Descriptions of spring 2005 courses in the Rutgers-New Brunswick Math Graduate Program

# Descriptions of proposed spring 2005 courses in the Rutgers-New Brunswick Math Graduate Program

Description-free course listings

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

Real Analysis II

This course will, of course, be a continuation of Math 501.

Topics to be covered: the Riesz representation theorem for positive linear functionals on C(X), the Birkhoff ergodic theorem, the Marcinkiewicz interpolation theorem, the Riesz-Thorin interpolation theorem, the Fourier transform and elementary theory of singular integrals.

The last several weeks of the course will discuss some topics which lead to current research interests concerning fractals, Hausdorff measure and Hausdorff dimension of fractals.

640:504    Theory Func Comp Vars II     X. Huang    HLL 423   MTh 3; 11:30-1:10

Introduction to the Theory of Riemann Surfaces

This course is a continuation of Math 503. We study holomorphic and meromorphic functions defined over Riemann surfaces. We also study the classification problem for Riemann surfaces under the action of biholomorphic transformations.

Riemann surfaces give the simplest examples of complex manifolds; and complex manifolds play essential roles in the recent development of many branches of mathematics such as Algebraic Geometry, Differential Geometry, Non-Linear Analysis, Number Theory, etc. Also, the research on Riemann surfaces provides an important source of motivations for the study of Complex Analysis of Several Variables.

The main topics that we will discuss include the following:
(A) Invariant metrics on domains in the complex plane, Ahlfors-Pick-Schwarz Lemma
(B) Concept of Riemann surfaces and the proof of the Riemann-Roch theorem through the method of the elliptic theory
(C) Non-compact Riemann surfaces and the proof of the uniformization theorem through the method of $\bar{\partial}$-equation
(D) Quasi-conformal mappings and complex dynamics (if time permiting)

References:

1. Xiaojun Huang, Lecture notes on compact and non-compact Riemann surfaces, 2002. (To be distributed during in the class).
2. C. L. Siegel, Topics in Complex Function Theory, Wiley-Intersciences, New York.
Vol 1, 1969, Elliptic Functions and Uniformization Theory
Vol 2, 1971, Automorphic Functions and Abelian Integrals.
3. H. Weyl, The concept of a Riemann surfaces, Addison-Wesley, Reading, Massachusetts, 1955.

640:508    Functional Analysis     A. Soffer HLL 423   TTh 4; 1:10-2:30

The goal of the course is a basic introduction to Functional Analysis and Hilbert space operator theory, with some applications to PDE and Mathematical-Physics.

We begin with a review of the basic facts about Hilbert and Banach spaces.
Then the notion of topological vector space with general topology will be discussed; dual spaces and distributions then follow.
Applications to Partial differential equations, in particular Evolution equations of mathematical physics.
Finally, the theory of linear operators in Hilbert spaces including the spectral theorem.

Book: mostly Reed Simon I -Functional analysis and Davies book on Differential Operators.

Prereq.: real analysis,including basic measure theory.

640:510    Sel Topics in Analysis    YY. Li     HLL 423   TF 2; 9:50-11:10

Nonlinear PDE's and the Yamabe Problem

The course will consist of two parts. In the first part, I will present a few useful methods in the study of some nonlinear partial differential equations. In the second part, I will present some results on the Yamabe problem and a fully nonlinear version of the Yamabe problem, and some results on the Monge-Ampere equations.

For the second part, emphasis will be on the ideas, methods and open problems----we do not carry out all the details.

The first part of the course will cover:
Leray-Schauder degree theory and some applications to PDEs (existence of solutions, bifurcation of solutions, etc.), degree theory for second order fully nonlinear elliptic equations and applications, concentrated compactness, compensated compactness and some applications, small energy implies regularity'' (a theorem of Morrey on the regularity minimizing harmonic maps from two dimensions, partial regularity for solutions of elliptic systems including harmonic maps), etc.

