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

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:

- Topological spaces (Locally compact spaces, Compact sets of functions, Stone-Weierstrass theorem);
- Introduction to functional analysis (normed vector spaces, continuous linear maps, Baire category theorem, Hilbert spaces, topological vector spaces);
- L
^{p}spaces (Integral inequalities, duality, integral operators); - Continuous functions and Radon measures on locally compact spaces;
- Introduction to Fourier analysis (convolutions, Fourier transform and Fourier series, Plancherel theorem, Poisson summation formula).
- 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**

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.

If you have any questions, please contact me at fluo@math.

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 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 L^{2}-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

Gromov-Hausdorff convergences and collapsing Riemannian manifolds with bounded curvature

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*.

^{2}-homology and cohomology, selecting topics from Wolfgang Lueck's new book

*L*.

^{2}Invariants: Applications to Geometry and K-theoryThe 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:

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:**

- S. Dascalescu, C. Nastasescu, S. Raianu: Hopf Algebras: An Introduction, Marcel Dekker Inc., 2001.
- C. Kassel: Quantum Groups, Springer-Verlag, 1995.
- 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:***Text**: David Marker,**Model Theory: An Introduction**- Springer Graduate Texts in Mathematics 217, © 2002

ISBN 0-387-98760-6 - 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.

(other titles may be added)

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):

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

#### Broad outline:

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

#### A partial list of references: