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:

1. Riemann mapping theorem, Picard's theorem, Kobayashi hyperbolic metric and the geometric interpretation of the Schwartz lemma;
2. 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.
3. 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

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^(n+2)/(n-2), 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.
• 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

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

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

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.

1. Basic transformation group theory
   1. Isometry groups.
   2. Slice lemma and homotopy lifting
   3. Isotropy representation and stratification.
   4. Fixed point theorems.
2. Positive curvature and totally geodesic submanifolds
   1. Morse theory of loop spaces.
   2. Synge type theorems and Wilking theorem.
   3. A uniformation: a connectedness principle.
   4. Applications.
3. Positive curvature and continuous symmetry
   1. Examples.
   2. Fixed point theorem in even-dimensions
   3. Fundamental groups in odd-dimensions.
4. Classification of positive curved manifolds with large symmetry rank.
   1. Classification of the homogeneous spaces.
   2. Diffeomorphism classification of the maximal rank.
   3. Homotopy classification of almost half maximal rank.
   4. 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

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

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:

1. Classical L-functions
   • functional properties
   • subconvexity bounds
   • distribution of zeros