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MATHEMATICS (MATH) MATH 505: Mathematical 3 Credits MATH 501: Real Kinematics, balance laws, constitutive ; , viscous 3 Credits flows, boundary layers, lubrication; dynamics.

Legesgue measure theory. Measurable sets and measurable functions. Prerequisite: MATH 402 or MATH 404 Legesgue integration, convergence theorems. Lp spaces. Decomposition MATH 506: Ergodic Theory and differentiation of measures. . The Fourier transform. MATH 501 I (3) This course develops Lebesgue measure 3 Credits and integration theory. This is a centerpiece of modern analysis, providing a key tool in many areas of pure and . The course Measure-preserving transformations and flows, ergodic theorems, covers the following topics: Lebesgue measure theory, measurable sets ergodicity, mixing, weak mixing, spectral invariants, measurable and measurable functions, Lebesgue integration, convergence theorems, partitions, entropy, ornstein isomorphism theory. Lp spaces, decomposition and differentiation of measures, convolutions, the Fourier transform. Prerequisite: MATH 502

Prerequisite: MATH 404 MATH 507: Dynamical Systems I

MATH 502: 3 Credits

3 Credits Fundamental concepts; extensive survey of examples; equivalence and classification of dynamical systems, principal classes of asymptotic Complex . Holomorphic functions. Cauchy's theorem. invariants, circle maps. Meromorphic functions. Laurent expansions, residue . Conformal maps, of the plane. MATH 502 Complex Analysis (3) This Prerequisite: MATH 502 course is devoted to the analysis of differentiable functions of a complex MATH 508: Dynamical Systems II . This is a central topic in , as well as a vital computational tool. The course covers the following topics: complex 3 Credits numbers, holomorphic functions, Cauchy's theorem, meromorphic functions, Laurent expansions, residue calculus, conformal maps, Hyperbolic theory; stable , hyperbolic sets, attractors, Anosov topology of the plane. systems, shadowing, structural stability, entropy, pressure, Lyapunov characteristic exponents and non-uniform hyperbolicity. Prerequisite: MATH 501 Prerequisite: MATH 507 MATH 503: MATH 511: Ordinary Differential Equations I 3 Credits 3 Credits Banach spaces and Hilbert spaces. Dual spaces. Linear operators. Distributors, weak . Sovolev spaces. Applications to linear Existence and uniqueness, linear systems, methods, Poincare- differential equations. MATH 503 Functional Analysis (3) This course Bendixson theory, stability. develops the theory needed to treat linear and differential equations, within the framework of infinite-dimensional linear . Prerequisite: MATH 411 or MATH 412 Applications to some classical equations are presented. The course MATH 513: Partial Differential Equations I covers the following topics: Banach and Hilbert spaces, dual spaces, linear operators, distributions, weak derivatives, Sobolev spaces, 3 Credits applications to linear differential equations. First order equations, the Cauchy problem, Cauchy-Kowalevski theorem, Prerequisite: MATH 501 Laplace , wave equation, .

MATH 504: Analysis in Prerequisite: MATH 411 or MATH 412

3 Credits MATH 514: Partial Differential Equations II

The Fourier transform in L1 and L2 and applications, of 3 Credits operators, Riesz and Marcinkiewics theorems, singular integral operators. Sobolev spaces and Elliptic boundary value problems, Schauder Prerequisite: MATH 502 estimates. Quasilinear symmetric hyperbolic systems, conservation laws.

Prerequisite: MATH 502 , MATH 513 2 Mathematics (MATH)

MATH 515: and Variational Methods solution of nonlinear systems and optimization. One- iterations, Newton's and quasi-Newton's method, Broyden's method, unconstrained 3 Credits optimization, -search methods.

