Graduate Courses

Courses marked with an asterisk (*) are special topics courses, designed by the instructor to lead graduate students to deeper study of a particular area that might lead to thesis research.  The other courses are offered at least every other year, and many are offered every year.

MATH 611 - Algebra I

The course will focus on rings (largely, but not exclusively, commutative ones) and modules,
including the structure theory of finitely generated modules over principal ideal domains.

MATH 612 - Algebra II

Description: This course is a continuation of Math 611. Topics covered will include group theory and Galois theory.

MATH 621 - Complex Analysis

This course will cover the basic theory of functions of one complex variable, at a pace that will allow for the inclusion of some non-elementary topics at the end. Basic Theory: Holomorphic and harmonic functions; conformal mappings; Cauchy's Theorem and consequences; Taylor and Laurent series; singularities; residues; elliptic functions and/or other topics as time permits.

MATH 623 - Real Analysis I

General theory of measure and integration and its specialization to Euclidean spaces and
Lebesgue measure; modes of convergence, L^1 spaces; product spaces; differentiation of measures and
functions, signed measures, Radon-Nikodym theorem.

MATH 624 - Real Analysis II

Continuation of Math 623. Signed measures and differentiation. Introduction to functional analysis including Hilbert spaces; Banach spaces and elementary operator theory.  L^p-spaces: duality and approximation theorems. Elementary Fourier Analysis and applications.

MATH 645 - ODE and Dynamical Systems

Classical theory of ordinary differential equations and some of its modern developments in
dynamical systems theory. Linear systems and exponential matrix solutions; Floquet theory for linear
periodic systems. Well-posedness for nonlinear systems. Qualitative theory: limit sets, invariant sets and
manifolds. Stability theory: linearization about equilibria and periodic orbits, Lyapunov functions.
Autonomous two-dimensional systems and other special systems.

*MATH 646 - Topics in Dynamical Systems: Ergodic Theory - Prof. L. Rey-Bellet [1]

Text: (No text required.) Recommended texts: An Introduction to Ergodic Theory''
by P. Walters (Springer) and
Intorduction to Ergodic Theory by Y. G. Sinai (Princeton Univ. Press)

Prerequisites: Students should be comfortable with mathematical analysis at the
level of beginning graduate level courses, like real and complex analysis.
Math 645 is helpful, but not a prerequisite.

Description: Consider a set L^P (the state space) and a map X (the time evolution). A trajectory of the system is the sequence T:X--> X. For interesting systems the question ``What is the state of the system at time x, T(x), T<sup>2</sup>(x) = T(T(x)), ...?'' is impractical, too hard, and, in a way irrelevant and we should ask instead ``What is the probability that at time x, T(x), T²(x), ... the system is in some specified subset of the state space?'' Of greatest interest is to study the asymptotics as x, T(x), T²(x), ... tends to infinity. This is the subject of ergodic theory. We will introduce and study the concepts of measure-preserving transformation, ergodicity, mixing, entropy. We will use examples taken from dynamical systems, probability, number theory and physics.

MATH 651 - Numerical Analysis I

The analysis and application of the fundamental numerical methods used to solve a common body
of problems in science. The following topics will be included: Methods for solving large linear systems
(direct and iterative methods), interpolation and least squares, nonlinear equations and systems, numerical
integration and the solving of initial value problems for ordinary differential equations. Grading will be
based on homework, exams and programs. The methods will be implemented by writing programs using a
scientific programming language of choice or in some cases MATLAB (no previous knowledge required).

MATH 652 - Numerical Solution of PDE's
Boundary value problems in ordinary differential equations (finite different and shooting methods), Finite difference methods for partial differential equations (consistency, stability and convergence), Finite Element Methods, Spectral Methods, Boundary effects, Iterative Methods for Linear systems, application to nonlinear systems. There will be assigned homework and programs.

MATH 671 - Topology I

Review of set theory. Metric spaces and topological spaces. Product and quotient topology.
Connectedness. Compactness, local compactness. Separation axioms. Complete metric spaces and function
spaces.

*MATH 697A - Number Theory - Prof. P. Gunnells (Spring 2004)

Texts: Elliptic Curves, by Anthony Knapp, Mathematical Notes, v. 40, Princeton Univ. Press.

Prerequisites: A good background in algebra at the level of 611-612 or consent of
instructor. Prior exposure to concepts in topology (671-781) and complex variables
621 will also be helpful

Description: This is a one-semester course on modular forms and their applications. Topics to be covered include the modular group and congruence subgroups, quotients of the upper halfplane, basic definition of modular forms, Fourier expansions of modular forms, cusp forms and Eisenstein series, Hecke operators, Petersson product, and modular curves. Additional possible topics include the connection between modular forms and the arithmetic of elliptic curves, modular symbols, and the Eichler-Selberg trace formula.

MATH 697AA - Convex Polytopes and Toric Varieties - Prof. T. Braden

Text: Combinatorial Convexity and Algebraic Geometry, by Günter Ewald
          Toric Varieties, by W. Fulton

Description: A toric variety is an algebraic variety built using data from a convex lattice polytope in Euclidean space. Toric varieties form a rather special class of varieties, but they are general enough to contain a lot of interesting examples, and many things such as cohomology and line bundles can be explicitly computed in terms of the geometry of the polytope. On the other hand, it is also possible to work the other way and study convex polytopes using the geometry of toric varieties.

We will use Ewald and Fulton as basic references, together with other sources including recent research papers. After we learn the basic definitions, the course will be run as a seminar. There are a lot of different applications of toric geometry, and which ones we spend the most time on will depend partly on student interest.

