Descriptions of spring 2004 courses in the Rutgers-New Brunswick Math Graduate Program

# Descriptions of proposed spring 2004 courses in the Rutgers-New Brunswick Math Graduate Program

Description-free course listings

640:502    Theor Func Real Vari    R. Goodman    HLL 425   MW 5; 2:50-4:10

Real Analysis II
Text (required): Gerald B. Foland, "Real Analysis: Modern Techniques and Their Applications" (2nd ed., ISBN \#0-471-31716-0, Wiley-Interscience/John Wiley Sons, Inc., 1999.

This course will be a countinuation of 640:501. The goal is to give an introduction to some core topics in real and functional analysis that every professional mathematician should know. In particular, the course will cover some of the written qualifying exam topics that were not in 640:501.

The syllabus will include:

1. Topological spaces (Locally compact spaces, Compact sets of functions, Stone-Weierstrass theorem);
2. Introduction to functional analysis (normed vector spaces, continuous linear maps, Baire category theorem, Hilbert spaces, topological vector spaces);
3. Lp spaces (Integral inequalities, duality, integral operators);
4. Continuous functions and Radon measures on locally compact spaces;
5. Introduction to Fourier analysis (convolutions, Fourier transform and Fourier series, Plancherel theorem, Poisson summation formula).
6. If time permits, further topics will include generalized functions (distributions) and wavelets (multiresolution analysis, orthonormal wavelet systems).

640:504    Theory Func Comp Vars    F. Luo    HLL 423   MTh 3; 11:30-1:10

Complex Analysis II
This course is an introduction to Riemann surfaces. We will emphasis its relationship to various fields in mathematics: algebraic geometry, geometric analysis, differential geometry, topology and algebra (discrete subgroups of SL(2,R)). Thus it may also be considered as an introduction to these fields by the examples of Riemann surfaces.

We shall begin with normal families, Riemann mapping theorem, elliptic function theory and theta functions. We then introduce the theory of Riemann surfaces and algebraic curves. These include the basic material like homomorphic maps between Riemann surfaces, meromorphic functions and meromorphic differentials on Riemann surfaces, and hyperbolic geometry. One of the goal of the course is to produce a proof of Riemann-Roch theorem. Another goal is to prove the uniformization theorem.

If time permits, we will also give a brief introduction to the theory of moduli space of Riemann surfaces and its relationship to physics. Applications of Riemann surfaces to various fields in mathematics will also be covered.

640:507    Functional Analysis    S. Greenfield HLL 525   TTh 4; 1:10-2:30

My very valuable notes from the last time I taught Math 507 have been lost. I am heartbroken and will use a text. Conway's A Course in Functional Analysis seems suitable. Math 507 will be a basic course on classical linear functional analysis, covering the beginnings of Hilbert space and Banach space, the elements of duality and linear operators, some definitions from LCTVS's (LCTVS=locally convex topological space) and some applications, and probably a few versions of the spectral theorem. Prerequisites are basic real analysis and complex analysis.
Students will have homework problems and will be asked to speak about a topic in class. There will probably be a written final exam.
Please communicate with the instructor for further information:
S.~Greenfield, Hill 542, 445-3074, greenfie@math.rutgers.edu

640:509    Sel Topics in Analysis    A. Bahri     HLL 423   MW 5; 2:50-4:10

I will show how to complete Morse Lemmas at infinity in Yamabe-type problems. I will descibe the topological arguments in these kind of problems, including the scalar-curvature problems.

640:518    Partial Diff Equations    Y. Li     HLL 525   MW 4; 1:10-2:30

Partial Differential Equations
In the first two thirds of the course, I will present some useful methods in the study of nonlinear partial differential equations. Results on Sobolev spaces, Sobolev embeding theorems, and some linear theories of partial differential equations will be stated without proof. The emphasis is on how to use them in the study of nonlinear problems.

In the second one third of the course I will present some results on the Yamabe problem and on the free boundary problem.
The topics for the first two thirds of the course will include:
Leray-Schauder degree theory and some applications in PDEs.
Weak lower semi-continuity in the calculus of variations
Solving nonlinear elliptic equations using sub and super solutions
Mountain Pass Lemma and applications to semilinear elliptic equations
Concentrated Compactness
Compensated compactness
A theorem of Morrey and small energy implies regularity''

640:519    Sel Topics Diff Equations    R. Wheeden     HLL 525   TTh 5; 2:50-4:10

The course will study regularity properties of solutions of some subelliptic, second order, linear, divergence form p.d.e.'s with rough coefficients. The coefficients are assumed to be only Lipschitz continuous, and consequently the well-known Hormander condition is not applicable. The primary goal is to show that the solutions are Holder continuous. The approach is a modified version of the Moser iteration method, adapted to a natural class of quasimetrics on Euclidean space. All necessary background facts will be treated. If time permits, applications to hypoellipticity of solutions of some nonlinear Monge-Ampere type equations will also be given.