The second part will cover:
I. Introduce the Yamabe problem ---- existence on a given compact Riemannian manifold of conformal metrics of constant scalar curvature. The problem was solved through the works of Yamabe (60), Trudinger (68), Aubin (76) and Schoen (84). We outline the proofs of these results, including the positive mass theorem of Schoen and Yau on which the proof of the last case relies.
II. Present some recent and ongoing joint work with Lei Zhang on compactness of solutions to the Yamabe problem.
III. Present some recent and ongoing joint work with Aobing Li on a very general fully nonlinear version of the Yamabe problem.
IV. Present Pogorelov's interior second derivative estimates for solutions to the Monge-Ampere equations, the classical work of Caffarelli, Nirenberg and Spruck on the Dirichlet problem for the Monge-Ampere equations, some aspects of the regularity theory of Caffarelli on Monge-Ampere equations, and some recent and ongoing joint work with Caffarelli,motivated by the SYZ conjecture in Mirror Symmetry, on multi-valued solutions to the Monge-Ampere equations.

640:511    Sel Topics in Analysis    Bahri     HLL 124     TTh 2; 9:50-11:10

Contact Form Geometry (in dimension 3)

This course is an introductory course to Contact Form Geometry and Legendrian curves.

It will start with the basic definition of a contact form, Reeb vector-fields, basic properties of a contact structure and a vector-field in its kernel, Gray's theorem about deformation of contact structures.

We will then study in detail the action functional defined by a contact form on a C1 (H1-) curve. We will compute the first variation of this action functional as well as the second variation, discuss the stationary points and the finiteness of the Morse index.

We will then consider a dual form β and constrain the variations to the space of Legendrian curves of β, Lβ. We will build on Lβ a pseudo-gradient for the action functional (construction from scratch), study the geometric and analytic properties of its ω-limit set. This will lead us to the formal definition of a homology.

Part of this homology is understood. Another part is related, as we will show, to the topology of the Configuration Spaces of the underlying manifold M3.

The course should bifurcate then in two directions: One one hand, we will study finite dimensional manifolds Γ2k which are intermediate between Configuration Spaces and Legendrian curves. We will explore whether these are symplectic manifolds and discuss the extension of the action functional to them (critical points, Morse index etc.). This is directly related to the homology introduced earlier. It also develops into an interesting relationship between singular and smooth solutions to the periodic orbit problem for the Reeb vector-field of a contact form (a strong analogy with Yamabe-type problems).

On the other hand, we will come back to some basic assumptions underlying the homology and describe how to get rid of them.
The computation of the homology follows in some simple cases.
Several open problems will be described, some quite within reach, others more difficult.

640:518    Partial Diff Equations    R. Wheeden     HLL 423   MW 4; 1:10-2:30

Partial Differential Equations

The course will focus primarily on developing subelliptic versions of the material in Chapters 8 and 10 of Gilbarg-Trudinger: Elliptic Partial Differential Equations of Second Order. Some appropriate generalizations of results from Chapter 7 will be needed and developed, including estimates of Poincaré-Sobolev type. Prior knowledge of the elliptic case is not necessary. The basic idea is to study regularity properties of solutions of some subelliptic, second order, linear, divergence form p.d.e.'s with rough coefficients.

The primary goal is to show that their solutions are Hölder 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. Motivation for such results comes from many places, including the question of hypoellipticity of solutions of degenerate nonlinear Monge-Ampere type equations with smooth data. Quasilinear equations arise naturally in this context. As time permits, some applications to nonlinear and quasilinear problems will be discussed.

640:519    Sel Topics Diff Equations    Z. Han     HLL 423   MW 2; 9:50-11:10

Linear and Nonlinear Heat Equations

Nonlinear heat/diffusion equations have played important roles in recent years in several active areas of research, including phase transitions and the heat flows in the study of deformation of metrics, surfaces. In this course I plan to introduce the basic background material for dealing with such type of equations, and demonstrate some of their applications.