*MATH 697B and MATH 697S- Introduction to Riemann Surfaces - Prof. E. Markman
 (Fall 2001 and Fall 2004)

Text: 1) Complex Algebraic Curves, by Frances Kirwan, Cambridge University Press 1992.
2) Lectures on Riemann Surfaces by Otto Forster, Springer-Verlag 1981.

Prerequisites: 1) Complex Analysis (at the level of Math 621).
2) Familiarity with fundamental groups and covering spaces.

Description: An introductory course on the theory of complex algebraic curves (Riemann Surfaces). Topics will include:

Part 1) Complex algebraic curves in the affine and projective plane, Bezout's Theorem and cubic curves.

Part 2) Topological properties: Branched covers, the Riemann-Hurwitz formula, and the degree-genus formula for plane curves.

Part 3) Sheaf Cohomology, Divisors and Line bundles, Linear systems, Serre's Duality and the Riemann-Roch theorems,

Part 4) The Jacobian Variety and Abel's and Jacobi's Theorems.

We will develop the modern language of sheaf cohomology, but the emphasis will be on examples and applications. This may force us to omit some of the details of the proofs of the main Theorems in part 3. Throughout the course, numerous examples of the geometry of curves of low genus in the projective plane and space will be discussed.

*MATH 697D - Introduction to Algebraic Goups - Prof. E. Sommers [9:30]

Texts: TBA

Prerequisites: MATH 611-612

Description: Algebraic groups sit at the intersection of algebraic geometry and group theory and are a useful tool in representation theory, number theory, and algebraic geometry. In this course we will cover the classification of reductive algebraic groups. We will then survey some applications, including the Borel-Bott-Weil Theorem.

*MATH 697G - Introduction to Algebraic Geometry - Prof. T. Weston (Spring 2006)

Texts: Basic Alebraic Geometry I by I. Shafarerich (Springer-Verlag)

Prerequisites: MATH 611

Description: This course will be an introduction to algebraic geometry, a subject which is fundamental to much of mathematics. Topics to be covered include: algebraic curves, algebraic varieties, rational functions, dimension, singularities, divisors, differential forms, the Riemann-Roch Theorem, intersection theory on surfaces.


*697I - Introduction to Financial Engineering - Prof. M. Sullivan (Spring 2006)

Texts: "Arbitrage Theory in Continuous Time , Tomas Bjork, 2nd Ed., 2004, Oxford

Prerequisites: Stat 607 or Stat 515

Description: The object of this course is to introduce arbitrage theory and stochastic differential equations (SDE's), and use them to determine the price of certain financial securities, notably options. The class will cover stochastic calculus, as well as some basic SDE techniques, including martingales, Feynman-Kac representation, and the Kolmogorov equations. We consider Black-Scholes theory and its extensions, as well as incomplete markets. We cover several interest rate theories: short rates and the Heath-Jarrow-Morton framework. Stochastic optimal control problems (utility maximization) may also be covered if time permits.

Grading: The grade for the course will be determined by timely completion of four to six homeworks and two exams. The homeworks will make up 35% of the grade, the first exam will be worth 25-30% and the second exam 35-40%. There will be no make-up exams.

*MATH 697R - Applicable Algebraic Geometry - Prof. F. Sottile (Fall 2000)

Text: Draft of Applicable Algebraic Geometry, by Sottile, J. Rosenthal, A. Wang. The relevant chapters will be distributed during the course.

Prerequisites: Some familiarity with abstract algebra, particularly linear algebra and rings of polynomials.

Algebraic geometry, which is the geometric study of solutions to systems of polynomial equations, has potentially important applications -- polynomial equations and geometric problems are ubiquitous in mathematics and the applied sciences. Recent developments (symbolic computation and Gröbner bases) from within algebraic geometry greatly facilitate its use in the applied sciences. This course will concern itself with complementary developments -- specific problems from the applied sciences whose understanding uses ideas from algebraic geometry in an essential way. These include problems of pole placement and stabilization
(linear systems theory) as well as matrix completion and eigenvalue inequalities (matrix theory). The primary goal of this course will be to provide a motivated and very concrete introduction to some parts of algebraic geometry, particularly Grassmann manifolds and the Schubert calculus which have been important in these applications. No familiarity with algebraic geometry is expected.

*MATH 697S - Automorphic Forms - Prof. F. Williams (Fall 1999)

Prerequisites: Advanced calculus, basic complex variables

The course is a self-contained introduction to one of the most important enterprises in all of mathematics: the
theory of automorphic forms (classical and modern), a subject whose presence is felt very much in numerous areas of mathematics and more recently in physics. More focus will be given to historic classical topics such as the following: modular forms, Eisenstein series (holomorphic and non-holomorphic), Fourier expansion of Eisenstein series, cusp forms, the modular group, Hecke operators, Hecke's converse theorem, Euler products, Poincaré series, and Ramunujan's conjecture. If time permits an introduction to the modern theory will be given, such as L-functions for GL(2,R) and Rankin-Selberg theory.

*MATH 697T - Solving Polynomial Equations: A Gentle Introduction to Algebraic Geometry - Prof. E. Cattani (Spring 2000)
Texts: Cox, Little, O'Shea: Ideals, Varieties and Algorithms;
           Cox, Little, O'Shea: Using Algebraic Geometry
           Sturmfels: Solving Systems of Polynomial Equations

Prerequisites: Basics of Ring Theory at the level of a good undergraduate course or Math 611. No familiarity with commutative algebra
or algebraic geometry will be assumed.