640:523    Functions Several Complex Vars    X. Huang     HLL 423   MTh 2; 9:50-11:10

Introduction to Complex Analysis of Several Variables
Several Complex Variables is the study of the properties and structure of holomorphic functions, complex manifolds and CR manifolds. A function of n complex variables z in C n is said to be holomorphic if it can be locally expanded as power series in z. An even dimensional smooth manifold is called a complex manifold if the transition functions can be chosen as holomorphic functions. A CR manifold is essentially a manifold that can realized as the boundary of a certain complex manifold.

Different from one complex variable, if $n>1$ one can never find a holomorphic function over the punctured ball that blows up at its center. This is the striking phenomenon that Hartogs discovered about 100 years ago, which opened up the first page of the subject. Then Poincar\'e, E. Cartan, Oka, etc, further explored this field and laid down its foundation. Nowadays as the subject is intensively interacting with other fields, providing important examples, methods and problems, the basic materials in Several Complex Variables have become mandatory for many investigations in pure mathematics.

This class tries to serve such a purpose, by presenting the following fundamental topics from Several Complex Variables.
(a) Holomorphic functions, plurisubharmonic functions, pseudoconvex domains and the Cauchy-Riemann structure on the boundary of complex manifolds
(b) H\"ormander's L2-estimates for the $\bar \partial$-equation, sub-elliptic estimates, and the Levi problem;
(c) Complex manifolds, K\"ahler manifolds, elliptic theory on complex manifolds
(d) Cauchy-Riemann geometry, Webster's pseudo-Hermitian Geometry and subelliptic analysis on CR manifolds

The course materials will be largely taken from the following:
[1] L. Hormander, An introduction to complex analysis in several variables, Third edition, North-Holland, 1990.
[2] James Morrow and K. Kodaira, Complex Manifolds, Rinehart and Winston, 1971.
[3] Xiaojun Huang, Equivalence Problems for Real Submanifolds in Complex Manifolds, Springer-Verlag Lecture Notes Series, 2003, to appear.
Prerequisites: One complex variable and the basic Hilbert space theory from real analysis

640:534:01    Sel Topics in Geometry    X. Rong     HLL 313    TTH 4; 1:10-2:30

Topics in Riemannian Geometry
Gromov-Hausdorff convergences and collapsing Riemannian manifolds with bounded curvature
We will give an introduction to the convergence and collapsing theory in Riemannian geometry. This theory was initiated by M. Gromov more than two decades ago.

This theory is to deal with an arbitrary' sequence of Riemannian manifolds of the same dimension whose curvature are bounded (in various ways). After passing to a subsequence, the sequence always converges to some `limit space'' (the convergence is taking place in the space of all metric spaces equipped with the Gromov-Hausdorff distance).

Our topics include the regularity and the singular structure of limit spaces, relations between geometrical and topological structures of the converging manifolds and that of their limit space, and some applications in Riemannian geometry.

Prerequisites: a basic knowledge of (Riemannian) manifolds (calculus on manifolds: such as differentiable structure, tangent bundles, connection, curvature, etc).

640:547    Topology of Manifolds    S. Ferry     HLL 525    MTh 2; 9:50-11:10

Topology of Manifolds

We will begin with a study of Whitehead torsion, which we will use

• To classify lens spaces, giving examples of manifolds which are homotopy equivalent but not homeomorphic.
• To prove the s-cobordism theorem, which gives the Poincare Conjecture in high dimensions.
• To give examples of polyhedra which are homeomorphic without being piecewise linearly homeomorphic, solving the classical Hauptvermutung.
Whitehead torsion will also serve as an introduction to algebraic K-theory. We will proceed to the study of L2-homology and cohomology, selecting topics from Wolfgang Lueck's new book L2 Invariants: Applications to Geometry and K-theory.
The selection of these topics is negotiable, depending on the interests of the class.

640:549    Lie Groups    C. Woodward     HLL 525   MTh 3; 11:30-12:50

"We will discuss semisimple Lie groups and their representations."

640:552    Abstract Algebra II    W. Vasconcelos     HLL 425   TF 2; 9:50-11:10

Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. Representative topics will be:

• Galois Theory: Finite algebraic extensions, resolutions of equations by radicals (and without radicals)
• Rings of Polynomials: Noetherian rings, Hilbert basis theorem, Noether normalization, Nullstellensatz
• Basic Module Theory: Projective and injective modules, resolutions, baby homological algebra, Hilbert syzygy theorem
• Category Theory: Basic categories and their functors

Any algebra text at the level and coverage of one of the following will do:
T. Hungerford, Algebra, Graduate Texts in Mathematics, Springer, 1989+.
N. Jacobson, Basic Algebra, Vols. I & II, Freeman and Co., 1974, 1980.
I am in Hill 228 or reachable at vasconce@math.rutgers.edu for any additional info.