I will start by discussing the basics of linear heat equations, most probably following the last three chapters of Lectures on Elliptic and Parabolic Equations in Holder Spaces by N. V. Krylov (Graduate Studies in Mathematics, vol. 12, AMS). Then I will use other sources to discuss the Lp estimates, Harnack estimates via Moser iteration and via Li-Yau. The rest of the semester will be used to discuss applications involving nonlinear heat equations. The particular choices of topics on applications will take into account students inputs.

The technical prerequisite for this course is the basics of Lebesgue integration theory and Lp spaces. No prior course on PDE is required.

640:534:01    Sel Topics in Geometry    X. Rong     HLL 423    MTh 2; 9:50-11:10

Positive Curvature, Symmetry and Topology

Positively curved manifolds have been a frequent subject in Riemannian geometry, where several disciplines of mathematics interact (differential geometry, analysis and topology).

This course will focus on a core issue of Riemannian geometry: interplay between curvature and topology. We will give a quick introdction to the area of (positive) curvature, symmetry and topology with a self prepared tools. We also hope to present some of the most recent advances in this field.

1. Chapter 1. Compact transformation groups
2. Chapter 2. Positively curved manifolds
3. Chapter 3. Positive curvature with abelian symmetry
4. Chapter 4. The connectedness principle in positive curvature
5. Chapter 5. Fundamental groups
6. Chapter 6. Classifications of positively curved manifolds with large symmetry (depending on progress of lectures)
Prerequisites: a basic knowledge in differential geometry that equivalent to the basic contents in the introduction level course, Introduction to differential geometry '' (math 533), taught by Professor Huang in the fall of 2004.

640:541    Intro Alg Topology II P. Landweber     HLL 525     T3 (11:30-12:50) and Th5 (2:50-4:10)

Introduction to Algebraic Topology II

This course will be a sequel to Math 540 being taught by Prof. Ferry in Fall 2004, but can also be viewed as a mostly independent course on cohomology and homotopy theory for students who already have had an introduction to homology.

The text will be Allen Hatcher's excellent new book Algebraic Topology, available for \$30 in paperback from Cambridge University Press, as well as online at hatcher's site

The plan is to start with cohomology in Chapter 3 (the extent of the coverage depending on how far Math 540 gets into this chapter). We will then cover basic results on homotopy groups in Chapter 4, such as the long exact sequences for pairs of spaces and fiber bundles, and will take up a number for further topics that relate homotopy groups to homology and cohomology. We will also select from the additional topics, and will study the homotopy groups of classical groups and the cohomology of fiber bundles.

Depending on available time, the course may end with an introduction to vector bundles and characteristic classes, following a further book in progress by Hatcher on his web site, or Milnor and Stasheff's book Characteristic Classes.

640:547    Topology of Manifolds    C. Woodward     HLL 525     MTh 3; 11:30-12:50

Symplectic Geometry

Description: This will be a course on symplectic geometry. Symplectic geometry is a kind of skew-symmetric sister to Riemannian geometry, in which one studies manifolds equipped with a skew-symmetric bilinear form on each tangent space.
Originally a framework for Hamiltonian mechanics, symplectic geometry now places a role in most areas of modern mathematics. Many of the recent activity has centered around Floer theory for symplectic manifolds, an infinite dimensional version of Morse theory, developed to a conjectural lower bound on the number of equilibria of a Hamiltonian dynamical system. Even more recently, Fukaya's categorical framework for Floer theory has been conjectured by Konstevich to play a role in mirror symmetry of quantum field theories.

The first part of the course is planned as an introduction to the basics,
the second will be a survey of various topics, including Fukaya-Floer theory.

Topics: Part I Introduction to symplectic geometry: Darboux's theorem, Poisson brackets, Hamiltonian flows, and examples in classical mechanics. Symmetries of symplectic manifolds: moment maps, symplectic reduction.
Part II Topics in Floer-Fukaya theory: J-holomorphic curves, Floer theory of Lagrangian submanifolds, Fukaya category for monotone symplectic manifolds.

