## Penn Graduate Math Course Descriptions

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GRADUATE MATHEMATICS COURSE DESCRIPTIONS:

500. TOPOLOGY.

(A) Staff. Prerequisite(s): Prerequisite(s): Math 240/241, Math 360 or 508, or with the permission of the instructor.

Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.

501. GEOMETRY. (A) Staff. Prerequisite(s): Prerequisite(s): Math 240/241, Math 312, and Math 361 (or 508), or with the permission of the instructor.

The course moves from a study of extrinsic geometry (curves and surfaces in n-space) to the intrinsic geometry of manifolds. After a section on tensor algebra, we study manifolds and intrinsic geometry, including metrics, connections, and the Riemann curvature tensor. Other topics as time permits.

502/503. ABSTRACT ALGEBRA.

Staff. Prerequisite for Math 502:Â Math 240 or 260; and Math 314/514 or permission of the instructor.Â Prerequisite for Math 503: Math 502 or permission of the instructor. Credit given for each semester.

An introduction to groups, rings, fields and other abstract algebraic systems, elementary Galois Theory, and linear algebra - a more theoretical course than Math 370/371.

504/505. GRADUATE PROSEMINAR IN MATHEMATICS.

Staff. Credit given for each semester.

This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.

508. ADVANCED ANALYSIS. Terms: A

Staff. Prerequisite(s): Math 240/241 or 260. A previous course with proofs (e.g. Math 116, 260, 202, 203) is recommended. Credit given for each semester.

Construction of the real numbers, the topology of the real line and the foundations of single variable calculus. Notions of convergence for sequences of functions. Basic approximation theorems for continuous functions and rigorous treatment of elementary transcendental functions. The course is intended to teach students how to read and construct rigorous formal proofs. A more theoretical course than Math 360.

509. ADVANCED ANALYSIS. Terms: B

Staff. Prerequisite: Math 508 or permission of the instructor. Linear algebra is also helpful.

Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods.

520/521. SELECTIONS FROM ALGEBRA.

Staff. Corequisite(s): Math 500/501, Math 502/503. Credit given for each semester.

Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.

524/525. TOPICS IN MODERN APPLIED ALGEBRA.

Staff. Prerequisite(s): Math 371 or Math 503. Credit given for each semester.

Topics such as automata, finite state languages, Boolean algebra, computers and logical design will be discussed.

530. MATHEMATICS OF FINANCE.

Staff. Prerequisite(s): Math 240, Stat 430. Credit given for each semester.

This course presents the basic mathematical tools to model financial markets and to make calculations about financial products, especially financial derivatives. Mathematical topics covered: stochastic processes, partial differential equations and their relationship. No background in finance is assumed.

540/541. SELECTIONS FROM CLASSICAL AND FUNCTIONAL ANALYSIS.

Staff. Corequisite(s): Math 500/501, Math 508/509. Credit given for each semester.

Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces.

542. CALCULUS OF VARIATIONS.

Staff. Prerequisite(s): Math 241.

Introduction to calculus of variations. The topics will include the variation of a functional, the Euler-Lagrange equations, parametric forms, end points, canonical transformations, the principle of least action and conservation laws, the Hamilton-Jacobi equation, the second variation.

546. PROBABILITY THEORY

The required background is (1) enough math background to understand proof techniques in real analysis (closed sets, uniform convergence, fourier series, etc.) and (2) some exposure to probability theory at an intuitive level (a course at the level of Ross's probability text or some exposure to probability in a statistics class).

After a summary of the necessary results from mesaure theory, we will learn the probabilist's lexicon (random variables, independence, etc.). We will then develoop the necessary techniques (Borel Cantelli lemmas, estimates on sums of independent random variables and truncation techniques) to prove the classical laws of large numbers. Next come Fourier techniques and the Central Limit Theorem, followed by combinatorial techniques and the study of random walks.

548/549. TOPICS IN ANALYSIS.

Staff. Prerequisite(s): Math 360/361 or Math 508/509 and Math 370 or Math 502. Credit given for each semester.

Topics may vary but typically will include an introduction to topological linear spaces and Banach spaces, and to Hilbert space and the spectral theorem. More advanced topics may include Banach algebras, Fourier analysis, differential equations and nonlinear functional analysis.

560/561. SELECTIONS FROM GEOMETRY AND TOPOLOGY.

Staff. Corequisite(s): Math 500/501, Math 502/503. Credit given for each semester.

Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory.

570/571. INTRODUCTION TO LOGIC AND COMPUTABILITY.

Staff. Prerequisite(s): Math 371 or Math 503. Credit given for each semester.

