Mathematics Department - Graduate Program - Spring 2008 Course Descriptions

Descriptions of Proposed Spring 2008 Courses


Mathematics Graduate Program

For less detail, see the description-free course listings.

640:502    Theor Func Real Vars    A. Shadi Tahvildar-Zadeh

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. We will use the textbook by Folland, "Real Analysis, modern techniques and and Applications" and supplement it with material from Lieb & Loss's "Analysis."


640:504    Theory Func Comp Vars     X. Huang


640:510    Sel Topics in Analysis    E.Carlen   

Topics in Optimal Mass Transportation

The Monge problem on optimal mass transportation concerns the deformation of one probability measure into another by transport of mass, subject to a cost for the transport of a unit mass from point x to point y. The problem is to determine the transportation plan that minimizes the total cost, should a minimizer exist. Important breakthroughs were made by Kantorovich in 1942 and Brenier in 1987. Since the work of Brenier especially, there have been many surprising developments in this very active area of analysis. Methods involving optimal mass transportation have been used to prove functional inequalities, new and classical, study the behavior of solutions of partial differential equations, and study a wide range of problems in the calculus of variations, for example. This course will provide an introduction to the theory and a number of recent applications. The only prerequisite will be a standard graduate course in real analysis.

Text: Topics in Optimal Transport by Cedric Villani, Published by the AMS as Volume 58 of Graduate Studies in Mathematics, 2003

ISBN-10: 0-8218-3312-X
ISBN-13: 978-0-8218-3312-4

(Available on the AMS website, $47 to members, $59 to non members.)


640:518    Partial Diff Equations    Z.C. Han    

Partial Differential Equations

This course will be a continuation of the PDE course, 517, offered in the fall semester. We will take a similar approach, i.e., we will emphasize the methods than the most general forms of the results. One main theme will be how to extend the results and methods for dealing with constant coefficients prototype equations to related variable coefficients and nonlinear equations. We will continue to use the text being used in 517, ``Partial Differential Equations" by Lawrence C. Evans. I am planning to focus my discussion around chapters 5--8, perhaps supplementing with some material not contained in the text. Students' input may affect my emphasis and order of presentation. Students should be prepared to do some reading and working of certain sections on their own.


640:519    Sel Topics Diff Equations    S. Chanillo    

The course will focus on developing the proof of Aubin-Yau for the existence of a Kahler-Einstein metric on Kahler manifolds with non-negative Chern class. Topics to be covered:

  1. Definition of Kahler manifolds.
  2. Basic properties of Kahler manifolds
  3. Ricci curvature
  4. The complex Monge-Ampere eqn.
  5. Solution of the Monge-Ampere eqn. by the continuity method.
  6. The case of positive Chern classes.
  7. Obstructions to the existence of Kahler-Einstein metrics, the Futaki invariant.
  8. Harnack inequalities.
  9. Existence of Kahler-Einstein metrics in the presence of symmetry.

7-9 are the topic of intense activity nowadays to establish the existence of Kahler- Einstein metrics when the Chern class is positive. This is not possible in general due to obstructions like the Kazdan-Warner obstruction in the Nirenberg Gauss curvature problem.

Pre-requisites: Knowledge of basic measure theory and complex analysis is required. It will be helpful if there is familiarity with Sobolev spaces but it is not essential.

There is no textbook for the course and we will generate enough notes culling the literature from many sources. However a good book is: Lectures on Hermitian-Einstein Metrics and Kahler-Einstein metrics, by Yum-Tong Siu, DMV seminar notes vol. 8. The book is out of print but a copy is in the library and will be placed on reserve if the course runs.


640:547    Topology of Manifolds    F. Luo    

topics on 3-manifolds and topological quantum field theory
This course focuses on a relationship between two-dimensional conformal field theory and three-dimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3-manifolds derived from Chern-Simons gauge theory.

We plan to cover the following topics.

  1. basic 3-manifold topology and geometry
  2. Jone¡¯s work on quantum invariants of knots
  3. Turaev-Viro¡¯s work on topological quantum invariants of 3-manifolds
  4. brief introduction to classical and quantum field theory, and topological quantum field theory
  5. introduction to conformal field theory
  6. Teichmuller theory and its quantization
  7. Witten¡¯s work on quantization of Chern-Simon theory
  8. relationships between conformal field theory and 2+1 topological quantum field theory.

The students should know some basic algebraic topology and differential geometry. We plan to make the lectures self-contained.

If you are interested in the course, please contact the instructor at fluo@math.