The aim of this course is to introduce students to the ideas of symbolic computation in algebra and geometry and to the basic concepts of algebraic geometry and the algebra of polynomial rings. Its central theme will be the solution of systems of polynomial equations. This is a very classical problem with applications in many different branches of mathematics and science (chemistry, robotics, statistics, economics, to name a few) and which motivates much of classical algebraic geometry and commutative algebra. The key tool will be Groebner bases and the Buchberger algorithm, which may be viewed as a generalization of Gaussian elimination for non-linear systems. This algorithm, which makes it possible to replace the given system of equations by a simpler one, is implemented in general symbolic computation programs - such as Mathematica
or Maple - and in specialized ones such as Macaulay2 and Singular.

*MATH 597U/697U - Stochastic Proc. and Applications - Prof. L. Rey-Bellet (Fall 2003 and Spring 2006)

Text: Introduction to Probability Models (8th ed.) by Sheldon M. Ross (Academic Press)

Prerequisites: Undergraduate probability class (515) or instructor consent.

This course is an introduction to the theory and applications of stochastic processes. Topics include Markov chains, Poisson processes, continuous-time Markov processes, Brownian motion. In addition, if time permits, selected applications from simulation of Markov processes, hidden Markov models, queuing theory, etc., will be discussed.

*MATH 697W - Symmetric Functions - Prof. F. Sottile (Fall 2003)

Text: Bruce Sagan, The Symmetric Group: Representations, Combinatorial Algorithms,
and Symmetric Function, Springer Verlag (Graduate Texts in Mathematics, 203)

Prerequisites: The undergraduate or the graduate sequence in Algebra
and some mathematical maturity should suffice.

Symmetric functions are a central object in enumerative combinatorics and have applications in representation theory (they are the characters of irreducible representations of the general linear and symmetric groups).

This course will begin with the elementary algebraic and combinatorial theory of symmetric functions, including Schur functions and the Littlewood-Richardson rule. The middle of the course will discuss their role in the representation theory of the symmetric group and of the general linear group. The final part of the course will discuss applications of symmetric functions and their cousins, quasi-symmetric functions, to enumerative combinatorics.

We will use (but not strictly follow) the text book by Bruce Sagan.

MATH 703M - Theory of Manifolds I

This course gives the students the opportunity to tie together the various fundamental
components of mathematics: analysis, topology, algebra and geometry. After briefly reviewing the basics on
n-dimensional calculus, we will develop the notion of manifolds, vector bundles, connections and curvature.
We will investigate the basics of transversality, and apply this to differential topology, including results like the Whitney embedding and immersion theorems. We will also study special manifolds such as Lie groups, and special bundles such as the tangent bundle and its associated ``tensor'' bundles. The theory of exact PDE's and Frobenius integrability will be discussed. A linear version of this theory is the Maurer-Cartan Lemma which we apply to the trivialization of flat connections. All fundamental theorems in manifold and submanifold geometry are based on this result.

MATH 704 - Theory of Manifolds II

Continuation of Math 703. Integration on manifolds. Cohomology. Elliptic Theory. Connections, curvature & Chern classes.  Riemannian Geometry.

*MATH 713 - Elementary and Algebraic Number Theory - Prof. D. Hayes (Spring 2000)

Text:  A classical introduction to modern number theory by Ireland Rosen

The course will start with the basics: congruences, prime numbers, Chinese Remainder Theorem. This material will be covered rather quickly. We will then prove the Quadratic Reciprocity Theorem. Next, we will discuss some simple algorithms for ``recognizing'' primes and factoring moderately sized numbers. Then we will discuss a little algebraic number theory for real and complex quadratic number fields. If time permits, we will also discuss the basics of the arithmetic of elliptic curves with a hint as to how they were used in the proof of Fermat's Last Theorem.

*MATH 713 - Introduction to Algebraic Number Theory - Prof. F. Hajir (Fall 2005)

Text: Primes of the form , by David Cox, Wiley

Prerequisites: Math 611-612 (or consent of instructor).

Description: After covering the basics of algebraic number rings (Dedekind domains, Dirichlet unit theorem, finiteness of ideal class groups, etc.), the major goals of the course are to motivate and state all the major theorems of class field theory, and to learn how to use them to solve concrete questions about properties of ordinary integers. Included in the course will be an algorithm for computing class fields of imaginary quadratic fields using elliptic modular functions. Three major themes (Galois representations, arithmetic of elliptic curves, and modular forms) as well as their interconnections via L-functions will be emphasized throughout the course, as will computer calculations and explicit examples. No previous exposure to number theory beyond Math 611-612 will be assumed.

Bi-weekly problem sets will be collected and graded and the students can choose between a final exam or a final project.

*MATH 725 - Introduction to Functional Analysis - Prof. A. Benyi (Spring 2004)

Text: No Text required. Certain books will be suggested for background reading

Prerequisites: Math 623. Also recommended: Math 621 and Math 624

Description: The course wishes to cover some important topics of functional analysis, namely: linear topological spaces, closed graph and Hahn-Banach theorems, duality in locally convex spaces, Riesz's representation theorem in Hilbert spaces, and the spectral theorem in Banach algebras. Time permitting we will discuss other topics of interest to the audience.

MATH 731 - Partial Differential Equations I

This course introduces the modern methods of analysis in partial differential equations.
Emphasis is placed on the theory of existence, uniqueness and stability of solutions to boundary-value
problems and initial-value problems. This theory is developed in the context of the prototypical PDEs
arising in mathematical physics - The Laplace/Poisson equation, the Heat/Diffusion equation, and the Wave
equation. Only linear equations are considered in this one semester course. The theory of distributions is
presented along with some functional analysis. Elliptic problems are studied in Hilbert-Sobolev spaces,
using the variational formulation, and the associated spectral theory is developed. Parabolic problems are
considered in the setting of the analytic theory of semigroups. Hyperbolic problems are treated with energy
methods.