Propositional logic: semantics, formal deductions, resolution method. First order logic: validity, models, formal deductions; Godel's completeness theorem, Lowenheim-Skolem theorem: cut-elimination, Herbrand's theorem, resolution method. Computability: finite automata, Turing machines, Godel's incompleteness theorems. Algorithmically unsolvable problems in mathematics.

574/575. MATHEMATICAL THEORY OF COMPUTATION.

Staff. Prerequisite(s): Math 320/321. Credit given for each semester.

This course will discuss advanced topics in Mathematical Theory of Computation.

580/581. COMBINATORIAL ANALYSIS AND GRAPH THEORY.

Staff. Prerequisite(s): Permission of the instructor. Credit given for each semester.

Generating functions, enumeration methods, Polya's theorem, combinatorial designs, discrete probability, extremal graphs, graph algorithms and spectral graph theory, combinatorial and computational geometry.

582/583. APPLIED MATHEMATICS AND COMPUTATION.

Staff. Credit given for each semester.

This course offers first-hand experience of coupling mathematics with computing and applications. Topics include: Random walks, randomized algorithms, information theory, coding theory, cryptography, combinatorial optimization, linear programming, permutation networks and parallel computing. Lectures will be supplemented by informal talks by guest speakers from industry about examples and their experience of using mathematics there.

584/585. THE MATHEMATICS OF MEDICAL IMAGING AND MEASUREMENT.

Staff. Prerequisite(s): Math 241, knowledge of linear algebra and basic physics. Credit given for each semester.

In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly. The data which is accessible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT). In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.

590/591. ADVANCED APPLIED MATHEMATICS.

Staff. Prerequisite(s): Math 241. Credit given for each semester.

This course offers first-hand experience of coupling mathematics with applications. Topics will vary from year to year. Among them are: Random walks and Markov chains, permutation networks and routing, graph expanders and randomized algorithms, communication and computational complexity, applied number theory and cryptography.

594(Phys500). ADVANCED METHODS IN APPLIED MATHEMATICS.

Staff. Prerequisite(s): Math 241 or Permission of instructor. Physics 151 would be helpful for undergraduates. Credit given for each semester.

Introduction to mathematics used in physics and engineering, with the goal of developing facility in classical techniques. Vector spaces, linear algebra, computation of eigenvalues and eigenvectors, boundary value problems, spectral theory of second order equations, asymptotic expansions, partial differential equations, differential operators and Green's functions, orthogonal functions, generating functions, contour integration, Fourier and Laplace transforms and an introduction to representation theory of SU(2) and SO(3). The course will draw on examples in continuum mechanics, electrostatics and transport problems.

599. MASTERS INDEPENDENT STUDY.

600. TOPOLOGY AND GEOMETRIC ANALYSIS.

(A) Staff. Prerequisite(s): Math 500/501 or with the permission of the instructor.

Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.

601. TOPOLOGY AND GEOMETRIC ANALYSIS.

(B) Staff. Prerequisite(s): Math 600 or with the permission of the instructor.

Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.

602/603. ALGEBRA.

Staff. Prerequisite: Math 370/371 or Math 502/503. Credit given for each semester.

Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal deals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations.

604. FIRST YEAR SEMINAR IN MATHEMATICS. Terms: A

Open to first year Mathematics graduate students. Others need permission of the instructor.

This is a seminar for first year Mathematics graduate students, supervised by faculty. Students give talks on topics from all areas of mathematics at a level appropriate for first year graduate students. Attendance and preparation will be expected by all participants, and learning how to present mathematics effectively is an important part of the seminar.

605. FIRST YEAR SEMINAR IN MATHEMATICS. Terms: B

Open to first year Mathematics graduate students. Others need permission of the instructor.

Continuation of Math 604.

MATH 608 ANALYSIS (First semester). TERM: A.

Staff. Prerequisite: Math 360/361 or Math 508/509.

Complex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, d-bar equation, Mittlag-Leffler theorem, harmonic and sub-harmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. Introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C^0 and the Riesz-Markov theorem, L^p spaces, Fubini theorem.

MATH 609 ANALYSIS (Second semester). TERM: A.

Staff. Prerequisite: Math 608.

Real analysis: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, Radon-Nikodym theorem, Fubini's theorem, Hilbert space and L^2-theory of the Fourier transform. Functional analysis: normed linear spaces, convexity, the Hahn-Banach Theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, L^p theory for the Fourier transform.

618/619. ALGEBRAIC TOPOLOGY, PART I.