References Graeme Segal, The definition of conformal field theory, Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, edited by U. Tillmann, Cambridge University Press 2004, p.421-577.

Toshitaki Kohno, Conformal Field Theory and Topology American Mathematical Society (March 1, 2002)


640:548    Intro Geometry & Topology    C. Woodward    

This course aims at introducing differential geometry. We plan to cover the following material: smooth manifolds, Riemannian manifolds, de-Rham cohomology and Harmonic forms, connections and covariant derivatives, geodesics and Jacobi fields, and comparison theorem. If time permits, we plan to cover Morse theory and Floer homology. Some introductory topology may be covered depending on the audience.

All students thinking of taking the course should get in touch with me as soon as possible at ctw@math.rutgers.edu


640:549    Lie Groups    R. Goodman    

This course will be an introduction to Lie groups and algebraic groups. The prerequisites are real analysis, linear algebra, and elementary topology. Students currently taking Prof. Buch's course 640:556 in representation theory and interested in learning more Lie theory will find this course provides a natural continuation and alternate points of view. However, no prior knowledge of Lie algebras, Lie groups, or representation theory will be assumed.

Tentative Outline:

  1. The classical linear groups (compact and complex forms).
  2. Closed subgroups of GL(n) as Lie groups.
  3. Linear algebraic groups and rational representations.
  4. Structure of complex classical groups: maximal torus, roots, adjoint representation.
  5. Semisimple Lie algebras: structure and classification.
  6. Highest weight theory for representations of semisimple Lie groups; reductivity of classical groups.

Depending on the interests of the class and the time available, additional topics from Lie groups and representation theory will be covered.

Text: Selected chapters from Roe Goodman and Nolan R. Wallach, "Representations and Invariants of the Classical Groups" (newly revised 2nd edition).

The chapters covered in the course can be downloaded from the course web page.


640:552    Abstract Algebra II    R. Wilson    

This is the continuation of Math 551. It will cover many fundamental algebraic structures. As in Math 551, we will consider many examples.

Modules: Artinian and Noetherian modules, tensor products of modules, bimodules, projective and injective modules (This is some of sections 3.1 . 3.8, 3.10, 3.11 of BA.2.)

Homological algebra: Complexes, resolutions, derived functors, Ext and Tor (This is some of sections 6.1 . 6.8 of BA-2.)

Ideal theory of commutative rings: Prime ideals, localization, Noetherian rings, Hilbert Nullstellensatz, primary decompositions ( This is some of sections 7.1 . 7.12 of BA-2.) Field theory: Algebraic and transcendental extensions, separability and normality, Galois theory (This is some of chapter 4 of BA-I and of sections 8.1 . 8.9, 8.12 of BA-2, but I will probably follow different developments of some of the topics.)

Representation theory of finite groups: Maschke's Theorem, characters

Prerequisite: 640:551 or equivalent

Text: Jacobson, Basic Algebra, Volumes 1 and 2, second edition (abbreviated as BA-I and BA-II below)

Note that these volumes are out of print, but (as in the fall semester) copies are available through the department.Topics:


640:555:01    Topics in Algebra    V. Retakh    

Modern trends in noncommutative algebra: Quasideterminants, noncommutative localizations and their application

In the last few years there were several attempts to develop a noncommutative version of algebraic geometry. The noncommutative case is much harder than the commutative one and we have to rethink our approach to basic results and also to introduce new tools and methods.

It is a common understanding that noncommutative localizations should play an important role in the new theory. We will study the theory of noncommutative localizations developed by P.M. Cohn, quasideterminants by I. Gelfand and V. Retakh as natural tools in noncommutative algebra and geometry and their relations to a theory of noncommutative symmetric functions and formal power series.

We will also discuss some application to Theoretical Computer Science, Noncommutative Birational Geometry, and Noncommutative Integrable Systems.

The course will be based on ``Lecture notes on quasideterminants and noncommutative localizations" by V. Retakh and R. Wilson. The notes will be distributed among participants.

Prerequisite: Math 551


640:555:02    Topics in Algebra     Y.Z Huang    

Conformal field theories and Geometric Langlands correspondence

This will be an introduction to the conformal-field-theoretic method to the geometric Langlands correspondence. The classical Langlands correspondence is on a deep connection between number theory and representation theory. If one replaces the number fields in the classical Langlands correspondence by the fields of functions on complex algebraic curves, one can formulate a geometric version of the classical Langlands correspondence. By reformulating the notion of vertex operator algebra in the framework of algebraic geometry, Beilinson and Drinfeld gave a new approach to the geometric Langlands correspondence using the methods developed in the study of conformal field theories.