640:555    Sel Topics in Algebra    J. Lepowsky     HLL 425   MTh 2; 9:50-11:10

Combinatorial Identities and Vertex Operators

This course will develop interesting relationships between combinatorial identities such as the Rogers-Ramanujan identities and vertex operator theory and infinite-dimensional Lie theory. Both early developments and more recent work will be discussed. Part of the course will include an introductory treatment of such combinatorial identities. If students who have not studied vertex operator theory are potentially interested in this course, they may wish to contact me.

640:556    Representation Theory of Rings    R. Wilson     HLL 525   TTh 6; 4:30-5:50

Introduction to Representation Theory of Lie Algebras
Text: Introduction to Lie algebras and representation theory by James Humphreys.

This course will continue the study of semisimple Lie algebras begun in Professor Sahi's fall course (640:550). We will complete the (Cartan-Weyl) classification of the irreducible finite-dimensional representations of semisimple Lie algebras (following the treatment in Humphreys' book). We will also pursue some additional topics, chosen from among: construction of Chevalley groups, representation theory of Kac-Moody Lie algebras, representation theory of quivers.

640:558    Theory of Algebras    E. Taft     HLL 425   TTh 5; 2:50-4:10

Hopf Algebras and Quantum Groups

Hopf algebras arose in topology and now play an important role in algebra and many other areas of mathematics. Basic examples include the universal enveloping algebra of a Lie algebra, the rational functions on an algebraic group, and certain deformations of these two examples which are now called quantum groups. Quantum groups have important connections to many fields in mathematics and physics.

We will study the algebraic structure of Hopf algebras. Quantum groups will be among the principal examples studied.

Prerequisites: Basic algebra, including linear algebra and tensor products.
Reference textbooks:

1. S. Dascalescu, C. Nastasescu, S. Raianu: Hopf Algebras: An Introduction, Marcel Dekker Inc., 2001.
2. C. Kassel: Quantum Groups, Springer-Verlag, 1995.
3. S. Montgomery: Hopf Algebras and Their Actions on Rings, Amer. Math. Soc., 1993.

640:567    Model Theory    G. Cherlin     HLL 423   TTh 4; 1:10-2:30

Model Theory 567

An introductory course, following the text by David Marker, Model Theory: An Introduction, whose particular emphasis is on the interaction between the very "pure" theory of Morley/Shelah, and applications to algebra and analysis (Tarski/Robinson/Macintyre). On the pure side, he arrives at a complete proof of Morley's difficult "categoricity theorem" which launched modern pure model theory (the original paper recently won the AMS Steele prize for a "seminal" work). The applied side is largely what Marker works on himself, and he covers a variety of topics, many of them in extended exercises.

At the same time, recent developments connect model theory, combinatorics (Ramsey Theory), and topological dynamics in a new way connected with the research of the group in logic at Rutgers, and we want to develop this direction as well. This ties in to the end of Marker's Chapter 2, and the beginning of his Chapter 5. Other applications not covered by Marker may be considered, such as Godel's "L" as a model of set theory.

References:
1. Text: David Marker, Model Theory: An Introduction
ISBN 0-387-98760-6
3. Kechris et al: pdf or ps format.

Prerequisites. No formal prerequisites. Some prior knowledge of mathematical logic is helpful.

640:574    Topics in Number Theory    R. Bumby     HLL 423   TF 3; 11:30-12:50

Diophantine Approximations
Text: J.W.S. Cassels, "Introduction to Diophantine Approximation", Cambridge Univ. Press, 1957
Prerequisites: No prior course in Number Theory will be assumed.

The course will be close to Cassels' book, modified to account for subsequent discoveries.

Reference books for the course will be put on reserve in the library. In addition to the primary text, these are
J. W. S. Cassels, "An Introduction to the Geometry of Numbers", second edition, Springer-Verlag, 1971.
Alan Baker, "Transcendental Number Theory", Cambridge University Press, 1975.

642:528    Methods of Appl Math II    O. Costin     HLL 423   TTh 6; 4:30-5:50

The course focuses on complex variable methods in applied mathematics, suitable especially for students of engineering or physics or related disciplines. It is a "how to" course, with a minimum of theoretical mathematics and a lot of problem solving and applications.

642:562    Intro Math Physics II    M. Kiessling    M 7 in HLL 425 (6:10-7:30)   Th 6&7 (4:30-7:30)

Description to appear.