(A) Staff. Prerequisite(s): Math 600/601. Credit given for each semester.

Simplicial and singular homology and cohomology, homotopy groups, Hurewicz theorem, Whitehead theorem, spectral sequences. Classification of vector bundles and fiber bundles. Characteristic classes and obstruction theory. Additional topics can include rational homotopy theory, cobordism, K-theory, Morse theory and the h-cobordism theorem, surgery theory.

620/621. ALGEBRAIC NUMBER THEORY.

Staff. Prerequisite: Math 602/603. Credit given for each semester.

Dedekind domains, local fields, basic ramification theory, product formula, Dirichlet unit theory, finiteness of class numbers, Hensel's Lemma, quadratic and cyclotomic fields, quadratic reciprocity, abelian extensions, zeta and L-functions, functional equations, introduction to local and global class field theory. Other topics may include: Diophantine equations, continued fractions, approximation of irrational numbers by rationals, Poisson summation, Hasse principle for binary quadratic forms, modular functions and forms, theta functions.

622/623. COMPLEX ALGEBRAIC GEOMETRY.

Staff. Prerequisite: Math 602/603. Credit given for each semester.

Algebraic geometry over the complex numbers, using ideas from topology, complex variable theory, and differential geometry. Topics include: Complex algebraic varieties, cohomology theories, line bundles, vanishing theorems, Riemann surfaces, Abel's theorem, linear systems, complex tori and abelian varieties, Jacobian varieties, currents, algebraic surfaces, adjunction formula, rational surfaces, residues.

624/625. ALGEBRAIC GEOMETRY.

Staff. Prerequisite: Math 602/603. Credit given for each semester.

Algebraic geometry over algebraically closed fields, using ideas from commutative algebra. Topics include: Affine and projective algebraic varieties, morphisms and rational maps, singularities and blowing up, rings of functions, algebraic curves, Riemann Roch theorem, elliptic curves, Jacobian varieties, sheaves, schemes, divisors, line bundles, cohomology of varieties, classification of surfaces.

626/627. COMMUTATIVE ALGEBRA.

Staff. Prerequisite(s): Math 602/603. Credit given for each semester.

Topics in commutative algebra taken from the literature. Material will vary from year to year depending upon the instructor's interests.

628/629. HOMOLOGICAL ALGEBRA.

Staff. Prerequisite(s): Math 602/603. Credit given for each semester.

Complexes and exact sequences, homology, categories, derived functors (especially Ext and Tor). Homology and cohomology arising from complexes in algebra and geometry, e.g. simplicial and singular theories, Cech cohomology, de Rham cohomology, group cohomology, Hochschild cohomology. Projective resolutions, cohomological dimension, derived categories, spectral sequences. Other topics may include: Lie algebra cohomology, Galois and etale cohomology, cyclic cohomology, l-adic cohomology. Algebraic deformation theory, quantum groups, Brauer groups, descent theory.

630/631. DIFFERENTIAL TOPOLOGY.

Staff. Prerequisite(s): Math 600/601. Credit given for each semester.

Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory. The instructor may elect to cover other topics such as Morse Theory, h-cobordism theorem, characteristic classes, cobordism theories.

632/633. TOPOLOGICAL GROUPS.

Staff. Prerequisite(s): Math 600/601 and Math 602/603. Credit given for each semester.

Fundamentals of topological groups. Haar measure. Representations of compact groups. Peter-Weyl theorem. Pontrjagin duality and structure theory of locally compact abelian groups.

638/639. ALGEBRAIC TOPOLOGY, PART II.

Staff. Prerequisite(s): Math 618/619. Credit given for each semester.

Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.

640/641. ORDINARY DIFFERENTIAL EQUATIONS.

Staff. Prerequisite(s): Math 508/509. Credit given for each semester.

General existence and uniqueness theorems for systems of ordinary differential equations and the dependence of solutions on initial conditions and parameters appearing in the equation. Proofs of existence and uniqueness are related to numerical algorithms for finding approximate solutions for systems of ODE's. Special properties of constant coefficient and linear systems, We then present the theory of linear equations with analytic coefficients, the theories of singular points, indicial roots and asymptotic solutions. Boundary value problems for second order equations with an emphasis on the eigenfunction expansions associated with self adjoint boundary conditions and the Sturm comparison theory; other topics. Hamiltonian systems and symplectic geometry, singular boundary value problems, perturbation theory, the Lyapounov-Schmidt theory and the Poincare-Bendixson theorem, the equations of mathematical physics, the calculus of variations, symmetries of ODE's and transformation groups.

642. TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS.

Staff. Prerequisite(s): Math 608/609. This course will not presume courses in Partial Differential Equations or Differential Geometry. Background will be covered in the course.

Problems in differential geometry, as well as those in physics and engineering, inevitable involve partial derivatives. This course will be an introduction to these problems and techniques. We will use P.D.E. as a tool. Some of the applications will be small, some large. The proof of the Hodge Theorem will be a small application. Discussion of the Yamabe problem and Ricci flow (used to prove the Poincare Conjecture) will be larger.

644/645. PARTIAL DIFFERENTIAL EQUATIONS.

Staff. Prerequisite(s): Math 608/609 or permission of the instructor. Credit given for each semester.

Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo-differential operators, and nonlinear problems. Sobolev spaces and the theory of distributions will be developed as needed.

646/647. SEVERAL COMPLEX VARIABLES.

Staff. Prerequisite(s): Math 600/601, Math 608/609. Credit given for each semester.

Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.

648. PROBABILITY THEORY

Staff. Prerequisite: STAT 430 OR STAT 510 OR MATH 608.

Measure theory and foundations of Probability theory. Zero-one Laws. Probability inequalities. Weak and strong laws of large numbers. Central limit theorems and the use of characteristic functions. Rates of convergence. Introduction to Martingales and random walk.

649. STOCHASTIC PROCESSES

Staff. Prerequisite: MATH 648 OR STAT 930.

Markov chains, Markov processes, and their limit theory. Renewal theory. Martingales and optimal stopping. Stable laws and processes with independent increments. Brownian motion and the theory of weak convergence. Point processes.

650/651. LIE ALGEBRAS.

Staff. Prerequisite(s): Mth 602/603. Credit given for each semester.

Connections with Lie grouups, universal enveloping algebras, Poincare-Birkhoff-Witt Theorem, Lie and Engels Theorems, free Lie algebras, Killing form, semisimple algebras, root systems, Dynkin diagrams, classification of complex simple Lie algebras, representation theory of Lie algebras, cohomology of Lie algebras.

652/653. OPERATOR THEORY.

Staff. Credit given for each semester.

Subject matter may include spectral theory of operators in Hilbert space, C*-algebras, von Neumann algebras.

654/655. LIE GROUPS.

Staff. Prerequisite(s): Math 600/601, Math 602/603. Credit given for each semester.

Connection of Lie groups with Lie algebras, Lie subgrops, exponential map. Algebraic Lie groups, compact and complex Lie groups, solvable and nilpotent groups. Other topics may include relations with symplectic geometry, the orbit method, moment map, symplectic reduction, geometric quantization, Poisson-Lie and quantum groups.

656/657. REPRESENTATION OF CONTINUOUS GROUPS.

Staff. Credit given for each semester.

Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups.

660/661. DIFFERENTIAL GEOMETRY.

Staff. Prerequisite(s): Math 600/601, Math 602/603. Credit given for each semester.

Riemannian metrics and connections, geodesics, completeness, Hopf-Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet-Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions, the soul theorem, Gromov-Hausdorff convergence.

670/671. TOPICS IN LOGIC.

Staff. Prerequisite(s): Math 570/571. Credit given for each semester.

Discusses advanced topics in logic.

676. ADVANCED GEOMETRIC METHODS IN COMPUTER SCIENCE.

Staff. Prerequisite(s): Math 312 or Math 412, or with the permission of the instructor. Credit given for each semester.

Advanced geometric methods used in geometric modeling,computer graphics, computer vision, and robotics.

680/681. APPLIED LINEAR ANALYSIS.

Staff. Prerequisite(s): Math 241 and one semester of: Math 360/361 or Math 508/509. Credit given for each semester.

Application of techniques from linear algebra to real problems in economics, engineering, physics, etc. and the difficulties involved in their implementation. Review of linear algebra from the point of view of applications and analysis. Particular emphasis placed on solving equations, the eigenvalue problem for symmetric matrices and the metric geometry of spaces of matrices. Application to problems such as options pricing, image reconstruction, airplane and ship design, oil prospecting, etc. (these topics will vary from year to year). Analysis of the numerical algorithms to solve such problems. We will consider rates of convergence, accuracy and stability.

690/691. TOPICS IN MATHEMATICAL FOUNDATIONS OF PROGRAM SEMANTICS.

Staff. Credit given for each semester.

The course discusses advanced topics in mathematical foundations of semantics of programming languages and programming structures.

692/693. NUMERICAL ANALYSIS.

Staff. Prerequisite(s): Math 320/321. Credit given for each semester.