I will start with the notion of vertex operator algebra and and the equivalent notion of chiral algebra of Beilinson-Drinfeld. Examples, especially those constructed from representations of affine Lie algebras, of these algebras and their representations will be discussed. Then I will discuss this approach of Beilinson and Drinfeld based on the lecture notes

Text: "Lectures on the Langlands Program and Conformal Field Theory, hep-th/0512172" by Edward Frenkel and some other papers.

Prerequisites: Basic algebra and complex analysis as covered in 640:551 and 640:503 or equivalent. Knowledge in the representation theory of Lie algebras is very helpful but is not required.


640:560     Homological Algebra    C. Weibel    

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.

Prerequisite: First-year graduate knowledge of groups and modules.
Textbook: An introduction to homological algebra, by C. Weibel, Cambridge U. Press, paperback edition (1995).


640:561     Intro to Math Logic    A. DeLoro    

This is an introductory course in Mathematical Logic aimed at graduate students in mathematics rather than prospective logicians.

In Set Theory, we will discuss basic topics such as ordinals, cardinals and the various equivalents of the Axiom of Choice; as well as more advanced topics such as the club filter and stationary sets. (The latter are needed to prove results such as the existence of $2^{\alelph_{1}}$ nonisomorphic dense linear orders of cardinality $\aleph_{1}$.)

In Model Theory, we will begin with basic results such as the Completeness and Compactness Theorems. Then we will cover some more advanced topics focussing on the problem of computing the number of nonisomorphic models of a complete theory in a countable language.

The course has no formal prerequisites, although a familiarity with some basic naive set theory would be helpful.

Text:
Basic Set Theory (Levy)
Model Theory: An Introduction (Marker)
Model Theory (Chang & Keisler)


640:573    Spec Topics in Number Theory     H. Iwaniec    

Sieve Methods

This will be a one semester course on sieve methods in number theory. Sieve methods were created ninety years ago with expectation to treat problems concerning the distribution of prime numbers in basic arithmetical sequences. This goal was not achieved until recently when some intrinsic barriers were crashed (the so called parity problem).

In this course I am going to present the modern state of the theory. Among many applications I shall address questions about representations of primes by polynomials.

The main topics are:

Brun.s sieve
Selberg.s sieve
Bombieri.s sieve
The linear sieve
Sublinear sieves
Asymptotic sieve for primes
Primes represented by polynomials
The least prime in an arithmetic progression
Primes in short intervals

The course is aimed for any graduate student who likes prime numbers. No advanced knowledge of analytic number theory is required with only a few exceptions when in particular applications we shall borrow some facts from literature without proofs. Essentially there is no need for a textbook, because I shall be distributing my notes before lectures to all students.


640:640    Experimental Mathematics    D. Zeilberger

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 we will explore Automated (symbolic!) Enumeration, that consists of teaching the computer how to find explicit formulas, and/or general algorithms, for enumerating combinatorial objects. But the actual content is not that important, it is mastering the methodology of computer-generated and computer-assisted research that is so crucial for your future.

There are no prerequisites, and no previous programming knowledge is assumed. Also, very little overlap with previous years. The final projects for this class may lead to journal publications.

Text: Handouts


642:528    Methods of Appl Math II    J. Song    

METHODS OF APPLIED MATHEMATICS
This is a second semester graduate course appropriate for students in mechanical and aerospace engineering, biomedical engineering, other engineering programs and physics. The topics to be covered will be, primarily, from complex variable theory, but there will also be some discussion of the calculus of variation. The topics from complex variable theory will include: the differential and integral calculus of functions of a complex variable, conformal mapping, Taylor series, Laurent series and the residue theorem. There is a minimum of theoretical mathematics in the course, the emphasis is on applications and calculations which graduate students in engineering may encounter in their courses.
Text: Advanced Engineering Mathematics (second edition) by M. Greenberg


642:564    Statistical Mechanics II    J. Lebowitz    

Statistical Mechanics II
Description to appear.


642:574    Numerical Analysis    Y-J. Lee    

Numerical Analysis

This course is a general survey of some the basic topics in numerical analysis -- the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications. Although it is the second semester of the general survey Math 573,574, its topics are mostly independent of those covered in Math 573 and thus can be taken by students who have not taken Math 573.

In Math 642:574, we will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations.

In Math 642:573, we considered the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations, and the relationship of all these problems.

Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.

To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, however, the use of Matlab will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.

PREREQUISITES: Advanced calculus, linear algebra, and familiarity with differential equations.

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