The course assumes that the student has some familiarity with the elementary methods of solution of linear
ODEs and PDEs. Modern analysis at the first-year graduate level is presumed.

MATH 732 - Linear/Nonlinear Waves - Prof. R. Young (Spring 2005)

Text: I will refer to the book we will work through Linear and Nonlinear Waves
by G. B. Whitham, although this is not required.

Prerequisites: The course should be fairly self-contained, and requires only a mastery
of calculus and some familiarity with basic ODEs and PDEs.

Description: Waves occur naturally in almost any physical system, and exhibit a large variety of behavior. Some major areas of application include: traffic flow, water waves, gas dynamics, electromagnetism, and ``universal equations''.

We will concentrate on the use of asymptotic expansions and heuristic descriptons of wave phenomena. The course is aimed at people interested in applications of mathematics, as well as those with an interest in analysis, PDE and/or differential equations.

Topics Include:

One-way wave equation, linear first-order systems, method of characteristics; second-order linear equations, Fourier expansions.

Nonlinear equations: Hopf equation, breakdown of solutions, shock formation.

Nonlinear hyperbolic systems: linearization; similarity; nonlinear waves; asymptotic states. Nonlinear interaction of waves. Local approximation of general solutions.

Multi-dimensional effects: symmetry, radial waves, effects of curvature.

Higher order effects: viscosity (dissipation), dispersion. Propagation of modes, dispersion relations and group velocity. Energy and amplitude propagation.

Applications to water waves, continuum physics, and other physical systems as time permits.

MATH 781 - Algebraic Topology

Fundamental group and covering spaces.  Homology of chain complexes and homology theories (singular
homology and cohomology, deRham cohomology), homology of CW complexes.  Applications to algebra,
analysis, and geometry.

*MATH 791A - Non-Linear Fourier Analysis - Prof. A. Nahmod (Spring 2003)

The purpose of this course is to give a self contained introduction to recent results and techniques concerning nonlinear evolution equations and the behavior of their solutions. We will mainly focus on nonlinear wave and Schrödinger equations. We will start with the classical local existence theory. Then we move to modern techniques involving nonlinear Fourier analysis and geometric consolidation that allow the regularity on the data to be weakened considerably and still get existence and uniqueness results. The course will mix lectures once a week with student's presentations. It will proceed as a working seminar.

*MATH 797A - Theory of Optimal Transport - Prof. J. Feng (Fall 2004)

Text: Topics in Optimal Transportation, Cédric Villani

Prerequisites: M624. Some elementary probability would be helpful.

Description: The origin of optimal mass transport theory can be traced to Monge's study of optimal allocation of mass in 1781, with applications to engineering. In 1942, Kantorovich reformulated the problem using an abstract linear programming, with applications to Economics. In the early 90s, using probability techniques, Rachev and Ruschendorf find applications in problems regarding minimal probability metric and stability of stochastic systems. In 1987, Brenier discovered an explicit map that describes the optimal transport plan, with application to fluid mechanics.

All these applications are connected and points to a common mathematical theme about optimal allocation of probability measures under a given cost functional.

This course will cover the basics of mass transport theory (Monge-Kantorovich duality using linear programming) and some latest developments (Brenier's map, Geometric, Gaussian and Entropy inequalities, displacement convexity...). The special emphasize will be probability and PDEs.

Prerequisites are real analysis and basics peoperties of general measures.

*MATH 797C/STAT 797C - Theory of Large Deviations and Applications - Prof. R. S. Ellis (Fall 2005)

Text: (Optional) A Weak Convergence Approach to the Theory of Large Deviations,
by Paul Dupuis and Richard S. Ellis. New York: John Wiley & Sons, 1997.

Prerequisites: Students should know basic measure theory (e.g., Math 623). While an acquaintance with basic measure-theoretic probability would be desirable (e.g., Stat 605), background in probability will be filled in as needed. The course will develop from scratch all other material that is needed.

Description: The theory of large deviations studies the exponential decay of probabilities in certain random systems. It is an extremely active branch of probability today, having applications to numerous areas including statistics, statistical mechanics, queueing systems, communication networks, information theory, and risk-sensitive control. This course will be an introduction to the theory. It will cover the following topics:

• Convex analysis and the Cramér theory for sums of i.i.d. random variables.
• The three levels of large deviations: sample means, empirical measures, empirical processes.
• Some generalities about large deviations.
• A new approach to large deviations based on the theory of weak convergence of probability measures.
• Applications to statistical mechanics and other areas.

A central concept in the course is entropy, which arises as a bridge between a microscopic and a macroscopic description of random systems. The course will begin by presenting the basic concepts of probability theory. Although the course is intended for non-experts, it will cover some exciting recent material that could lead to a dissertation.

All reading materials used in the course will be handed out by the instructor. The optional text will not be used extensively.

*MATH 797D - Topology and Geometry of Singular Spaces - Prof. P. Gunnells (Spring 2006)

Prerequisites: Completion of Math 703-704 or equivalent. Experience with Algebraic Topology (Math 781) and Math 611 also helpful but not as essential.

Description: Singular spaces arise naturally in many contexts, including algebraic geometry and representation theory. Any singular space has a decomposition into manifolds, and so the study of them is a mixture of topology, geometry, and combinatorics. Topics to be studied possibly include manifolds with boundary, manifolds with corners, Whitney stratifications, links, normal slices, the Thom presentation of a stratified space, orbifolds, hypersurface singularities, symmetric spaces and their compactifications.