Text: Foundations of Mechanics by Abraham and Marsden, Perseus Publishing, 2nd Edition (1994).

640:548    Differential Topology    S. Ferry     HLL 124     TTh 4; 1:10-2:30

Introduction to Geometry & Topology (New!)

The goal of this NEW! course is to introduce students to the basic concepts and examples in the field of differential geometry and topology. It will cover the point set topology required in the qualify exam. It will also provide a preparation for taking both Algebraic Topology and Differential Geometry courses.

1. Point-set Topology (2-3 weeks). Compactness, connectedness of metric and other topological spaces.
2. Basic Examples (4 weeks). Manifolds, Riemann surfaces, Stokes Theorem.
3. Algebraic Topology (3-4 weeks). Fundamental group, basic homotopy theory, applications.
4. Differential Geometry (4 weeks). Metrics, length, connections, curvature.
5. Summary, or Catch-up (1 week). Maybe the syllabus is too ambitious?

640:549    Lie Groups    R. Goodman     HLL 425    MTh 5; 2:50-4:10

This course will be an introduction to Lie groups, beginning with the general linear group and the other classical groups (the unitary, orthogonal and symplectic groups) and finishing with the Weyl character formula and the Borel-Weil theorem for the irreducible representations of a compact, connected Lie group such as U(n). The prerequisites are advanced calculus (differentiation and integration of functions of several real variables), basic linear algebra and elementary ideas from topology (such as covering spaces). The point of view will be much more analytic than in the Fall 2004 Lie algebra course 640:550, and students with no prior knowledge of Lie algebras should not be at a disadvantage in the Lie groups course.

Topics will include the following:

1. The exponential map and exponential coordinates for matrices; the Campbell-Baker-Hausdorff series.
2. Linear Lie groups and their Lie algebras; correspondence between Lie subalgebras and Lie subgroups.
3. The classical linear groups as Lie groups.
4. Homogeneous spaces for Lie groups.
5. Integration on manifolds, Lie groups, and homogeneous spaces; Weyl integral formula for compact Lie groups.
6. Representations of compact Lie groups; Peter-Weyl theorem; Weyl character formula; Borel-Weil theorem
Main Text: Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups, ISBN 0-19-859683-9, Oxford University Press, 2002.
Supplementary Text: Roe Goodman and Nolan R. Wallach, Representations and Invariants of the Classical Groups, (3rd printing), ISBN 0-521-66348-2, Cambridge University Press, 2003.

640:552    Abstract Algebra II    R. Wilson     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:

4. Galois Theory: Finite algebraic extensions, resolutions of equations by radicals (and without radicals)
5. Rings of Polynomials: Noetherian rings, Hilbert basis theorem, Noether normalization, Nullstellensatz
6. Basic Module Theory: Projective and injective modules, resolutions, baby homological algebra, Hilbert syzygy theorem
7. 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.

640:555    Sel Topics in Algebra    D. Maclagan     HLL 425    MW6; 4:30-5:50

Combinatorial Commutative Algebra

The interface between combinatorics and commutative algebra/algebraic geometry has been fruitful for both subjects. We will discuss some of the impact of commutative algebra in combinatorics (a little) and the use of combinatorial techniques in commutative algebra and related areas (more).

The main tools include simplicial complexes and polyhedral techniques (on the combinatorial side) and monomial ideals, resolutions, Hilbert functions, and computational techniques (on the algebraic side). The only prerequisite is basic algebra.
Useful books include Stanley's Combinatorics and Commutative Algebra, Bruns and Herzog's Cohen Macaulay Rings, and the forthcoming Combinatorial Commutative Algebra by Miller and Sturmfels.

640:556    Representation Theory    F. Knop     HLL 525   TTh 6; 4:30-5:50

Representation Theory

Representation theory studies the ways a given group or algebra can be represented as a group (or algebra) of matrices. Knowing these has often profound consequences since representations appear often as linearizations of non-linear (e.g. discrete) objects.

