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

16:640:503Theory 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, residue theorem. Prerequisite: Advanced calculus. |

16:640:504Theory 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. |

16:640:507,508Functional Analysis (3,3) First term: 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 term: Banach algebra, spectral theory of operators on Hilbert space, other selected topics and applications. Prerequisites: 16:640:502, 503, 540. |

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

16:640:513Numerical Functional Analysis (3) Bases in Banach spaces. Constructive solution of equations involving symmetric and nonsymmetric linear operators and nonlinear compact, P-compact, monotone-accretive, and A-proper operators. Constructive fixed-point theory. Eigenvalue problems. Application to differential and integral equations. Prerequisite: Permission of instructor. |

16:640:515Ordinary 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. |

16:640:517,518Partial 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, hyperbolic equations. Elliptic boundary value problems and Sobolev spaces. Pre- or corequisites: 16:640:502, 503, 507. |

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

16:640:520Distribution Theory (3) Spaces of distribution; temperated distributions; Sobolov spaces; spaces of test functions; topology and duality of these spaces. Kernel theorems. Growth conditions; the Fourier transform. Prerequisites: 16:640:501,502. |

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

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

16:640:529Potential Theory (3) Harmonic and superharmonic functions in Rn; polar sets, potentials, capacities, Green`s functions, balayage, thin sets, and the 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. |

16:640:532Differential Geometry (3) Differentiable manifolds, connections, Riemannian manifolds. |

16:640:533Introduction 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. |

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

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

16:640:537Selected Topics in Geometry (3) |

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

16:640:542,543Algebraic Topology (3,3) Further topics of algebraic and differential topology, including duality theorems, homotopy theory, vector bundles, characteristic classes, and applications to geometric problems. Prerequisites: 16:640:504, 541. |

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

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

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

16:640:548Differential Topology (3) Vector bundles, differentiable manifolds. Sard`s theorem and applications to imbedding problems. Tubular neighborhoods. Other selected topics. Prerequisites: 16:640:540,541. |

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

16:640:550Lie 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, 16:640:551,552. |

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

16:640:553Theory 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, infinite groups. Prerequisite: 16:640:551. |

16:640:555Selected Topics in Algebra (3) Prerequisite: Permission of instructor. |

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

16:640:558Theory 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. |

16:640:559Commutative Algebra (3) Ideal theory, Noetherian rings, local rings, regular local rings, valuation theory, polynomial and power series rings, Gröbner bases, computations in polynomial rings. Prerequisite: 16:640:552. |

16:640:560Homological Algebra (3) Projective and injective modules, the derived functions Ext and Tor, categories and functors. Morita theorems, homological dimension. Prerequisite: 16:640:552. |

16:640:561Mathematical 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. |

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

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

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

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

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

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

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

16:640:616,617Seminar in Mathematics (1,1) Two-term 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. |

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

16:640:663Topics in Mathematical Physics (3) |

16:640:699Nonthesis Study (1) |

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