*MATH 797E/MATH 797R- Homological Algebra - Prof. I. Mirkovic  (Spring 2002, Spring 2006)

Prerequisites: Algebra 611

Description: Homological algebra is a general tool useful in various areas of mathematics. One tries to apply it to constructions that morally should contain more information then meets the eye. The homological algebra, if it applies, produces "derived" versions of the construction ("the higher cohomology"), which contain the "hidden " information. The goal is to understand the usefulness of homological ideas in applications and to use this process as an excuse to visit various interesting topics in mathematics.

Some Topics: (1) Algebraic topology. (2) Duality of abelian groups. (3) Derived functors, Ext and Tor. (4) Sheaves and cohomology of sheaves. (5) Derived categories.

MATH 797F - Spectral Theory - Prof. W-M. Wang [11:15]

Texts: A useful reference: Schödinger Operators, H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon,
Springer-Verlag 1987

Prerequisites:

Description: We start from introducing basic notions in spectral theory. Some emphasis will be put on matrix operators on  l²(Z^d), which has the advantage of illuminating the basic ideas without the distraction of some technical issues.

We will then study equations coming from quantum mechanics, where there is an underlying dynamical system. This has connections to nonlinear Schrödinger equations. We will work with concrete objects such as the quantum harmonic oscillator and the discrete Laplacian.

The purpose of this course is to bring to attention some of the newer ideas and tools arising in the subject, which as of now are mainly scattered in the literature.

*MATH 797J - Riemann Surfaces - Prof. F. Pedit (Fall 1999)

Riemann surfaces are fundamental in modern mathematics.  Their structure and function theory contain
2-dimensional differential geometry/topology, complex variable theory, analysis of the Laplace equation,
algebraic geometry of complex curves, and mathematical physics. In other words, all of the fundamental pillars of mathematics come to bear. The present course will develop Riemann surface theory along the lines of the above topic list. We will use the geometrical language of line bundles (rather than divisors) to state and prove the basic results such as Serre duality, the Riemann-Roch theorem, the Abel-Jacobi theorem and the Kodaira embedding theorem.

Participants need to have a solid understanding of complex variable theory, topology and abstract linear algebra. Basic knowledge of the differential geometry of curves and surfaces in space would be helpful.

I will not follow a specific book. The following are books which develop the basic material and are quite readable for the most part:

O. Forster: Riemann surfaces, Springer.
R. Narashiman: Compact Riemann surfaces, Birkhauser.
R. C. Gunning: Lectures on Riemann surfaces, Princeton Univ. Press.
D. Mumford: Tata lectures on Theta, Vol 1 & 2, Birkhauser.
D. Mumford: Curves and their Jacobians, Univ of Michigan Press, 1975.

Those with a sense of beauty, poetry and leisure must read B. Riemann, Collected Works, H. Weyl, Die Idee der Reimannschen Flaeche, Teubner 1913. A more informal, physicists friendly introduction to the subject can be found in M. Cornalba, X. Gomez-Mont,  A. Verjovsky: Lectures on Riemann Surfaces, International Centre for Theoretical Physics, Trieste, Italy, Dec 1987. If you like the Russian school's view read the beautiful, comprehensive article by B. A. Dubrovin: Theta functions and non-linear equations, Russian Math. Surveys, 36 (1982), 11-92. The bible on all of this still is: P. A. Griffiths and J. Harris: Principles of Algebraic Geometry. If you want to know everything you need to know about Sheaf Theory in 15 pages look at the relevant chapter in: F. Hirzebruch: Topological methods in algebraic geometry, Springer.

*MATH 797M - Quantum Field Theory - Prof. Ivan Mirkovic (Spring 2000)

We will try to understand parts of the mathematical structure underlying the Quantum Field Theory. For us the
goal is not the physics of elementary particles but some understanding of the pervasive usefulness of Quantum Field Theory in mathematics. Over the last decade various aspects of Quantum Field Theory have penetrated several areas of mathematics. The Topological Quantum Field Theory gave a systematic approach to producing invariants that would help distinguish distinct manifolds of dimension 3 and 4. The Conformal Field Theory revolutionarized the representation theory and the related parts of number theory and algebraic geometry. Even more impressively, the String Theory led to completely unexpected fundamental
connections between previously unrelated classical questions in algebraic geometry. Possibly the most recent of these influences appears in Kontsevich's proof of the formality conjecture. Half of the Fields prizes last year (analogue of Nobel in mathematics) were awarded for the work of this kind (Borcherds, Kontsevich).

Topics: Formalism of Quantum Field Theory, Conformal Field Theory (a well developed special case), String Theory (what happens when we assume that a particle is a circle rather than a point), Quantum corrections (we will be interested in examples of quantum cohomology and Kontsevich's proof of the formality conjecture).

*MATH 797N - Topics in Harmonic Analysis - Prof. A. Nahmod (Spring 2001)

Text:  Fourier Analysis by Javier Duoandikoetzea Zuazo (Translated by David Cruz-Uribe).

The goal of this course is to study the real variable methods in Fourier Analysis introduced by Calderón and Zygmund and their
modern generalizations and applications. We start with a review of Fourier series, integrals, convergence and summability methods. Next, we introduce two operators which are basic to the field: the Hardy-Littlewood's maximal function and the Hilbert transform. These are the basic examples of Calderón  and Zygmund's theory of singular integrals and we use them as an introduction to understand and develop their techniques. The goal being to study analogues in higher dimension which are of great interest in applications (eg, in elliptic PDE's). We next consider one of the main contributions of the 70's: the spaces H^1 and BM and its connections to the above. We discuss Littlewood-Paley theory, almost orthogonality techniques and square functions. Littlewood-Paley theory had its origins in the 30's but has had an extensive development more recently as it is the core idea behind a number of applications not only to PDE's but also in engineering, computational mathematics, signal and image processing. In particular it's the precursor to the theory of wavelets. In this vein we will review unconditional bases and study the characterization of function spaces using wavelets. We end with a recent crucial result in the field, the so-called T(1)-theorem, which brings together many of the basic ideas above in a powerful and elegant way.

