# Descriptions of proposed spring 2003 courses in the Rutgers-New Brunswick Math Graduate Program

640:502 `Theor Func Real Vari`
M. Kiessling
HLL 525 TTh 6; 4:30-5:50

Description to appear.

640:504 `Theor Func Comp
Vari` X. Huang
HLL 423 MW 5; 2:50-4:10

This is a continuation of Math 640:503. I intend in this course to cover some materials which are basic for those students working on topics related to geometric analysis, algebraic geometry and dynamics. More precisely, I will be focusing on the following:

- Riemann mapping theorem, Picard's theorem, Kobayashi hyperbolic metric and the geometric interpretation of the Schwartz lemma;
- Riemann surfaces, Divisors, sheaf and cohomology, Riemann-Roch theorem on compact Riemann surfaces, Hodge theory, and the uniformization theorem for non compact Riemann surfaces, if time permits.
- Quasiconformal mappings and its application to complex dynamics of one variable

640:508
`Functional Analysis` R. Nussbaum
HLL 423 TTh 2; 9:50-11:10

**Linear and Nonlinear Positive Operators with Applications**

The simplest example of a positive linear operator is a square matrix with nonnegative entries. More generally, a positive operator is a map which leaves invariant a "cone" in a Banach space. There is a rich theory of such operators and many applications in analysis. We shall begin with the classical Perron-Frobenius theory of nonnegative matrices and then develop some of the theory of positive linear operators in Banach spaces, particularly the Krein-Rutman theorem and some of its less well-known generalizations. Selected results in the nonlinear theory will also be developed. Applications will be given to PDE's (existence of positive eigenvalues and eigenvectors for second order elliptic PDE's), differential equations, differential-delay equations and max-plus operators.

640:509:02
`Sel Topics in Analysis`
Y. Li
HLL 423 MW 2; 9:50-11:10

In this course I plan to give an introduction to some areas in nonlinear elliptic partial differential equations, and will present a number of open problems in these areas. Ideas of proofs and outlines of proofs will be given, but details of proofs will not be emphasized. References will be given. The topics will include the following.

- Some recent joint work with Nirenberg on distance functions and cut locus in Riemannian and Finsler geometry. Two related subjects : The size of singular sets of viscosity solutions of Hamilton-Jacobian equations and a conjecture of Ambrose in differential geometry. Will introduce some basic concepts concerning viscosity solutions. A few open problems will be mentioned.
- The Yamabe problem, the Liouville type theorems of Obata,
Gidas-Ni-Nirenberg, and Caffarelli-Gidas-Spruck concerning
*-Delta u=u*, and some joint work with Aobing Li which establish existence and compactness results for a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds and also extend the Liouville type theorems to certain conformally invariant fully nonlinear elliptic equations. A number of open problems will be mentioned.^{(n+2)/(n-2)} - The Weyl problem in differential geometry. Local isometric embedding of two dimensional Riemannian surfaces into three dimensional Euclidean space. A theorem of Efimov concerning isometric immersions of complete Riemannian surfaces. Some recent progress in these areas and some open problems.

640:518
`Partial Diff Equations`
A. Shadi Tahvildar-Zadeh
HLL 423 TTh 5; 2:50-4:10

**Evolution equations and wave phenomena**

Beginning with the linear wave equation, we will study the wave phenomena in various hyperbolic PDEs arising in physical systems, including geometric field theories, electromagnetics, continuum mechanics, and general relativity. No prior knowledge of hyperbolic equations is assumed, and functional analytic tools that are needed will be developed along the way.

640:534:01
`Sel Topics in Geometry`
C. Woodward
HLL 425 MW 2; 9:50-11:10

**Symplectic Geometry**

The first half of the course will be an introduction to symplectic geometry, including Darboux's theorem, Poisson brackets, Hamiltonian flows, and examples in classical mechanics.

In the second half, we will discuss two dimensional gauge theory, and relations with quantum three-manifold invariants.

Prerequisite for the course is familiarity with differential forms on manifolds.

640:534:02
`Sel Topics in Geometry`
X. Rong
HLL 124 TTh 2; 9:50-11:10

**Curvature, Symmetry, and Topology**

In this course, we will focus on a core issue in Riemannian geometry: interplay between curvature and the topology. We will tour the field of (positive) curvature, symmetry and topology along preparing the basic tools. We will also discuss the most recent advances in this field.

The requirement for this course is a basic knowledge in Riemannian geometry, or amounts to the contents in the introduction level course, "Introduction to Riemannian geometry - A Metric Entrance", which I taught in the Fall of 2002.

