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

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640:502    Theor Func Real Vars    R. Goodman    HLL 423   MW 6; 5:00-6:20

This course is a continuation of 640:501. It will cover coure topics in real and functional analysis, including locally compact topological spaces, normed spaces and bounded linear transformations, Lp spaces, Fourier analysis, and integration on locally compact spaces.

The textbook is
G. B. Folland: Real Analysis (2nd edition)

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640:504    Theory Func Comp Vars     H. Sussmann

CANCELED    NOTE: This will be a continuation of the Mathematics 503 course taught by Prof. S. Chanillo in the Fall of 2006.

Some of the topics listed below, especially the first four items, overlap with Prof. Chanillo's syllabus. If any those of topics are covered by Prof. Chanillo, then we will only review them briefly before we move on to the rest of the material.

1. Weierstrass products. Functions of finite order.
2. Meromorphic functions, the Mittag-Lefler theorem.
3. The Phragmen-Lindelof principle.
4. The D-bar operator.
5. Schwarz-Christoffel maps.
6. The Gamma function.
7. The Riemann Zeta function.
8. The prime number theorem.
9. Elliptic functions. Theta functions.
10. Picard's theorem.
11. Introduction to Riemann surfaces.

Text: Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton Lectures in Analysis II

Prerequisite: A first graduate course on complex variables, such as our Mathematics 503.

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640:508    Functional Analysis II     R. Nussbaum    HLL 423   TTh 5; 3:20-4:40
(Note: This course was incorrectly listed previously as 640:507.)

Theory of Positive Operators

There is a beautiful classical theory, sometimes called Perron-Frobenius theory, concerning eigenvalues and eigenvectors of nxn matrices with nonnegative entries. Perron-Frobenius theory has inspired a large literature concerning "positive operators", i.e., maps f (linear or nonlinear) which take a closed cone K in a Banach space X to itself. The Krein-Rutman theorem, which treats compact linear maps L:X-->X which maps a closed, total cone K into itself, is one example. These results have had many applications in analysis. More recently, several authors have obtained surprisingly detailed results concerning the behaviour of iterates of nonlinear maps which map a finite dimensional cone into itself.

This course will be self-contained introduction to the field and to a variety of open questions. The reader should know some very basic functional analysis; all else will be provided. Topics to be covered include:

(1) A quick summary of classical Perron-Frobenius theory.
(2) Cones in Banach spaces. Normal cones, reproducing cones, and some basic theorems concerning them.
(3) Uses of fixed point theory and degree theory in studying positive operators.
(4) The Krein-Rutman theorem and its generalizations.
(5) Hilbert's projective metric, Thompson's metric and their application to the study of positive operators. The Birkhoff-Hopf theorem.
(6) Nonlinear Perron-Frobenius theory in finite dimensions, e.g., the theory of nonlinear analogues of stochastic matrices.

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

General Relativity and Geometro dynamics

This is an introductory course on Einstein's theory of general relativity and gravitation, emphasizing the geometric-PDEs point of view of the equations and therecent advances in the field. Although welcomed and very much appreciated, no previous knowledge of physics or of partial differential equations is assumed. Some knowledge of differential geometry and special relativity is helpful.

This course is designed to be taken concurrently with Professor Kiessling's Quantum Gravity course 642:662, which both relies on and complements the material covered here.

The following is an outline of the course:

I. The Geometry of Space-time: Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit, The Penrose singularity theorem. Black holes and cosmic censorship conjectures. Homogeneous and isotropic solutions.

II. The Cauchy problem for Einstein Vacuum Equations (EVE): The symbol and the characteristics of EVE. The local existence theorem in wave coordinates. Local existence using maximal hypersurfaces.

III. Conservation Laws and Noether's Theorem: Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps, and for sections of vector bundles. Asymptotic flatness. The definition of global energy, momentum and angular momentum. The Positive Energy Theorem.

IV. Reduction under Symmetry: Kerr and Newman solutions. Wave maps and cosmological solutions of Kasner and Gawdy. The B-K-L conjectures. Spherically symmetric solutions: Schwarzschild, Reissner-Nordstrom, Tolman-Oppenheimer-Volkov.

