# Descriptions of proposed spring 2006 courses in the Rutgers-New Brunswick Math Graduate Program

640:502
`Theor Func Real Vars`
S. Tahvildar-Zadeh
HLL 425 TF 3; 12:00-1:20

This course will, of course, be a continuation of Math 501, so we
will begin with a quick review of 501.

Then it's on to differentiation
theory, followed by an introduction to functional analysis, with the
goal of covering the Sobolev embedding theorems and their basic
applications in PDEs by the end of the semester, as well as some basic
probability if time permits.
**Text:**We will use the textbook by Folland, and
supplement it with material from Lieb's Analysis.

640:504
`Theory Func Comp Vars`
R. Nussbaum
HLL 423 MTh 3; 12:00-1:20

This course will be a topics course and a continuation of Math 503. If these topics are not covered in Math 503, we will begin with the Riemann mapping theorem and results about the gamma function (including the Bohr-Mollerup axiomatic characterization of the gamma function) and the zeta function.We will then present a geometric approach to Picard's great theorem: the range of a non-constant entire function omits at most one value. We will use the Poincare metric in the proof, and so will be led naturally to a discussion of the so-called Denjoy-Wolff theorem and fixed points of analytic maps.

Other topics to be covered include the Weierstrass and Hadamard
factorization theorems for entire functions, Hardy spaces and
connections between Fourier transforms and analytic functions, elliptic
functions, and the prime number theorem.

The text will be *Function Theory of One Complex Variable*
by Robert Greene and Steven Krantz.

640:508
`Functional Analysis II`
M. Kiessling
HLL 423 TTh 5; 3:20-4:40

**Text:** J. Glimm and A. Jaffe, *Quantum Physics: A Functional
Integral Point of View*, Springer, 2nd ed.
(available at www.amazon.com and other online book retailers; also
ordered to be in the Rutgers Bookstore).

The course will be based on the book *Quantum Physics: A Functional
Integral Point of View*. I first review some of the
basics of functional analysis, which part as usual overlaps with
functional analysis I. This should help refreshing the memories of
those who took functional analysis I and will level the ground for
those who haven't. Next I review the basic concepts of quantum
theory and quantum field theory and how the idea of functional integrals
emerges as a tool in one of the approaches. The concept of functional
integrals is made rigorous, and finally applied to a variety of problems
in mathematical physics, not at all all from quantum field theory. (sic)
The goal is to give the participants in the course a working knowledge of the
powerful mathematical tool of functional integrals.

640:510
`Sel Topics in Analysis`
S. Chanillo
HLL 423 TF 2; 10:20-11:40

The course will focus on the theory of minimal surfaces and free boundaries. We will begin with developing basic results for minimal surfaces and in particular the monotonicity formula for such surfaces. We will study then the Weierstrass parameters for a minimal surface. Next we will introduce the obstacle problem. We then develop some basic results for the free boundary for the obstacle problem. Next we prove a monotonicity formula for free boundaries due to Alt-Caffarelli-Friedman. This monotonicity formula will then allow us to study the fine regularity properties of the free boundary in the obstacle problem. If time exists we will study the free boundary of a new type of problem: the problem of composite materials.

640:518
`Partial Diff Equations`
Y. Li
HLL 425 MW 5; 3:20-4:40

**Partial Differential Equations**

In this course I plan to cover the following:

- Sobolev spaces and some embedding theorems.
- L² theory for divergence form linear second order elliptic equations and systems.
- Scaling, blow up, and "small energy implies regularity": a theorem of Morrey on the regularity of minimizers of a class of quadratic functionals in dimension two, partial regularity for solutions of some nonlinear elliptic systems, harmonic maps with prescribed singularity (presenting in the simplest special case of the results in [Y.Y. Li and G. Tian, Comm. Math. Phy., 1992]).
- Elliptic equations and systems from composite material (presenting in the simplest special case of the results in [Y.Y. Li and M. Vogelius, Arch. Rat. Mech. Anal., 2000] and [Y.Y. Li and L. Nirenberg, Comm. Pure Appl. Math., 2003].

