, division, calculating the greatest common divisor of polynomials.

Effective: Spring 2008

Prerequisite:
CMPSC 465

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

Effective: Spring 1987

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.

Effective: Spring 1987

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.

Effective: Spring 1987

MATH 598
**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 semester.

Effective: Spring 1994

MATH 599
(IL)
**Foreign Studies ** (1-12 per semester, maximum of 24) Full-time graduate-level foreign study at an overseas institution with whom linkages have been established.

Effective: Summer 2005

MATH 600
**Thesis Research** (1-15) No description.

Effective: Fall 1983

MATH 601
**Ph.D. Dissertation Full-Time** (0) No description.

Effective: Winter 1978

MATH 602
**Supervised Experience in College Teaching** (1-3 per semester/maximum of 6) Teaching of mathematics undergraduate recitation classes with senior faculty instruction supervision.

Effective: Fall 1983

MATH 610
**Thesis Research Off Campus** (1-15) No description.

Effective: Fall 1983

MATH 611
**Ph.D. Dissertation Part-Time** (0) No description.

Effective: Winter 1978

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

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