642:564    Statistical Mechanics II    J. Lebowitz    HLL 124   By arrangement

Description to appear.

642:581    Graph Theory    J. Kahn     HLL 124   TF 3; 11:30-12:50

Applied Graph Theory

This course is intended to provide an introduction to most of the basic topics in graph theory, including:

• connectivity
• matching theory and minimax theorems
• coloring problems
• minors
• planar graphs
• some extremal graph theory, Ramsey theory, random graphs (some of this is also covered in 582-3, but we'll try to avoid a lot of overlap)
• polyhedral issues if time allows

Prerequisites: The course is mostly self-contained, though some previous combinatorics, linear algebra, rudimentary probability are all occasionally helpful. See me if in doubt.
Text R. Diestel, Graph Theory

642:583    Combinatorics II    M. Saks    RUTCOR 166   TTh 4; 1:10-2:30

This is a continuation of Combinatorics I, 642:582. In unusual circumstances, with approval of the instructor, a student may be allowed to register for this without having had 642:582.

642:587    Sel Topics Discrete Math    G. Kindler     HLL 423   TTh 5; 2:50-4:10
Also listed as CS course 198:540.

Analysis of Boolean functions, with computer science applications

Somewhat surprisingly, harmonic analysis turns out to be a very effective tool in the study of Boolean function. It has numerous applications in various areas, such as property-testing, learning, complexity theory, hardness of approximation, and the theory of social choice. The course will cover some of these applications, as well as some theoretical results.
Students are only assumed to have elementary knowledge of linear algebra and normed spaces.
Here is a list of possible topics (not all can be covered in one course):

1. The Fourier transform over the discrete cube.
2. A three-query test for the Hadamard code and for the long-code, and their relation to noise-sensitivity.
3. Hardness of approximating exact 3-LIN over Z_2.
4. A test for being dependent on at most k variables (a junta-test).
5. Responsibility is indivisible (into very little pieces) [Kahn, Kalai, and Linial].
6. Every monotone property has a sharp threshold [Friedgut and Kalai].
7. A function with constant average sensitivity is a junta [Friedgut].
8. Learning Juntas [Mossel, O'Donnell and Servedio].
9. Vertex-Cover is hard to approximate within 4/3 [Dinur and Safra].
10. How much are monotone functions positively correlated [Talagrand].

642:612    Sel Topics Applied Math    P. Feehan     HLL 124   TTh 4; 1:10-2:30

Financial Math II

The course will be a continuation of Selected Topics in Applied Mathematics (Mathematical Finance I), though we shall try to accommodate students who wish to take this course and have not taken the autumn semester course. A broad course outline and a tentative list of detailed topics is given below, though this may be modified depending on student background and interests.

The course content will differ from that of the autumn 2003 semester by emphasizing recent research literature on topics such as extensions of the Black-Scholes model, heavy-tailed stochastic processes, numerical methods for pricing American, exotic and path-dependent options, interest-rate models, statistical analysis of financial data and model building and calibration. When required in order to understand mathematical finance research literature, we shall also include excursions into associated theoretical topics in statistical analysis and time series, stochastic processes, and numerical methods.

#### A partial list of topics

• Extension of Black-Scholes
• Volatility smiles
• Numerical and computational methods in finance
• Techniques for pricing American options
• Techniques for pricing exotic and path-dependent options
• Interest rate modelsS
• Swaps
• Yield curve fitting
• Interest rate derivatives
• Hull & White model
• Ho & Lee model
• Heath, Jarrow & Morton model
• Cox, Ingersoll, & Ross model
• Risk measurement and management
• Portfolio theory
• Portfolio insurance
• Value-at-Risk
• Credit risk
• Credit derivatives
• RiskMetrics
• Numerical methods
• Hedging
• Statistical analysis of financial data, parameter estimation, time series, GARCH
• Extreme value theory
• Advanced topics, including applications of stochastic control theory and path-integral methods to finance

#### A partial list of references:

• Brockwell & Davis, Time series: theory and methods
• Clewlow & Strickland, Implementing derivatives models
• Duffie, Dynamic asset pricing theory
• Embrechts, Kluppelberg, & Mikosch, Modeling extreme events
• Hull, Options, futures, and other derivatives
• Karlin & Taylor I, II, A first (second) course on stochastic processes
• Karatzas & Shreve, Brownian motion and stochastic calculus
• Øksendal, Stochastic differential equations
• Shreve , Lecture notes on stochastic calculus and finance
• Taleb, Dynamic hedging
• Wilmott, Derivatives
• Recent research papers by Peter Carr, Mark Broadie, Paul Glasserman, Patrick Hagan, Marco Avallandra, and others.