640:521:01
`Harmonic Analysis`
R. Wheeden
HLL 425 MTh 2; 10:20-11:40

**Harmonic Analysis**

^{n}. The main topics will include the following as time permits: Hardy-Littlewood maximal function, Marcinkiewicz interpolation theorem, Hardy-Littlewood-Sobolev theorem about Riesz fractional integrals, John-Nirenberg space of functions of bounded mean oscillations, endpoint results for fractional integrals, Poincaré-Sobolev estimates, Calderòn-Zygmund singular integrals, Littlewood-Paley theory, elements of weighted norm inequalities.

640:541
`Intro Alg Topology II`
P. Landweber
HLL 525 TF 2; 10:20-11:40

**Introduction to Algebraic Topology II**

This course will be a sequel to Math 540 being taught by Prof. Ferry in Fall 2005, but can also be viewed as a mostly independent course on cohomology, vector bundles, and characteristic classes for students who have already had an introduction to homology. The main point will be to show the use of cohomology for solving geometric problems.

The text (for coverage of cohomology) will be Allen Hatcher's excellent new book "Algebraic Topology", available for about $30 in paperback from Cambridge University Press, as well as online at http://www.math.cornell.edu/~hatcher The second part of the course will be an introduction to vector bundles and characteristic classes, following a further book in progress by Hatcher on his web site, as well as Milnor and Stasheff's classic book "Characteristic Classes".

640:548
`Intro Geometry & Topology`
F. Luo
HLL 423 MW 5; 3:20-4:40

The goal of this one-semester course is to introduce students the basic concepts and examples in the fields of geometry and topology. It is example-based, namely, we want the students know: what is a manifold, a Riemannian metric, a Riemann surface, a classical Lie group, fundamental groups, covering spaces, the Euler characteristic and few others.

The following is a list of topics intended to be covered.

0. Point Set Topology

0.1. metric spaces, topological spaces and continuous maps

0.2. compactness theorem, contraction mapping theorem

0.3. connectedness

1. Basic examples in Topology and Geometry.

1.1. inverse function theorem.

1.2. manifolds, smooth manifolds, Lie groups, O(n),
GL(n), SL(n), S^n, RP^n, CP^n, Riemann surfaces,
relationship to algebraic geometry.

1.3. tangent and cotangent bundles, vector fields,
Lie derivatives,differential forms, integrations and Stokes' theorem
and the definition of de Rham cohomology.

2. Introduction to Topology

2.1. fundamental group and elementary homotopy theory.

2.2. covering spaces.

2.3. The Euler characteristic.

3. Introduction to Geometry

3.1. Riemannian metrics, length, volume and the Riemannian connection.

3.2. covariant derivatives, Riemann, Ricci, scalar
curvature tensors, and computations on surfaces.

3.3. lengths of curves, geodesics, first and second
variations, and the Hopf-Rinow theorem.

640:552
`Abstract Algebra II`
S. Thomas
HLL 425 MW 4; 1:40-3:00

**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.

640:555
`Topics in Algebra`
Y-Z. Huang
HLL 525 MW 5; 3:20-4:40

**FUSION RULES, MODULAR TRANSFORMATIONS AND THE VERLINDE FORMULA**

One of the most important discoveries by physicists in two-dimentional
conformal field theory is the famous relation between the fusion rules
and the action of the modular transformation *t → -1/t* on
the space of vacuum characters. It states that this action of the
modular transformation diagonalizes the matrices formed by the fusion
rules. This relation was first conjectured by E. Verlinde. Combined
with axioms for higher-genus rational conforman field theories, the
Verlinde conjecture lead to a Verlinde formula for the dimensions of
the spaces of conformal blocks on higher-genus Riemann surfaces. In
the particular case of the conformal field theories associated to
affine Lie algebras (the Wess-Zumino-Novikov-Witten models), this
Verlinde formula gives a surprising formula for the dimensions of the
spaces of the "generalizedtheta divisors" and has led to many exciting
developments and deep mathematics.

