MATH G4073x. Quantitative Methods In Investment Management. 3 pts. Prerequisites: Knowlege of statistics basics and programming skills in any programming language.

Surveys the field of quantitative investment strategies from a "buy side" perspective, through the eyes of portfolio managers, analysts and investors. Financial modeling there often involves avoiding complexity in favor of simplicity and practical compromise. All necessary material scattered in finance, computer science and statistics is combined into a project-based curriculum, which give students hands-on experience to solve real world problems in portfolio management. Students will work with market and historical data to develop and test trading and risk management strategies. Programming projects are required to complete this course.

MATH G4151x. Analysis and Probability. 4.5 pts.

Measure theory; elements of probability; elements of Fourier analysis; Brownian motion.

MATH G4152y. Analysis, II. 4.5 pts.

Continuation of Mathematics G4151 (see Fall listing).

MATH G4153y. Probability, II. 4.5 pts.

Continuation of Mathematics G4152 (see fall listing).

MATH G4175x. Complex Analysis and Riemann Surfaces.. Topics include holomorphic functions; analytic continuation; Riemann surfaces; theta functions and modular forms.

MATH G4176y. Complex Analysis and Riemann Surfaces.. 4.5 pts. Continuation of Mathematics G4175 (see Fall listing).

MATH G4180x-G4181. Analytical Number Theory On the Group Gl(N,Z). 4.5 pts. Not offered in 2009-2010. Prerequisites: complex variables, graduate algebra. A systematic development of the theory of automorphic forms for the group GL(n,Z). Topics include: Spectral decomposition into Maass forms and Eisenstein series, L-functions with Euler products, and applications to classical problems in number theory.

MATH G4261x. Commutative Algebra. 4.5 pts. Commutative rings; modules; localization; primary decoposition; integral extensions; Noetherian and Artinian rings; Nullstellensatz; Dedekind domains; dimension theory; regular local rings.

MATH G4262y. Algebraic Geometry. 4.5 pts. Affine and projective varieties; schemes; morphisms; sheaves; divisors; cohomology theory; curves; Riemann-Roch theorem.

MATH G4307x. Algebraic Topology. Topics include homology and homotopy theory; covering spaces; homology with local coefficients; cohomology; Chech cohomology.

MATH G4308y. Algebraic Topology. 4.5 pts. Continuation of Mathematics G4307 (see Fall listing).

MATH G4343x. Lie Groups and Representations. Topics include basic notions of groups with algebraic and geometric examples; symmetry; Lie algebras and groups; representations of finite and compact Lie groups; finite groups and counting principles; maximal tori of a compact Lie group.

MATH G4344y. Lie Groups and Representations. Continuation of Mathematics G4343 (see Fall listing).

MATH W4386x-W4387y. Geometrical Concepts In Physics. 3 pts. Not offered in 2009-2010. Prerequisites:MATH V1202 or the equivalent and V2010.

Material from topology and differential geometry with illustrations of their use in electrodynamics, general relativity, and Yang-Mills theory. In particular topological and differential manifolds, tensors, vector bundles, connections, and Lie groups are covered.

MATH G4402x. Modern Geometry. 4.5 pts.

Manifold theory; differential forms, tensors and curvature; homology and cohomology; Lie groups and Lie algebras; fiber bundles; homotopy theory and defects in quantum field theory; geometry and string theory.

MATH G4403y. Modern Geometry. 4.5 pts.

Continuation of Mathematics G4401 (see Fall listing).

MATH G4472x. Hyperbolic Geometry. 4.5 pts. Not offered in 2009-2010. Prerequisites: advanced algebra Models for hyperbolic geometry, isometries, fractional linear transformations, discrete groups of isometries, fundamental domains, hyperbolic manifolds, Teichmueller space, Moscow-Prasad rigidity theorems.

MATH G4560x. Large Deviations. 4.5 pts. Not offered in 2009-2010. Prerequisites:MATH G4151

The study of rare events as in Theorems of Cramer, Schilder, Gartner-Ellis, Varadhan, Bryc, Sanov, Freidlin-Wentzell, up to entropy methods and Markov Processes. Applications to Statistics, Numerical Methods, Statistical Mechanics according to interest.

MATH G6071y. Numerical Methods In Finance. 4.5 pts. Prerequisites: some familiarity with the basic principles of partial differential equations, probability and stochastic processes, and of mathematical finance as provided, e.g., in Mathematics W4071. Review of the basic numerical methods for partial differential equations, variational inequalities and free-boundary problems. Numerical methods for solving stochastic differential equations; random number generation, Monte Carlo techniques for evaluating path-integrals, numerical techniques for the valuation of American, path-dependent and barrier options.