In this course we will be mostly concerned with representations of finite groups. This theory is beautiful, very useful, quite elementary, and paradigmatic for other branches of representation theory.

At the end, the representations of symmetric groups will be presented in more detail.

Prerequisites: basically none beyond Linear Algebra.

References: Jean-Pierre Serre: Linear representations of finite groups Springer GTM 42

640:558    Theory of Algebras    E. Taft     HLL 423     TF 3; 11:30-12:50

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.

640:566    Axiomatic Set Theory    S. Thomas     Freyl B-3 (CAC)     TTh 6; 4:30-5:50

An introduction to Borel equivalence relations

This course will give an introduction to the currently most active area in classical descriptive set theory: namely, the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related theory of Borel and analytic equivalence relations. This study will give us techniques for measuring the relative complexity of classification problems in numerous areas of mathematics and the nature of their possible complete invariants. For example, we shall see that the problem of classifying finitely generated groups up to isomorphism is strictly easier than that of classifying arbitrary countable groups. Similarly, the classification problem for finitely branching trees is strictly easier than that of arbitrary countable trees. We shall also discuss some of the many open problems in this exciting new area.

Prerequisites: a basic knowledge of Borel sets, Lebesgue measure, the notions of a meager and a comeager set, basic model theory.

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

640:574    Topics in Number Theory    J. Tunnell     HLL 525    MW 4; 1:10-2:30

Modular Forms and Modular Curves

Modular forms are functions on sets of elliptic curves with extra structure, and modular curves are algebraic curves parameterizing these sets. The complex analytic aspects involve holomorphic functions on the upper half-plane satisfying transformation laws and complex curves which are quotients of the upper half-plane by discrete groups. The number theoretic aspects stem from the interesting Fourier coefficients of modular forms and the rational points on modular curves.

Topics will include the following :

{1.}Modular forms as higher differentials on quotients of the upper half plane
{2.}Examples of modular forms and modular curves
{3.}Dimensions of spaces of forms via the Riemann-Roch theorem
{4.}Hecke theory and correspondences on modular curves
{5.}L-series of modular forms
{6.}Number theoretic applications of modular forms and curves

Text: Course notes will be distributed. Standard references will be on library reserve.

Course Format: There will be periodic problem assignments and term projects involving modular forms and curves.

642:528    Methods of Appl Math II    T. Butler     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.

642:583    Combinatorics II    J. Kahn     HLL 525    TF 2; 9:50-11:10

Introduction to Combinatorics

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

- Theory of finite sets, hypergraphs
- Ramsey theory
- Combinatorial discrepancy
- Partially ordered sets and lattices
- Correlation inequalities
- Algebraic methods
- Entropy methods

Prerequisites: 642.582 or some reasonable substitute (see me if in doubt).

Text: van Lint and Wilson, A Course in Combinatorics.
(Optional. We won't really follow it, but it's a nice book and has significant overlap with the course.
It and other relevant books will be on reserve.)

642:587    Sel Topics Discrete Math    D. Zeilberger     M3 in ARC PC IML Room 116 (11:30-12:50);     TH 3 in HLL 423

Algorithmic Discrete Math

Many proofs in Discrete Math are really algorithms in disguise. We will cover selected topics in discrete math from an algorithmic viewpoint, using Maple. The format will follow the succesful format of Spring 2003 Math 583. This means one class per week, traditional, and one class per week: in the computer lab. No former knowledge of Maple is needed. Students with little or no Maple background will be tutured. People who take this class will have a very good chance of getting a publication!
TEXT: Handouts.

642:622    Financial Math    T. Petrie     HLL 423    MW 5; 2:50-4:10

Financial Math II

This course is a continuation from the fall semester; however, it is open to new participants having the background mentioned below. New participants should check with the instructor at petrie@rci.rutgers.edu.