Assuming the axioms for rational conformal field theories, Moore and Seiberg proved this Verlinde conjecture by deriving a fundamental set of polynomial equations. However, since there is no construction of rational conformal field theories satisfying the axioms needed in the arguments of Moore and Seiberg, precise formulations and mathematical proofs of the Verlinde conjecture and the Verlinde formula is still needed. In the particular case of the Wess-Zumino-Novikov-Witten models, the Verlinde formula was studied by many people and was proved by Beauville-Laszlo and Faltings using the work of Tsuchiya-Ueno-Yamada and Kumar-Narasimhan-Ramanathan.

Recently a precise formulation and a proof of the Verlinde conjecture in the general case has been obtained. In this course, I will discuss the formulation, the proof and the application of this general version of the Verlinde conjecture.

**Prerequisites:** I will assume that the students have some basic
knowledge in algebra and complex variables, as covered in the
first-year graduate courses, and some basic knowledge in vertex
operator algebra theory, as covered in the course given by Lepowsky in
Fall 2005.

**Text:** There is no textbook available. Some expository and
research papers will be distributed.

640:556
`Representation Theory`
V. Retakh
HLL 525 TTh 6; 5:00-6:20

**Representation Theory**

Representation theory is one of the cornerstones of modern mathematics. It provides a mathematical formalism for studying symmetry, and has a very wide range of applications to other mathematical disciplines and other branches of science (physics, chemistry, economics, etc.). The course focuses on finite-dimensional representations of finite groups, semisimple complex Lie groups and Lie algebras.

Despite being one of the best developed parts of mathematics, the representation field is still full of natural open problems and is a subject of an active current research.

**Prerequisites:**
The required background is some basic algebra (main concepts of linear
algebra and the theory of groups, rings and modules) and analysis. No
knowledge of representation theory is assumed; the course will provide
an introduction to its basic concepts and techniques. An emphasis
will be made on a detailed study of specific examples such as the
symmetric group, the general linear group, other classical groups and
their Lie algebras.

**Textbook:** W. Fulton and J. Harris,
*Representation Theory. A First Course*

640:560
`Homological Algebra`
C. Weibel
HLL 423 TTh 4; 1:40-3:00

**Intro to Homological Algebra**

This will be an introduction to the subject of Homological Algebra. Homological Algebra is a tool used in many branches of mathematics, especially in Algebra, Topology and Algebraic Geometry.

The first part of the course will cover Chain Complexes, Projective
and Injective Modules, Derived Functors, *Ext* and *Tor*. In
addition, some basic notions of Category Theory will be presented:
adjoint functors, abelian categories, natural transformations, limits
and colimits.

The second part of the course will study Spectral Sequences,
and apply this to several topics such as Homology of Groups and
Lie Algebras. Which topics we cover will be determined by the
interests of the students in the class. No homework will be assigned,
but problems will be suggested on a weekly basis.

**Prerequisite:** First-year knowledge of groups and modules.

**Textbook:** *An introduction to homological algebra*,
by C. Weibel, Cambridge U. Press, paperback edition (1995).

640:567
`Model Theory`
G. Cherlin
HLL 425 TTh 6; 5:00-6:20

**Model Theory**

This course is intended to serve as an efficient introduction to model
theory, while arriving at some topics on or close to the research
frontier.

We aim to develop three topics.

- Models of arithmetic
- Automorphism groups
- Stability

Automorphism groups of infinite structures tend to be hugely complicated. There is a general sense that if the structures are sufficiently homogeneous then in a rough sense the entire structure should be recoverable from its automorphism group. This is an active research area whose central techniques involve a mixture of the model theoretic and descriptive set theoretic points of view (and some finite combinatorics). Stability theory is firmly established as the core technology of contemporary model theory and we will attempt to deal with it systematically following Buecher's treatment in Essential Stability Theory. This begins with Morley's categoricity theorem and its refinements (Baldwin-Lachlan, Zilber) before taking up the general theory.