Introduction to the , variational formulation of Prerequisite: MATH 456 , symmetry in mechanical systems, Legendre transformation, , completely integrable systems. MATH 524: Numerical

Prerequisite: MATH 401 , MATH 411 , or MATH 412 3 Credits

MATH 516: Stochastic Processes decompositions. Direct method of . Eigenvalue computations. Iterative methods. MATH 524 Numerical Linear 3 Credits Algebra (3) This course provides a graduate level foundation in numerical linear algebra. It covers the mathematical theory behind numerical Markov chains; generating functions; limit theorems; continuous for the solution of linear systems of equations and eigenvalue time and renewal processes; martingales, submartingales, and problems. Specific topics include: matrix decompositions, direct methods supermartingales; diffusion processes; applications. of numerical linear algebra, eigenvalue computations, iterative methods. Prerequisite: MATH 416 Prerequisite: MATH 535 MATH 517: Theory MATH 527: Topology 3 Credits 3 Credits Measure theoretic foundation of probability, distribution functions and This course provides an overview of the fundamental concepts of laws, types of convergence, central limit problem, conditional probability, Geometric and and presents examples of calculations special topics. of principal topological invariants. It starts with review of general Prerequisite: MATH 403 topology and covers the following topics: , Cross-listed with: STAT 517 theories, index theory, CW complexes, and examples of calculations.

MATH 518: Prerequisite: MATH 429

3 Credits MATH 528: Differentiable Manifolds

Measure theoretic foundation of probability, distribution functions and 3 Credits laws, types of convergence, central limit problem, conditional probability, Smooth manifolds, smooth maps, Sard's theorem. The tangent bundle, special topics. vector fields, differential forms, integration on manifolds. Foliations. De Prerequisite: STAT 517 Rham ; simple applications. Lie groups, smooth actions, Cross-listed with: STAT 518 quotient spaces, examples. MATH 528 Differentiable Manifolds (3) This course covers the foundations of , developing the MATH 519: Topics in Stochastic Processes theory of differentiation and integration on manifolds. It provides tools for the study of nonlinear problems, combining techniques in analysis and 3 Credits geometry. Concepts and tools from differential geometry have found wide Selected topics in stochastic processes, including Markov and Wiener use in different , including nonlinear differential processes; stochastic , optimization, and control; optimal equations, control and optimization problems, and . filtering. The goal is to cover the most important techniques of differential geometry in a concise way. The course will appeal not only to students Prerequisite: STAT 516 , STAT 517 who plan to do research in geometry, but also to those interested in Cross-listed with: STAT 519 analysis, or applied and computational mathematics, as well. It covers the following topics: smooth manifolds, smooth maps, Sard’s MATH 523: Numerical Analysis I theorem, the tangent bundle, vector fields, differential forms, integration on manifolds, foliations, de Rham cohomology, Lie groups, smooth 3 Credits actions, quotient spaces, examples. Approximation and interpolation, numerical , direct methods Prerequisite: MATH 527 of numerical linear algebra, numerical solutions of nonlinear systems and optimization. MATH 523 Numerical Analysis I (3) 1. Approximation MATH 529: Algebraic Topology and interpolation. Weierstrass theorem, Bernstein , Jackson theorems, Lagrange interpolation, approximation, 3 Credits , piecewise Lagrange and Hermite interpolation, spline interpolation, the .2. Numerical quadrature. Manifolds, Poincare duality, vector bundles, Thom isomorphism, Newton-Cotes rules, Peano Theorem, -Maclaurin expansion, characteristic classes, classifying spaces for vector bundles, discussion Romberg integration, , adaptive quadrature.3. Direct of bordism, as time allows. methods of numerical linear algebra. with pivoting, Prerequisite: MATH 528 backward error analysis, conditioning of linear systems.4. Numerical Mathematics (MATH) 3

MATH 530: Differential Geometry MATH 538:

3 Credits 3 Credits

Distributions and Frobenius theorem, of and surfaces, Topics selected from Noetherian rings and modules, primary , connections, curvature, Gauss-Bonnet theorem, decompositions, Dedekind domains and ideal theory, other special types geodesic and completeness. of commutative rings or fields.

Prerequisite: MATH 528 Prerequisite: MATH 536

MATH 533: I MATH 547: I

3 Credits 3 Credits

Lie groups, lie , exponential mappings, subgroups, subalgebras, Affine and projective algebraic varieties; Zariski topology; Hilbert simply connected groups, adjoint representation, semisimple groups, Nullstellensatz; regular functions and maps; birationality; smooth infinitesimal theory, Cartan's criterion. varieties normalization; .

Prerequisite: MATH 528 Prerequisite: MATH 536

MATH 534: Lie Theory II MATH 548: Algebraic Geometry II

3 Credits 3 Credits

Representations of compact lie groups and semisimple lie algebras, Topics may include algebraic curves, Riemann-Roch theorem, linear characters, , Peter-Weyl theorem, Cartan-Weyl highest systems and divisors, intersectino theory, schemes, sheaf cohomology, weight theory. algebraic groups.