This course is an introduction to the field of financial mathematics. One of the chief aims is to provide the theoritical framework in which to value securities especially derivative securities. (Basic securities are stocks and bonds. A derivative security is any security whose value is derived from a basic security. Examples are stock indicies, options and futures.) Another aim is to provide the mathematical tools and develop mathematical models which will lead to valuation of securities and to trading and hedging strategies. Another aim of the course is to provide the framework to evaluate and manage risk in holding securities.

The course will begin with a discussion of the financial aspects of the course. This involves definitions of various securities and the details and operation of the markets in which they trade. Here are some examples: Forward and Future Contracts. Future Markets and Hedging, Options, Swaps, Leaps.

The central mathematical feature of this field concerns modeling the individual markets mathematically. Serious work in this area requires applied propability, statistics, some computer skills and stochastic partial differential equations. This material will be developed in the course. For background, participants should have some probability background like a good undergraduate course in probability and or statistics and some ability with a computer such as connecting to the internet and downloading files. Excel is an important computer skill which is often used in practical applications.

Any model in this area begins with some probability assumptions. The valuations which occur may come from probabilitic arguments or equilibria arguments. Both approaches will be discussed. A famous example of this is the derivation of the Black-Scholes formula for the price of a European Option. We will treat the model for this and the derivation in detail from both points of view. In the same spirit we will also treat other securities mentioned above in the same light. A basic tool in managing risk is the Capital Asset Pricing Model. This too will be treated.

In addition to presenting the theory behind the models and pricing of financial assets, we will present trading strategies for buying and selling stocks and bonds. We will present statistical tests for evaluating the trading strategies and we will emperically evaluate the trading strategies using current data available on the internet using an excel based program.

Refs: Options, Futures and Other Derivative Securities by J. Hull ; An introduction to probability and its applications vols. I and II by W. Feller; Arbitrage Pricing-notes by Musiela and Rutkowski. The Econometrics of Financial Markets by Campbell et. al.

642:662    Topics Math Physics     M. Kiessling     HLL 425    TF 3; 11:30-12:50

Relativistic Fields with Point Defects

Course description: "This course introduces students to one of the most profound --- and notorious --- problems in mathematical (and theoretical) physics, namely the construction of a consistent dynamical theory of relativistic fields with point sources. We will limit the discussion to the two "classical" fields, i.e. the electromagnetic and the gravitational field, but that doesn't mean we will be just concerned with classical physics. In fact, the material covered is part of an alternate (non-mainstream) quest for a quantum field theory which is mathematically well-defined and physically coherent.

Highlights:
1) In the beginning: Classical electron theory
a) Abraham-Lorentz's formal manipulations
b) Formal manipulation is a touchy business, or: `100 years of nonsense in the physics textbooks'
c) Recent developments (Spohn et al., Appel-Kiessling)
2) Nonlinear relativistic field theory (electromagnetic and gravitational)
a) Mie's program (Mie, Born, Hilbert, Weyl)
b) Einstein's program (Einstein, Hilbert, Schroedinger)
c) Born's program (Born, Infeld, Schroedinger, Dirac)
d) Something's missing: an intermediate assessment
3) What is quantum mechanics (with hindsight)?
a) Schroedinger's equation -- where is the physics?
b) Bohmian mechanics in a nutshell (de Broglie, Bohm, Goldstein et al.)
4) Motion of point defects in electromagnetic fields
a) Classical motion of point charges
b) Quantum motion of point charges without spin
c) Quantum motion of point charges with spin
5) Motion of point defects in gravitational fields
a) Classical motion of naked singularities
b) Quantum motion of naked singularities
c) Classical and quantum motion of Black Holes?
6) Motion of point defects in combined electromagneto-gravitational fields
a) Classical motion of charged naked singularities
b) Quantum motion of charged naked singularities without spin
c) Quantum motion of charged naked singularities with spin
d) Einstein: the old fool?