**Prerequisites:** First order logic as far as the Completeness Theorem.

**Reference:** S. Buechler, *Essential Stability Theory,*
Springer, Perspectives in Mathematical Logic, 355 pp., ISBN 3540610111.

640:571
`Number Theory`
J. Tunnell
HLL 425 TTh 5; 3:20-4:40

This will be a introductory course in Algebraic Number Theory. The subject matter of the course should be useful to students in areas of algebra and discrete mathematics, which often have a number theoretic component to problems, as well as students in number theory and algebraic geometry.

The basic invariants of field extensions of finite degree over the rational field (so-called number fields) will be introduced --- ring of integers, class number, units group, zeta functions, adele rings and group of ideles. The relation of these abstract invariants to the problem of solving polynomial equations in integers will be developed. Special examples of number fields such as quadratic and cyclotomic fields which have a rich structure will be used to illuminate the theoretical aspects. Algorithmic computation of these invariants will be analyzed, and open questions detailed. Weekly problem sets will investigate applications and extensions of the material.

For more information consult the course website .

640:573
`Spec Topics in Number Theory`
H. Iwaniec
HLL 124 TF 3; 11:30-12:50

**Prime Numbers II**

This course is a continuation from the Fall 2005. For new students I shall distribute some of my notes from the Fall semester which may help to enter the course in the middle. The lectures will be on Tuesdays and Fridays, 12:00 - 13:20 in Hill #124 .

Questions about prime numbers lay in the heart of analytic number theory. Many subjects of the theory were created for solving problems of asymptotic distribution of primes, and later became useful for other topics. In particular the theory of the zeta and L-functions grew from these inspirations. In this course I will present a large panorama of problems, methods and results. Proofs will be given for the most fundamental results, while more advanced arguments will be surveyed in considerable details. The central areas of the course are:

- Prime Number Theorem, elementary and analytic proofs

- Primes in arithmetic progressions

- Gaps between primes

- The Grand Riemann Hypothesis and its substitutes

- Density theorems for zeros off the critical line

- The Pair Correlation Theory, and random matrix interpretations

- The least prime in arithmetic progression

- Primes represented by polynomials

- Equidistribution of roots of quadratic congruences to prime moduli

- Irregularities in the distribution of prime numbers.

- Goldbach conjecture for three primes

640:640
`Experimental Mathematics`
D. Zeilberger ARC 119 (PC IML lab)
HLL 124 MTh 3; 12:00-1:20

**Bioinformatics Algorithms**

**TEXT:** *An Introduction to Bioinformatics Algorithms*
by Neil C. Jones and Pavel A. Pevzner and handouts

* E-mail: zeilberg at math dot rutgers dot edu

* Classroom: ARC 119 (IML room inside computer lab)

* Dr. Zeilberger's Office: Hill Center 704 (Phone: (732) 445-1326)

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in this direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they'll decide to do research in.

We will first learn Maple, and how to program in it. This semester the focus will be on bioinformatics, using the Jones/Pevzner excellent and very readable book. So in addition to becoming Maple whizes, students will learn the fundamentals of bioinformatics, in the best possible way, by programming it! In addition to its intrinsic interest and beauty, knowing the basics of both Maple programming and bioinformatics is an excellent backup plan for budding academicians.

642:528
`Methods of Appl Math II`
T. Butler
HLL 423 TTh 6; 5:00-6:20

**METHODS OF APPLIED MATHEMATICS**

**Text:**

*Advanced Engineering Mathematics*(second edition) by M. Greenberg

642:575
`Numerical Solutions of Partial
Differential Equations`
R. Falk
HLL 124 TTh 6; 5:00-6:20

**NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS**

In this course, we study finite difference, finite element, and finite volume
methods for the numerical solution of elliptic, parabolic, and hyperbolic
partial differential equations. The course will concentrate on the key ideas
underlying the derivation of numerical schemes and a study of their stability
and accuracy. Students will have the opportunity to gain computational
experience with numerical methods with a minimal of programming by the use of
* Matlab's PDE Toolbox* software.