Prerequisite: MATH 533 Prerequisite: MATH 547

MATH 535: Linear Algebra and Its Applications MATH 550: Numerical Linear Algebra

3 Credits 3 Credits

Vector spaces. Linear transformations. Inner products and quadratic Solution of linear systems, techniques, , forms. Theory of endomorphisms of a finite-dimensional . decomposition, numerical computation of eigenvalues and Orthogonal bases, spectral theorem and applications. eigenvectors.

Prerequisite: MATH 436 Prerequisite: MATH 441 or MATH 456 Cross-listed with: CSE 550 MATH 536: MATH 551: Numerical Solution of Ordinary Differential Equations 3 Credits 3 Credits Groups. Sylow's theorems. Rings. Ideals, unique factorization domains. Finitely generated modules. Fields. Algebraic and transcendental field Methods for initial value and boundary value problems; convergence and extensions, . MATH 536 Abstract Algebra (3) This course stability analysis, automatic error control, stiff systems, boundary value covers fundamental concepts, needed toward the study of advanced problems. areas in abstract algebra. The course covers the following topics: groups, Sylow's theorems, rings, ideals, unique factorization domains, finitely Prerequisite: MATH 451 or MATH 456 generated modules, fields, algebraic and transcendental field extensions, Cross-listed with: CSE 551 Galois theory. MATH 552: Numerical Solution Of Partial Differential Equations Prerequisite: MATH 535 3 Credits MATH 537: Theory methods for elliptic, parabolic, and hyperbolic differential 3 Credits equations; solutions techniques for discretized systems; finite element methods for elliptic problems. Finite and infinite algebraic extensions; cyclotomic fields; transcendental extensions; bases of transcendence, Luroth's theorem, ordered fields, Prerequisite: MATH 402 or MATH 404 ; MATH 451 or MATH 456 valuations; formally real fields. Cross-listed with: CSE 552

Prerequisite: MATH 536 4 Mathematics (MATH)

MATH 553: Introduction to Prerequisite: MATH 557 or MATH 558

3 Credits MATH 565: Foundations of Mathematics II

Interpolation; remainder theory; approximation of functions; error 3 Credits analysis; orthogonal polynomials; approximation of linear functionals; functional analysis applied to numerical analysis. Subsystems of second order ; existence ; reverse mathematics; foundations of analysis and algebra. Prerequisite: MATH 401 , 3 credits in Science and Prerequisite: MATH 557 , MATH 558 MATH 555: Numerical Optimization Techniques MATH 567: Theory I 3 Credits 3 Credits Unconstrained and constrained optimization methods, linear and quadratic programming, issues, ellipsoid and Karmarkar's Congruences, quadratic residues, arithmetic functions, partitions, , , parallelism in optimization. classical multiplicative ideal theory, valuations and p-adic numbers; primes in arithmetic progression, distribution of primes. Prerequisite: CMPSC456 Cross-listed with: CSE 555 Prerequisite: MATH 421

MATH 556: Finite Element Methods MATH 568: II

3 Credits 3 Credits

Sobolev spaces, variational formulations of boundary value problems; Congruences, quadratic residues, arithmetic functions, partitions, piecewise approximation theory, convergence and stability, classical multiplicative ideal theory, valuations and p-adic numbers; special methods and applications. primes in arithmetic progression, distribution of primes.

Prerequisite: MATH 502 , MATH 552 Prerequisite: MATH 421 Cross-listed with: CSE 556 MATH 569: Theory I MATH 557: 3 Credits 3 Credits Dedekind rings; cyclotomic and Kummer extensions; valuations; The predicate calculus; completeness and compactness; Godel's first ramification, decomposition, inertial groups; Galois extensions; locally and second incompleteness theorems; introduction to ; compact groups of number theory. introduction to . Prerequisite: MATH 536 , MATH 568 Prerequisite: MATH 435 or MATH 457 MATH 570: II MATH 558: Foundations of Mathematics I 3 Credits 3 Credits Topics chosen from class field theroy; integral quadratic forms; algebraic Decidability of the real numbers; computability; undecidability of the and arithmetic groups; algebraic of one variable. natural numbers; models of ; of choice; continuum Prerequisite: MATH 569 hypothesis. MATH 571: I Prerequisite: any 400 level math course 3 Credits MATH 559: Recursion Theory I Improvements of the prime number theorem, L-functions and class 3 Credits numbers, asymptotic and arithmetic properties of coefficients of modular Recursive functions; degrees of unsolvability; hyperarithmetic theory; forms. applications to Borel . Computational complexity. Prerequisite: MATH 421 Combinatory logic and the Lambda calculus. MATH 572: Analytic Number Theory II Prerequisite: MATH 557 , or MATH 558 3 Credits MATH 561: Set Theory I Distribution of primes, analytic number theory in algebraic number fields, 3 Credits transcendental numbers, advanced theory of partitions. Models of set theory. Inner models, forcing, large cardinals, determinacy. Prerequisite: MATH 571 Descriptive set theory. Applications to analysis. Mathematics (MATH) 5