*MATH 797P - Linear/Nonlinear Waves - Prof. R. Young (Spring 2000)

Text: I will refer to the book we will work through, Linear and Nonlinear Waves by G. B. Whitham, although this is not required.

Waves occur naturally in almost any physical system, and exhibit a large variety of behavior. Some major areas of application include: traffic flow, water waves, gas dynamics, electromagnetism, and ``universal equations.''

We will concentrate on the use of asymptotic expansions and heuristic descriptions of wave phenomena. The course is aimed at people interested in applications of mathematics, as well as those with an interest in analysis, PDE and/or differential equations.

Possible selection of topics:

•  One way wave equation, linear first-order systems, method of characteristics.
•  Poission, heat & wave equations, classification of second-order linear equations, Fourier expansions.
•  Hopf equation, breakdown of solutions, shock formation and propagation.
•  Nonlinear hyperbolic systems. Linearization. Riemann problem & self-similarity. Asymptotic states, Interaction of waves.  Local approximation of solution.
• Linear wave equation in higher dimensions. Effects of symmetry & radial waves.
• Burger's equation and effects of viscosity (dissipation). Cole-Hopf transform. Similarity transforms.
• Higher order equations and dispersion. Propagation of modes, dispersion relations and group velocity. Energy and amplitude propagation.
• Asymptotic expansions & weakly nonlinear waves. Geometric optics.

*MATH 797R - Representations of finite groups and compact Lie groups - Prof. Humphreys (Spring 2001)

Text: No single text covers this material, but for the first half of the course the book by J. P. Serre,
Linear Representations of Finite Groups (Springer) would be useful.

The representation theory of finite groups (over the field of complex numbers) was developed a century ago by Frobenius and Schur. The theory itself is a beautiful synthesis of ideas from all parts of algebra. It has many applications in mathematics, physics and chemistry. We will develop the main theorems, illustrated especially by finite groups of Lie type. Representations of compact Lie groups can be treated in parallel ways, but we will consider only some of the easire examples such as SU(2) and SU(3).

The goal of the course is to introduce two of the deep connections in Lie theory between representation theory and algebraic geometry: the McKay Correspondence (relating finite subgroups of SU(2) to Dynkin diagrams and Kleinian singularities) and the Springer Correspondence (relating Weyl groups to the flag varieties of semisimple groups).

*MATH 797S - Topics in Applied Mathematics - Prof. B. Turkington (offered Spring 2002)

We will learn how mathematics is applied to model the dynamics of the atmosphere or oceans. This topic is known as geophysical fluid dynamics, which describes the motion of a shallow fluid on a rotating sphere. First, we will derive the governing equations from basic physical principles and introduce the key approximations used to simplify these equations. We will then connect fundamental geophysical phenomena to special solutions of the simplified governing equations. Such phenomena include the large-scale patterns of circulation of the atmosphere and the mid-ocean gyres. Finally, we will consider the role of nonlinearity, and will take the first steps toward developing low-dimensional models of complex systems by means of statistical closures.

The philosophy of the course is to introduce the tools of applied mathematics, which are used throughout science, in a particular context. Among the important tools in the context of geophysical fluid dynamics are singular perturbations, dispersive waves, and statistical descriptions of complex dynamical systems.

*MATH 797T - Topics in Interacting Particle Systems -  Prof. M. Katsoulakis (offered Fall 2002)

In this course we introduce some of the main mathematical methods for studying Interacting Particle Systems and their scaling limits to deterministic and stochastic Partial Differential Equations. We also discuss relevant applications in materials science and fluid dynamics.

*MATH 797U - Representations of semisimple Lie algebras - Prof. J. Humphreys (Spring 2003)

The representation theory of semisimple Lie algebras (over the real or complex numbers) has become part
of the standard core of modern mathematics, with connections to physics and to many areas of pure and applied mathematics. The classical Cartan-Weyl theory is well-developed but several closely parallel theories (Kac-Moody algebras, -adic representations, quantum groups, modular Lie algebras) are less completely understood.

To approach the newer theories through the old one, it is best to work in the module category introduced around 1970 by Joseph Bernstein, I. M. Gelfand, S. I. Gelfand (the BGG category).  Starting with a quick overview of simple Lie algebras (based on standard examples), we will derive the classical theorems in an efficient way by studying composition factor multiplicities in Verma modules. Weyl's character formula and Kostant's equivalent formula are easy special cases. But the complete story about multiplicities involves the 1979 Kazhdan - Lusztig Conjecture, which was proved in 1980 using deep results from algebraic geometry (intersection cohomology, perverse sheaves) and has led to analogous open questions in other parts of Lie theory.

Further algebraic details about the BGG category will be discussed: projective modules, Jantzen's translation functors and filtrations, Loewy series. This leads toward more recent work on Koszul duality due to Beilinson - Ginzburg - Soergel. Students will be asked to write up some exercises and to make presentations in class.

