16:640:501,502 Theory of Functions of a Real Variable I,II (3,3) Real number system, measure theory, and Lebesgue integration in Euclidean and abstract spaces, set functions, bounded variation, absolute continuity, and differentiation of the indefinite integral; radon measure, and Lp�spaces. Prerequisite: Advanced calculus.

16:640:503 Theory of Functions of a Complex Variable I (3) Elementary properties of complex numbers, analytic functions, the exponential function and logarithm, conformal mapping, Cauchy integral formula, maximum modulus principle, Laurent series, classification of isolated singularities, and residue theorem. Prerequisite: Advanced calculus.

16:640:504 Theory of Functions of a Complex Variable II (3) Analytic continuation and the monodromy theorem, normal families and Riemann mapping theorem, Picard theorems, and other topics. Prerequisite: 16:640:503.

16:640:507,508 Functional Analysis I,II (3,3) First semester: introductory Hilbert space theory, including the Riesz representation theorem and the spectral theory of compact symmetric operators. Fundamental principles of linear analysis: Hahn-Banach, uniform boundedness, and closed-graph theorems. Weak topologies, Riesz theory of compact operators. Second semester:�Banach algebra, spectral theory of operators on Hilbert space, and other selected topics and applications. Prerequisites: 16:640:502, 503.

16:640:509,510,511 Selected Topics in Analysis (3,3,3) Prerequisites: 16:640:502 and permission of instructor.

16:640:515 Ordinary Differential Equations (3) Theory of ordinary and functional differential equations: basic existence theorems, linear systems, stability theory, periodic and almost-periodic solutions. Applications to biology, medicine, and physics. Prerequisites: Linear algebra and advanced calculus.

16:640:517,518 Partial Differential Equations I,II (3,3) Theory of distributions, Fourier transform, fundamental solutions of the heat, wave, and Laplace equations. The Cauchy problem, theorems of Cauchy-Kovalevska and Holmgren, and hyperbolic equations. Elliptic boundary value problems and Sobolev spaces. Pre- or corequisites: 16:640:502, 503, 507.

16:640:519 Selected Topics in Differential Equations (3) Topics in ordinary and partial differential equations chosen by the instructor. Prerequisite: Permission of instructor.

16:640:521 Harmonic Analysis on Euclidean Spaces (3) Maximal functions, fractional integrals, singular integrals, multipliers, Littlewood-Paley theory, Hp�spaces, weighted norm inequalities, Fourier series, and differentiation. Pre- or corequisite: 16:640:502.

16:640:523,524 Functions of Several Complex Variables I,II (3,3) Elementary theorems (Hartogs, Osgood), statement of Cousin and Levi problems, complex differential geometry, complex manifolds, and holomorphic convexity. Pre- or corequisites: 16:640:502, 503, 507.

16:640:529 Potential Theory (3) Harmonic and superharmonic functions in Rn; polar sets, potentials, capacities, Green's functions, balayage, thin sets, and fine topology. Energy and the Dirichlet integral. The Dirichlet problem in Rn. Lp boundary values and nontangential maximal functions for C1 and Lip1�boundaries. Ideal boundaries. Prerequisites: 16:640:502, 504.

16:640:532 Introduction to Differential Geometry (3) Riemannian manifolds, variational methods and theorems on geodesics, connections on vector and principal bundles, curvature, Euler, and other characteristic numbers and classes.

16:640:533 Differential Geometry (3) Differentiable manifolds, connections, and Riemannian manifolds.

16:640:534, 537 Selected Topics in Geometry I, II (3, 3) Selected topics, including Lie groups, representation theory, homogeneous spaces, and semi-Riemannian manifolds. Prerequisite: Permission of instructor.

16:640:535,536 Algebraic Geometry I,II (3,3) Geometry of projective spaces; cohomology of coherent sheaves and schemes. Prerequisite: Permission of instructor.

16:640:540,541 Introduction to Algebraic Topology I,II (3,3) Fundamental group, homology, and cohomology theory; elements of differentiable manifolds. Prerequisite: Basic concepts of point set topology.

16:640:544 Transformation Groups (3) Actions of compact Lie groups on manifolds. Prerequisite: 16:640:541. Corequisite: 16:640:549.

16:640:546 Topics in Algebraic Topology (3) K-theory, spectral sequences, cohomology operations, various other topics.