A study of numerical methods for matrix problems, ordinary and partial differential equations, quadrature and the solution of algebraic or transcendental equations. Emphasis will be on the analysis of those methods which are particularly suited to automatic high-speed computation.

694/695. MATHEMATICAL FOUNDATIONS OF THEORETICAL PHYSICS.

Staff. Credit given for each semester.

Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.

702/703. TOPICS IN ALGEBRA.

Staff. Credit given for each semester.

Topics from the literature. The specific subjects will vary from year to year.

720/721. ADVANCED NUMBER THEORY.

Staff. Prerequisite(s): Math 620/621. Credit given for each semester.

Ramification theory, adeles and ideles, Tate's thesis, group cohomology and Galois cohomology, class field theory in terms of ideles and cohomology, Lubin-Tate formal groups, Artin and Swan conductors, central simple algebras over local and global fields, general Hasse principles. Other topics may include the following: zero-dimensional Arakelov theory, Tate duality, introduction to arithmetic of elliptic curves, local and global epsilon factors in functional equations, p-adic L-functions and Iwasawa theory, modular forms and functions and modular curves.

724/725. TOPICS IN ALGEBRAIC GEOMETRY.

Staff. Prerequisite(s): Either Math 622/623 or Math 624/625. Credit given for each semester.

Topics from the literature. The specific subjects will vary from year to year.

730/731. TOPICS IN ALGEBRAIC AND DIFFERENTIAL TOPOLOGY.

Staff. Prerequisite(s): Math 618/619. Credit given for each semester.

Topics from the literature. The specific subjects will vary from year to year.

748/749. TOPICS IN CLASSICAL ANALYSIS.

Staff. Prerequisite(s): Math 608 and 609 and permission from the instructor. Credit is given for each semester.

Harmonic analysis in Euclidean space, Riemann surfaces, discontinuous groups and harmonic analysis in hyperbolic space, pseudodifferential operators and index theorems, variational methods in non-linear PDE, hyperbolic equations and conservation laws, probability and stochastic processes, geometric measure theory, applications of analysis to problems in differential geometry. The specific subjects will vary from year to year.

750/751. TOPICS IN FUNCTIONAL ANALYSIS.

Staff. Credit given for each semester.

Topics from the literature. The specific subjects will vary from year to year.

752/753. TOPICS IN OPERATOR THEORY.

Staff. Credit is given for each semester.

Topics from the literature. The specific subjects will vary from year to year.

760/761. TOPICS IN DIFFERENTIAL GEOMETRY.

Staff. Prerequisite(s): Math 660/661. Credit is given for each semester.

Topics from the literature. The specific subjects will vary from year to year.

794. PHYSICS FOR MATHEMATICIANS.

Staff. Prerequisite(s): Math 694. Corequisite(s): Math 695. Credit given for each semester.

This course is designed to bring mathematicians with no physics background up to speed on the basic theories of physics: Mechanics, relativity, quantum mechanics, classical fields, quantum filed theory, the standard model, strings, superstrings, and M-theory.

810. READING SEMINAR.

Staff.

May be taken for multiple course unit credit. Study under the direction of a faculty member.

820/821. ALGEBRA SEMINAR.

Staff.

Seminar on current and recent literature in algebra.

824/825. SEMINAR IN ALGEBRA, ALGEGRAIC GEOMETRY, NUMBER THEORY.

Staff.

Seminar on current and recent literature in algebra, algebraic geometry, and number theory.

830/831. GEOMETRY-TOPOLOGY SEMINAR.

Staff.

Seminar on current and recent literature in geometry-topology.

840/841. ANALYSIS SEMINAR.

Staff.

Seminar on current and recent literature in analysis.

844/845. SEMINAR IN PARTIAL DIFFERENTIAL EQUATIONS.

Staff.

Seminar on current and recent literature in partial differential equations.

850/851. SEMINAR IN FUNCTIONAL ANALYSIS.

Staff.

Seminar on current and recent literature in functional analysis.

860/861. SEMINAR IN RIEMANNIAN GEOMETRY.

Staff.

Seminar on current and recent literature in Riemannian Geometry.

870/871. LOGIC SEMINAR.

Staff.

Seminar on current and recent literature in logic.

872/873. SEMINAR IN LOGIC AND COMPUTATION.

Staff.

Seminar on current and recent literature in logic and computation.

880/881. COMBINATORICS SEMINAR.

Staff.

Seminar on current and recent literature in combinatorics.

999. INDEPENDENT STUDY AND RESEARCH.

Staff.

May be taken for multiple course unit credit. Study under the direction of a faculty member.