*MATH 797W - Algebraic Geometry - Prof. I. Mirkovic (Spring 2004)

Text: A source with classical and elementary flavor is:
Shafarevich, Igor R. Basic algebraic geometry
1. Varieties in projective space and
2.Schemes and Complex manifolds, Springer-Verlag.
A (more) modern treatment is:
Hartshorne, Robin Algebraic geometry. Graduate
Texts in Mathematics, No. 52. Spring-Verlag

Prerequisites: Some familiarity with algebra at the level of 611 will be helpful

Description: This course is an introduction to the vocabulary and methods of algebraic geometry, geared towards the use of algebraic geometry in various areas of mathematics: number theory, representation theory, combinatorics, mathematical physics. The basic vocabulary will evolve from systems of polynomial equations to algebraic varieties and schemes. We will also get introduced to the topology of algebraic varieties: cohomology and algebraic cycles. Our first example will be algebraic curves. This is the best understood part of algebraic geometry since it deals with one-dimensional objects. The highlights: Riemann Roch theorem and the relation to number theory (``geometric class field theory''). The second example will be the flag varieties, i.e., homogeneous compact algebraic varieties. We will consider maps of curves into flag varieties, a topic of current interest related to mathematical physics (``quantum cohomology'') and representation theory.

*MATH 797X - Riemann Surfaces and Integrable Systems - Prof. F. Pedit (Spring 2005)

Text: TBA

Prerequisites:

Description: This course will explore the relevant role Riemann surfaces play in integrable system theory. We will start with a simple geometric problem of polygon evolution in the plane and attach a finite genus Riemann surface to any closed polygon. The evolution is then governed by the Jacobi variety of this Riemann surface and theta functions explicitly parameterize the flow. This is a prototypical situation for the kind of integrable systems related to Riemann surfaces: the phase space consists of Riemann surfaces (of some type) together with their Jacobians as energy shells. If time permits, we will also discuss how problems of smooth geometry (Harmonic maps, self-duality equations, conformal surfaces, KdV equation) fit into this picture. The students are expected to have taken a Riemann surfaces class and preferably at least one semester of the manifold course. Other than that the course is self-contained with emphasis on the basic ideas rather then the most general techniques.

*MATH 861 - Complex Manifolds & Physics - Prof. F. Pedit (Spring 2003)
co-instructor Prof. David Kastor (Physics)

Text: None. Reading material will be distributed at the beginning of class

Prerequisites: Manifolds I and preferably II - Exposure to Riemann Surfaces would help.

Description: The course will study the basics of complex manifolds as they arise in geometry and physics. Physical motivations will alternate with mathematical constructions. The goal is to get an idea about recent conjectures in this area (mirror symmetry etc.).

*MATH 861 - Advanced Geometry - Prof. R. Kusner (Spring 1999)

This course runs in parallel with the GANG seminar, and involves students reporting on current research topics,
as well as lectures by both outside speakers and the faculty member in charge.  This spring we plan to work on;

• geometric analysis of knots and links, with applications in topology, fluid dynamics and microbiology;
• geometric Brownian motion, with applications in potential theory, Reimann surfaces and minimal surfaces.

*STAT 597E - Baysian Data Analysis - Prof. P. Sebastiani (Fall 2001)

Text: Recommended Readings: Lee, P. (1998). Bayesian Statistics: an introduction. Halsted Press, N.Y.
       A. O'Hagan (1994). Bayesian Inference, Halsted Press, N.Y. 1994.

Bayesian methods for data analysis are becoming increasingly popular in many fields, including Computer Science and Artificial Intelligence, Finance, Epidemiology and Bioinformatics. The goal of this course is to provide an updated overview of Bayesian statistics and its use in data analysis to senior/graduate students in Mathematics, Computer Science, Biostatistics, Economics, Physics and generally to students interested in modern techniques for data analysis. The course will first cover the fundamental aspect of Bayesian statistics, focusing, on prior to posterior analysis with conjugate priors, Bayesian estimation and hypothesis testing. The core of the course will be Bayesian modeling using linear models, hierarchical linear models, generalized
linear models and graphical models. Attention will be given to computational aspects and students will be taught to use computer programs as Bugs and Discoverer for analyzing real data sets.

STAT 605 - Probability Theory

A modern treatment of probability theory based on abstract measure and integration. Topics to be covered include the following:

       Random variables, expectations, independence, laws of large numbers, weak convergence, central limit theorems.
       The concepts of conditional probability and conditional expectation (via the Radon-Nikodym theorem).
       Basic properties of certain classes of random processes, such as martingales and random walks.

STAT 607 Mathematical Statistics I

Probability theory, including random variables, independence, laws of large numbers, central
limit theorem; statistical models; introduction to point estimation, confidence intervals, and hypothesis
testing.

STAT 608 - Mathematical Statistics II

The second part of a two-semester graduate level sequence in probability and mathematical statistics suitable for students in a wide variety of disciplines (statistics, mathematics, biostatistics, engineering, physical sciences, computer science, economics, etc.). Second semester: point and interval estimation, hypothesis testing, large sample results in estimation and testing; some of the following topics: basic decision theory, Bayesian methods, analysis of discrete data, nonparametric methods, sequential methods, regression, analysis of variance, resampling and simulation.

STAT 640 - Sampling Theory

Introduction to the theory and practice of sampling from finite populations. Designs covered include simple random sampling, systematic sampling, cluster sampling, stratified sampling, sampling with unequal probabilities, multi-stage and double sampling. Coverage will also include ratio and regression estimators (generally the use of auxiliary variables), the jackknife, and determination of samples sizes and optimal allocations. The course will include applications and data analysis, with computations carried out using SAS surveyselect, surveymeans and surveyreg and possibly Sudaan.

STAT 697F - Topics in Regression (offered every other year)

Prerequisites: A previous course on regression analysis covering multiple linear regression (e.g., ST505, BioEpi744, RESEC702) with some prior exposure to regression models in matrix form. Basic matrix algebra (those without any prior experience with matrix algebra will need to put in a little extra time at the beginning). Some prior exposure to computing is desirable.