- Basic transformation group theory
- Isometry groups.
- Slice lemma and homotopy lifting
- Isotropy representation and stratification.
- Fixed point theorems.

- Positive curvature and totally geodesic submanifolds
- Morse theory of loop spaces.
- Synge type theorems and Wilking theorem.
- A uniformation: a connectedness principle.
- Applications.

- Positive curvature and continuous symmetry
- Examples.
- Fixed point theorem in even-dimensions
- Fundamental groups in odd-dimensions.

- Classification of positive curved manifolds with large symmetry rank.
- Classification of the homogeneous spaces.
- Diffeomorphism classification of the maximal rank.
- Homotopy classification of almost half maximal rank.
- Homeomorphism classification of almost maximal rank.

640:536
`Algebraic Geometry`
J. Tunnell
HLL 525 MW 4; 1:10-2:30

This course will begin with the algebraic geometry of schemes as an extension of the classical theory of algebraic varieties. The tools of cohomology and sheaf theory will be developed and applied to obtain invariants of classical algebraic geometric objects such as curves and surfaces. Applications in areas such as arithmetical algebraic geometry, toric varieties, commutative algebra and number theory will be discussed. The course web site has more details.

640:540
`Intro Alg Topology`
F. Luo
HLL 425 MTh 3; 11:30-12:50

This will be an introduction to algebraic and differential topology. We plan to cover the basics on manifolds, singular homology theory, covering spaces and fundamental groups.

640:552
`Abstract Algebra`
F. Knop
HLL 124 TTh 4; 1:10-2:30

Module theory, Galois theory, representations of finite groups.

640:555:01
`Topics in Algebra`
W. Vasconcelos
HLL 425 MW5; 2:50-4:10

The aim of the course is the study of two rich structures of algebra: Hilbert Functions and Cohen-Macaulay Algebras. Both occur in Combinatorics, Commutative Algebra and Algebraic Geometry and we want to illustrate their uses in these areas. The treatment will be fairly elementary [i.e. direct] and preparatory material will be given whenever needed beyond the basic background of the Math 551-552 sequence [no additional course is really required]. We will also seek applications in Computational Algebra [and there are many].

Some of the texts we want to make use of are:

W. Bruns and J. Herzog, *Cohen-Macaulay Rings*, Cambridge
University Press, 1993

R. Stanley, *Hilbert functions of graded algebras*, Adv. Math. 28
(1978), 57-83.

640:555:02
`Topics in Algebra`
S. Sahi
HLL 423 MTh 3; 11:30-12:50

**Introduction to Kac-Moody algebras and their representations**

The course will be an introduction to the theory of Kac-Moody Lie algebras and their representations based on the book by V. Kac. Some prior knowledge of Lie theory will be helpful but not strictly necessary.

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

**L-functions**

This course is for advanced graduate students interested in number
theory, particularly in analytic and algebraic topics. Thge theory of
L-functions began from the work of Dirichlet on primes in arithmetic
progressions. Today the **ZOO** of L-functions spreads over all
territories of modern arithmetic (distribution of prime numbers,
arithmetic of number fields, automorphic forms, elliptic
curves). These L-functions are often used as a language for expressing
relations between remote objects, but first of all they provide
powerful tools for proving theorems. In the course I shall try to
cover many aspects, and I will touch the front line of current
research. The following subjects will be discussed:

- Classical L-functions
- functional properties
- subconvexity bounds
- distribution of zeros

- functional equation
- Riemann hypothesis

- Class number formula
- Gross-Zagier formula

- Equidistribution problems
- Small eigenvalues
- Quantum chaos

642:527
`Methods of Appl Math`
T. Butler
HLL 124 TTh 7; 6:10-7:30

This is a first-semester graduate course appropriate for students of
mechanical and aerospace engineering, biomedical engineering or other
engineering areas, materials science, or physics. We begin with power
series expansion, the method of Frobenius, and Bessel functions, and
go on to nonlinear differential equations, phase plane methods, and an
introduction to perturbation techniques. We then study vector spaces
of functions, including the L^{2} inner product, orthogonal bases,
Sturm-Liouville theory, Fourier series and integrals, and the Fourier
and Laplace transform. These ideas are applied using the method of
separation of variables to solve partial differential equations,
including the heat equation, the wave equation, and the Laplace
equation. The course focuses mainly on applied techniques and
conceptual understanding, rather than on theorems and rigorous proofs.

642:528
`Methods of Appl Math`
G. Goldin
HLL 124 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.