Since there are many sophisticated computer packages available for solving partial differential equations, one might think that a thorough understanding of the numerical methods employed is no longer necessary. A striking example of why naive use of such codes can lead to disastrous results is the sinking of the Sleipner A offshore oil platform in Norway in 1991, resulting in an economic loss of about $700 million. The post accident investigation traced the problem to inaccurate finite element approximation of the linear elastic model of the structure (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design. More careful finite element analysis, made after the accident, predicted that failure would occur with this design at a depth of 62m, which matches well with the actual occurrence at 65m.

For more information on this disaster, see the web site: Disaster

642:581
`Graph Theory`
R. Radoicic
HLL 525 TF 3; 12:00-1:20

**Applied Graph Theory**

* connectivity

* matching theory

* planar graphs

* coloring problems

* flows

* extremal graph theory

* some basic results in Ramsey theory and random graphs

* minors

* hamiltonicity

* random walks on graphs

* Tutte polynomial

Occasionally, if time permits, we may throw in some rubber band embeddings or spectral graph theory.

**Prerequisites:**The course is mostly self-contained, although some previous combinatorics, linear algebra, rudimentary probability are all sometimes helpful.

**Main text:**R. Diestel,

*Graph Theory*.

We will cover some topics from B. Bollobas,

*Modern Graph Theory*; or the lecture notes of Lovasz http://research.microsoft.com/users/lovasz/course.htm

642:583
`Combinatorics II`
J. Beck
HLL 425 WF 1; 8:50-10:10

**Combinatorics**

This is the second part of a two-semester course surveying basic topics in combinatorics.

642:587
`Sel Topics Discrete Math`
J. Kahn
HLL 124 TF 2; 10:20-11:40

**Algebraic Methods in Discrete Math**

This course will survey applications of ideas from algebra (mostly linear) to problems in discrete mathematics and related areas. Areas of application include extremal problems for finite sets and the n-cube; theoretical computer science; discrete geometry; graph theory; probability; additive number theory and group theory; etc. theory. Various open problems will be discussed.

**Prerequisites:** I will try to make the course self-contained except for
basic combinatorics and linear algebra. A course in each of these
would be helpful. See me if in doubt.

**Text:** Babai-Frankl, *Linear Algebra Methods in Combinatorics.*
(This is actually only a manuscript. It's not mandatory: we won't really
follow it, but will overlap it to some extent; on the other hand, it has
lots of nice material and is relatively cheap.)

642:591
`Topics in Probability and Ergodic Theory`
V. Vu
HLL 525 WTh 2; 10:20-11:40

**Random structures**

We will discuss several random structures such as random graphs, random
matrices and random polytopes and their relations with other fields
(internet graphs, theoretical computer science, mathematical physics).

Through the study of these structures, we also introduce some of the
most useful tools in probabilistic combinatorics (large deviation,
coupling method etc).

**Prerequisites:** Basic probability.
If you have taken Kahn's course in the Fall, then you must be well prepared.

**Textbook** (optional, but useful)
Alon-Spencer: The probabilistic method,

Janson-Luczak-Rucinski: Random graphs.

642:622
`Financial Math`
D. Ocone
HLL 425 TTh 4; 1:40-3:00

**Financial Math II**

This is the second semester of 642:621. It is an introduction to modern mathematical analysis of financial markets and financial instruments. The finance concepts, such as financial derivatives and no arbitrage, and the basic probabilistic ideas for their analysis will be introduced first and briefly for discrete time models. After this introduction, the course will move to continuous time models. It will cover Brownian motion, martingales, stochastic calculus, diffusions and their related partial differential equations, and apply these to modeling financial markets and to the valuation of derivatives. Major goals are the Black-Scholes option pricing formula, risk neutral pricing, hedging, and the study of American and exotic options. Last Modified 10/24/2005.