Description: This course examines a variety of additional topics in regression beyond those traditionally found in a first course. Within the multiple linear regression framework, topics will include models with changing variances and the use of generalized/weightedleast squares, models with correlated errors (data over time and repeated measures/random effect models), boostrapping in regression, smoothing, and covariate measurement error. Non-linear regression and generalized linear models (logistic and Poisson models) will also be introduced. 

This is an applied course. The focus is on a careful description of the models and methods and how they are applied, and on computational aspects. This is not a theory course concentrating on proving why things work (most of which would require more advanced backgrounds), but we do make liberal use of matrix/vector notation in expressing the models and methods. There is plenty of data anlysis using examples from a variety of disciplines. computing will be done in SAS using (but not limited to) procs REG, NLIN, LOGISTIC, PROBIT, MIXED and TSCSREG. In addition students will learn to use SAS-IML (interactive matrix language), a general and powerful computing environment. 

*STAT 697G - Survival Analysis - Prof. H. K. Hsieh (Spring 2000, Fall 2005)

Text: Statistical Methods for Survival Data Analysis, 3rd ed. by Lee and Wang, Wiley 2003.

Prerequisites: Solid knowledge of basic probability and statistical inference
or taking the second part concurrently.

*STAT 697J - Working Seminar - Prof. P. Sebastiani (Spring 2003)

Text: Selected readings from scientific journals

Prerequisites: Very good background in Statistics and Multivariate analysis.

The course will cover foundations of gene expression, microarray technology and statistical methods for the modelling of gene expression data measured with microarray technology.

*STAT 697K - Bioinformatics - Prof. E. Conlon (Fall 2004)
This course is cross listed with Microbiology - Prof. J. Blanchard
and Computer Science - Prof. D. Kulp

Text: TBA

Prerequisites: See description

Description: This course is an introduction to the biological, computational and statistical foundations necessary for bioinformatics-related research. Areas covered will include; biological databases, DNA and protein sequence analysis, structure-based analysis, expression analysis, and genetic mapping. This course is cross-listed in Mathematics and Statistics, Microbiology and Computer Science. While there are no formal prerequisites, some level of familiarity with molecular biology, statistics, and/or computer programming is recommended.

STAT 697R - Regression Modeling - Prof. J. Buonaccorsi (Fall 2005)

Text: Applied Linear Regression Models by Kutner, Nachsteim and Neter (4th ed.)

Prerequisites: Previous coursework in Probability and Statistics, including knowledge
of estimation, confidence intervals, and hypothesis testing.

Description: Topics covered include simple and multiple linear regression; correlation; inverse prediction; the use of dummy variables; residuals and diagnostics; model building and variable selection; weighted least squares; the development of regression models and methods in matrix form; an introduction to nonlinear regression. Focus is on a careful understanding and presentation of regression models and methods, interpretation of results and data analysis/statistical computation using SAS (no prior experience assumed).

A matrix formulation of the linear regression model is developed during the course to facilitate the presentation of models and results. The necessary basic matrix tools will be given in the course.

NOTE: While ST697R covers many of the same topics as ST505 it will differ in a number of ways. The pace of the course will be a bit quicker to allow coverage of additionial topics. More importantly, there will be more in the way of applications and computing, including more emphasis on choosing models and interpreting analyses.

TAT 705 - Linear Models I

ST705 is the first of two courses which treat the theory of Linear Models. This course
provides some general background for handling linear models covering i) a review and introduction of some
topics in linear algebra, (including generalized inverses,) ii) a discussion of random vectors, the
multivariate normal, linear and quadratic forms including their distributions. We then develop the basic
theory for inferences (estimation, confidence intervals and hypothesis testing, power) for the general
linear model, with a focus on design matrices of full rank as arise in regression. The applied part of the
course is not directed towards extensive data analysis which is available in many other regression courses.
Instead the emphasis is on understanding and choosing models and computational implementation of the
methods. Computing will be done via SAS including the use of IML.

STAT 706 - Linear Models II

This course follows ST705 and will make use of the general theory developed in the earlier part of that course. It differs from Stat 705 in that the focus is now on ``analysis of variance/design of experiments'' models. Coverage will include one-way and multi-way factorial designs (including balanced and unbalanced designs, notions of interaction, etc.); estimability and testability; randomization; randomized block designs; latin squares; incomplete designs; random effects and mixed models; nested models. As in ST705 this is to a large extent a theory course with the applications focusing on modeling and computational issues and not on data analysis. We will use SAS for computing weekly homework, a midterm and a final exam.

STAT 708 - Applied Stochastic Models and Methods

Stochastic processes and their applications to engineering, the physical, life, and social sciences, and finance. Specific topics will depend on the composition of the class. General topics: discrete-time Markov chains (with applications, for instance, to random walks, reliability, branching processes, Markov chain Monte Carlo); continuous-time Markov chains (applications to queueing theory, risk theory, population processes); counting processes (e.g., Poisson processes in time and space); and Brownian motion (applications to mathematical finance).  Additional possible topics include: renewal theory, stochastic algorithms in optimization, Markov decision processes.

*STAT 725 - Advanced Theory of Statistics - Prof. J. Horowitz (Fall 2000) Prof. J. Staudenmayer (Fall 2005)

Point and interval estimation, small sample and asymptotic properties of maximum likelihood estimators, generalized estimating equations; bootstrap estimates and confidence intervals; hypothesis testing, optimality properties; basic decision theory, Bayes rules, minimaxity. Applications to nonlinear regression, generalized linear models.

STAT 797I - Time Series

A one-semester course studying both theoretical and practical aspects of time series. Topics
include: Stationarity, ARMA models, Spectral Representation, Parameter Estimation for the models,
Prediction, and possibly Kalman Filtering if time permits. The course will also contain a computing
component.