MATH 573: Partitions Prerequisite: MATH 503

3 Credits MATH 583: Introduction to K-Theory

This course provides an overview of theory of partitions. This course 3 Credits focuses on the partition function p(n) and its number theoretic behavior and combinatorial structure, along with related problems. To achieve the K-theory groups for compact spaces and *-algebras. Long exact main goal, generating function theory, basic hypergeometric series, q- sequences, Bott periodicity, index theory and the Pimsner-Voiculescu series, and some related combinatorial theory are discussed. Building on theorem. these relevant theories, students will be able to understand and prove the Prerequisite: MATH 503 arithmetic properties of p(n) and related identities including the Rogers- Ramanujan identities. Students will also be able to recognize and apply MATH 584: Introduction to von Neumann Algebras the same methods to similar functions and identities. 3 Credits MATH 574: Topics in Logic and Foundations Comparison of projections, traces, products, ITPFI factors and 3-6 Credits/Maximum of 6 crossed products, the Jones index, modular theory, free probability. MATH 584 Introduction to von Neumann Algebras (3) A concise Topics in mathematical logic and the foundations of mathematics. introduction to von Neumann algebra theory, beginning with the basic Prerequisite: MATH 558 definitions and proceeding through modular theory. The currently important subjects of index theory and free probability theory will be MATH 577: Stochastic Systems for Science and Engineering introduced.

3 Credits Prerequisite: MATH 503

The course develops the theory of stochastic processes and linear and MATH 585: Topics in Mathematical Modeling nonlinear stochastic differential equations for applications to science and engineering. 3 Credits

Prerequisite: MATH 414 or MATH 418 ; M E 550 or MATH 501 Introduction to mathematical modeling, covering the basic modeling and Cross-listed with: ME 577 common mathematical techniques for problems from physical, biological and social sciences. MATH 578: Theory and Applications of Wavelets Prerequisite: MATH 403 , MATH 411 , and MATH 412 3 Credits MATH 588: Complexity in Theory and physical interpretation of continuous and discrete wavelet transforms for applications in different engineering disciplines. 3 Credits

Prerequisite: M E 550 or MATH 501 Complexity of , polynomial multiplication, fast Cross-listed with: ME 578 Fourier transform, , calculating the greatest common divisor of polynomials. MATH 580: Introduction to Applied Mathematics I Cross-Listed 3 Credits MATH 596: Individual Studies A graduate course of fundamental techniques including tensor, ordinary and partial differential equations, and linear transforms. 1-9 Credits/Maximum of 9

Prerequisite: Basic knowledge of linear algebra, and ODE, Creative projects, including nonthesis research, which are supervised on MATH 405 an individual and which fall outside the scope of formal courses.

MATH 581: Introduction to Applied Mathematics II MATH 597: Special Topics

3 Credits 1-9 Credits/Maximum of 9

A graduate course of fundamental techniques including Ordinary, Partial, Formal courses given on a topical or special interest subject which may and Stochastic Differential Equations, Wavelet Analysis, and Perturbation be offered infrequently; several different topics may be taught in one year Theory. or term.

MATH 582: Introduction to C* Algebra Theory MATH 600: Thesis Research

3 Credits 1-15 Credits/Maximum of 999

Basic properties of C* algebras, , group C* algebras No description. and crossed products, tensor products, nuclearity and exactness. 6 Mathematics (MATH)

MATH 601: Ph.. Dissertation Full-Time

0 Credits/Maximum of 999

No description.

MATH 610: Thesis Research Off Campus

1-15 Credits/Maximum of 999

No description.