MATH G6116x. The Selberg Trace Formula I. 4.5 pts. Not offered in 2009-2010. Prerequisites: Lie groups and representations (G4343)and elementary Number Theory. Automorphic representations of GL(2). Analytical aspects of the trace formula. Applications to the principle of functoriality and Artin conjecture.

MATH G6117y. The Selberg Trace Formula II. 4.5 pts. Not offered in 2009-2010. Prerequisites: The Selberg Trace Formula I (G6116) Arithmetic aspects of the trace formula. Applications to the construction of 1-adic representations attached to modular forms.

MATH G6209x-G6210y (Section 1). Partial Differential Equations. 4.5 pts. Topics of linear and non-linear partial differential equations of second order, with particular emphasis to Elliptic and Parabolic equations and modern approaches.

MATH G6238x. Enumerative Combinatorics. 4.5 pts. Not offered in 2009-2010. Techniques of generating functions, partially ordered sets and lattices, Mobius inversion, topological aspects of posets and applications to commulative algebra.

MATH G6248x. p-adic Eisenstein series. 4.5 pts. p-adic modular forms. p-adic Eisenstein series. Generalizations.

MATH G6267x. Floer Homology. 4.5 pts. Not offered in 2009-2010. A novel application of ideas from Morse theory in certain infinite-dimensional situations. Serves as an introduction to this theory, with an emphasis on topological applications and calculations of Floer homology groups. Assumes some familiarity with ideas from algebraic geometry and differential topology.

MATH G6325. Topics In Geometric Topology. 4.5 pts. Not offered in 2009-2010. Prerequisites: 1st year graduate course in modern geometry. First year course in analysis helpful but not required. A one semester course covering Perelman's recent proof of the Poincare Conjecture using the Ricci flow on the space of metrics. The course will begin with a brief outline of Thurston's Geometrization Conjecture for 3-manifolds, and a brief introduction to the basics of Ricci flow as developed by Hamilton. The course will concentrate on the parts of Perelman's two papers and the Colding-Minicozzi paper needed to prove the Poincare Conjecture.

MATH G6428x-G6429y. Partial Differential Equations. 4.5 pts. Analytic and geometric methods in the study of partial differential equations, in particular maximum principles, Harnack inequalities, isoperimetric inequalities, formation and singularities. Emphasis on non-linear heat equations and geometric evolution equations

MATH G6490x. 3-Manifolds. 4.5 pts. Not offered in 2009-2010. Corequisites: A year of abstract algebra is recommended. Hyperbolic manifolds; rigidity; arithmeticity and non-arithmeticity; arithmetic manifolds derived from quadratic forms, from quaternion algebras; commensurability and Borel's theorem; arithmetic invariants of hyperbolic manifolds.

MATH G6495x-G6496. Transformation Groups On Manifolds. 4.5 pts. Not offered in 2009-2010. A self-contained introduction to the theory of group actions (transfromation groups) on manifolds. Topics include: representations and G-spaces; vector bundles and characteristics classes; bordism theory and universal formal group; equivalent bordism theory; localization theorems.

MATH G6500x. The Poincaré Conjecture. 4.5 pts. Prerequisites: Modern Geometry and Analysis I. History of the Poincaré conjecture. Geometric flow equation. Ricci flow. Proof of the conjecture.

MATH G6761x or y. Topics In Arithmetic Geometry. 4.5 pts.

MATH G8000x or y. The Teaching of Mathematics. May be repeated. A seminar required of all incoming graduate students, designed to instill effective teaching techniques.

MATH G8210y. Math Finance Practitioner's Seminar. 3 pts. Prerequisites: MATH 4071 or knowledge of J. Hull's book Options, futures. Seminar consists of presentations and mini-courses by leading industry specialists in quantitative finance. Topics include portfolio optimization, exotic derivatives, high frequency analysis of data and numerical methods. While most talks require knowledge of mathematical methods in finance, some talks are accessible to general audience.

MATH G8680x. Advanced Topic - Analytic Number Theory. 4.5 pts. Not offered in 2009-2010. Prerequisites: complex variables and some familiarity with the spectral theory of GL(2)-automorphic forms. Some recent developments to classical problems in analytic number theory. Topics include: Integral Moments of the Riemann zeta-function- a new approach to the fourth moment, the non-existence of Landau-Seigel zeros for GL(2)-automorphic L-functions.

MATH G9902x. Research Seminar in Geometric Topology. 3 pts. Prerequisites: Instructor's permission. Discussion of current research activity in Geometic Topology.