16:640:547 Topology of Manifolds (3) Selected topics from the theory of topological and combinatorial manifolds. Prerequisite: 16:640:541.

16:640:548 Introduction to Topology and Geometry (3) Introduction to the basic concepts and examples in topology and differential geometry, possibly including point set topology, differentiable manifolds, and elementary homotopy theory.
Prerequisite: 16:640:503.

16:640:549 Lie Groups (3) Lie groups. Lie algebras, elements of representation theory. Prerequisites: 16:640:532, 541.

16:640:550 Lie Algebras (3) Introduction to the general structure theory of Lie algebras and to the structure theory of finite-dimensional semisimple Lie algebras over the complex numbers. Prerequisites: Linear algebra, 16:640:551.

16:640:551,552 Abstract Algebra I,II (3,3) Introductory topics in groups, rings, modules, linear algebra, fields, Galois theory, and homological algebra.

16:640:553 Theory of Groups (3) Solvable groups, Nilpotent groups, p-groups, transfer and fusion, permutation groups. Topics chosen from among group representations and character theory, primitive permutation groups, local groups, theoretic analysis of simple groups, and infinite groups. Prerequisite: 16:640:551.

16:640:554 Topics in Algebra I (3) Topics of current interest.
Prerequisite: Permission of instructor.

16:640:555 Topics in Algebra II (3) Topics of current interest.
Prerequisite: Permission of instructor.

16:640:556 Representation Theory (3) Irreducible modules, representations of rings, and radicals of rings; Artinian, semisimple, and quotient rings. Prerequisite: 16:640:552.

16:640:558 Theory of Algebras (3) General theory of not necessarily associative algebras and rings. Topics selected from the theory of associative, Lie, alternative, and Jordan algebras. Structure and representation theory. Prerequisite: Permission of instructor.

16:640:559 Commutative Algebra (3) Ideal theory, Noetherian rings, local rings, regular local rings, valuation theory, Gr�bner bases, and computations in polynomial rings. Prerequisite: 16:640:552.

16:640:560 Homological Algebra (3) Projective and injective modules; the derived functions Ext and Tor; categories and functors; spectral sequences, and derived categories. Prerequisite: 16:640:552.

16:640:561 Mathematical Logic (3) Metamathematics and first-order arithmetic and analysis, with emphasis on the questions of consistency and completeness. Introduction to model theory and its application to the study of formal systems.

16:640:566 Axiomatic Set Theory (3) Axioms of Zermelo-Fraenkel, axioms of infinity consistency and independence of the continuum hypothesis, Dedekind-finite cardinals, and large cardinals. Prerequisite: 16:640:561.

16:640:567 Model Theory (3) Types of elements, prime and saturated models, methods of constructing models, the two-cardinal problem, categoricity, and power. Prerequisite: 16:640:561.

16:640:569 Selected Topics in Logic (3) Topics of current interest. Prerequisite: Permission of instructor.

16:640:571,572 Number Theory I,II (3,3) Integrated, yearlong introduction to ideas in algebraic and analytic number theory. Prerequisites: 16:640:551,552.

16:640:573 Special Topics in Number Theory (3) Iwaniec. Prerequisite: Permission of instructor.

16:640:574 Topics in Number Theory (3) Topics of current interest.
Prerequisite: Permission of instructor.

16:640:601 (S) Mathematics TA Instructional Training (0) Issues in teaching college-level mathematics courses; practical teaching sessions videotaped and critiqued by faculty instructors. Butler, Russell, Weingart. Open only to Ph.D. students in mathematics or other graduate students appointed to teaching assistantships in mathematics.

16:640:615 Special Studies in Advanced Mathematics (BA)

16:640:616,617 Seminar in Mathematics I,II (1,1) Two-semester participation in one of the seminars conducted by the department required of all candidates for the Ph.D. Prerequisite: Two years of graduate study in mathematics.

16:640:640 Experimental Mathematics (3) Use of computer algebra systems to design and conduct mathematical experiments leading to rigorous proofs. Emphasis on algorithmic and constructive aspects. Prerequisite: Permission of instructor.

16:640:651 Category Theory (3) Basic theory of categories, functors, and natural transformations. Abstract theory interpreted and illustrated through examples. Prerequisite: Some background in algebra and topology.

16:640:699 Nonthesis Study (1)

16:640:701,702 Research in Mathematics (BA,BA)