642:581
`Applied Graph Theory`
M. Saks
HLL 423 TF 3; 11:30-12:50

**Graph Theory**

This course will be an "advanced introduction" to graph theory. Topics include matching theory, connectivity, graph coloring, planarity, extremal graph theory, and the main techniques (elementary, probabilistic, algebraic, and polyhedral) for analyzing the structure and properties of graphs.

Text: Reinhard Diestel, *Graph Theory*, Springer-Verlag, 2nd Edition, 2000
(Note this edition is viewable--though not printable--at the site:
http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/download.html)

642:583
`Combinatorics`
D. Zeilberger
HLL 525 MTh 2; 9:50-11:10

Combinatorics is the most fundamental, and hence the most important,
branch of mathematics, since it deals with FINITE structures,
and the world is finite.

We will learn all the aspects of combinatorics: how to count,
how to construct, how to estimate, and how to prove existence,
and of course how to deduce interesting properties of interesting
structures like graphs, posets, lattices, and circuits.

The only prerequisite is love of the discrete.

642:587
`Sel Topics Discrete Math`
J. Kahn
HLL 425 TF 2; 9:50-11:10

**Combinatorial issues arising in probability**

We'll discuss some of the many neat things that are going on at the boundary of combinatorics and discrete probability, with an emphasis on what's of interest for the combinatorialist.

642:612:01
`Sel Top Appl Math`
O. Costin
HLL 124 MW 6; 4:30-5:50

Asymptotic methods are ubiquitous in pure and applied mathematics. The course provides a rigorous treatment of the subject, while building upon the invaluable classical intuition. Major recent developments in the subject will be discussed in the later part of the course.

In the first part we will look at asymptotic expansions and their applications in the study of differential, difference and functional equations; asymptotic expansions of integrals; the saddle point and stationary phase methods; WKB analysis; singular perturbations; adiabatic invariance.

The last part introduces the new tools of exponential asymptotics, analyzable functions (a vast extension of analytic functions) and illustrates their application to studying nonlinear phenomena in ordinary and partial differential equations. Further applications will be discussed in the specific areas of interest of the students.

Familiarity with basic real and complex analysis is assumed.

640:662:01
`Topics Math Physics`
J. Lebowitz HLL 124 M 4; 1:10-2:30 & W 2;
9:50-11:10

**Complex systems: physical reality and mathematical models**

There is an increasing recognition that as one proceeds from
systems with a few components to those with many, the latter may exhibit
complex behavior whose origin or specific form cannot be deduced, in any
meaningful way, from the properties of its constituents. Examples include:
the weather, the global economy, the biosphere, the brain, and life itself.
In this course we will investigate interesting features of such complex
systems or phenomena which poses some kind of universality.

Our approach to this problem will be based on ideas coming from
statistical mechanics. This discipline provides a framework for describing
how well-defined higher level patterns of organized behavior may emerge
from the nondirected activities of a multitude of interacting simple
entities. Examples of emergent phenomena, well explained by statistical
mechanics, are phase transitions in macroscopic systems - for example the
boiling or freezing of a liquid. Here dramatic changes in structure and
behavior of the composite systems are brought about by small changes in the
temperature or pressure. This has no counterpart in the behavior of the
individual atoms or molecules making up the system which in fact do not
change at all in the process.

How might we adapt the methods of statistical mechanics to higher
level collective systems in which the relevant basic constituents are
themselves more complex than those for which the theory was developed? In
particular, how can we best model the dynamic responses of complex systems
in an environment subject to a variety of regular and irregular (sometimes
best modeled as random) influences? What may be particularly important in
some of these systems are rare, but powerful events (avalanches,
catastrophes) which have long term effects.

Subjects covered will include modeling of: vehicular traffic,
ecological systems, social systems, the vertebrate immune system and the
brain.

**Requirements** The course will be informal and interactive. Some
familiarity with statistical mechanics and/or probability theory,
dynamical systems theory is desirable but not essential. If you are
interested and have any questions, please contact me.

642:662:02
`Topics Math Physics`
Y. Huang
HLL 423 MW 4; 1:10-2:30

**Superconformal field theories**

Superconformal symmetry play a fundamental role in string theory. It is also deeply related to Kahler structures on complex manifolds and the moonshine module. We shall study N=1 and N=2 superconformal algebras and their representations. Vertex operator superalgebras and modules associated to these representations will be constructed and studied and their "world-sheet" geometry will be discussed. We shall also discuss Gepner models and the conjectured N=2 superconformal field theories associated to Calabi-Yau manifolds.