V. Advanced Topics (one or two will be discussed, as time permits): Non-globally hyperbolic space-times and quantum probing of singularities. Decay Estimates for scalar fields. Quasi-local Mass and the Penrose Inequality. Christodoulou's Two-phase model for a self-gravitating star. Spherically symmetric Einstein-massless scalar field equations: Formation of Black Holes and the Instability of Naked Singularities. Cauchy Horizons and their stability/instability. Einstein-Vlasov and Einstein-Complex scalar field equations: global existence results. Nonlinear Stability of the Minkowski Space-time: Strategy of the proof.

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640:519    Sel Topics Diff Equations    Y. Li     HLL 425   MW 2; 10:20-11:40

The course will consist of two parts.

In the first part, I will present some standard material on linear parabolic equations of second order. This will take one half to two thirds of the time.
In the second part I will outline some important results on the Ricci flow, some with full details but most without. I will only discuss results before the work of Perelman.

Part One consists of the following material for second order linear parabolic equations: L2-theory (Energy method, Rothe method, Galerkin method), Harnack inequality, Schauder theory.

Part Two consists of some of the following material for Ricci flow: Hamilton's classical result on three dimensional manifolds with positive Ricci curvature, the local derivative estimates for the Ricci flow on complete noncompact manifolds, the convergence on surfaces, pinching toward nonnegative curvature in dimension three,the Harnack estimate, and blow-up limits of singularities.

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640:524    Functions Several Complex Vars    X. Huang     HLL 124   Th 2&3; 10:20-1:20

The first half of the course covers the basic materials from Complex Geometry. We will start with the concepts of complex manifolds, holomorphic vector bundles, Hermitian metrics, Kahler metrics, Chern classes, etc.

We then proceed to study the harmonic theory on complex manifolds, Kodaira's vanishing theorem, Serre duality, Kodaira's embedding theorem, and Hormander's $L2$ method.

The second half of the course will be on more recent studies in Kahler Geometry and Algebraic Geometry.

Two topics to be covered include:

(1). Yau's solution to the Calabi conjecture on the existence of the Kahler-Einstein metric on a compact algebraic manifold with its first Chern class negative;

(2) Siu's solution to the invariance of plurigenera conjecture for a family of algebraic manifolds

1. James Morrow and K. Kodaira, {\it Complex Manifolds}, Rinehart and Winston, 1971.

2. J. P. Demailly, $L2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes in Mathematics on Transcendental Methods in Algebraic Geometry, Springer 1646, pp 1-98.

3. Y. T. Siu, Multiplier ideal sheaves in complex and algebraic geometry, www.arxiv.org (math.AG/0504259), 2005.

4. Y. T. Siu, A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring, www.arxiv.org (math.AG/0610740), October, 2006. (68 pages)

Prerequisites: basic materials from real analysis, complex analysis and the theory of differential manifolds

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640:532    Intro Differential Geometry     F. Luo     HLL 124   MW 2; 10:20-11:40

Textbook: Riemannian Geometry and Geometric Analysis by Jurgen Jost.

The book can be brought here.

This course aims at introducing differential geometry to graduate students. 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.

The students should know basic topology and multi-variable calculus.

If you have questions on the course, please contact me at: fluo@math.

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640:534    Sel Topics in Geometry    C. Woodward     HLL 423    M 3; 12:00-1:20; HLL 705    T 2; 10:20-11:40

Symplectic geometry and pseudoholomorphic curves

This course will cover

(i) some basic symplectic geometry;
(ii) moduli spaces of pseudoholomorphic curves and applications to quantum cohomology and Lagrangian Floer cohomology;
(iii) applications to dynamical systems, such as the Arnold conjecture on number of periodic trajectories of Hamiltonian systems;
(iv) symplectic aspects of homological mirror symmetry;

depending on the interests of the audience. Please e-mail me if you are considering taking the course at ctw@math.rutgers.edu.

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640:536     Algebraic Geometry II    A. Buch     HLL 124     MW 6; 5:00-6:20

Introduction to Algebraic Geometry

This course continues the study of algebraic geometry from the fall by replacing algebraic varieties with the more general theory of schemes, which makes it possible to assign geometric meaning to an arbitrary commutative ring. One major advantage of schemes is the availability of a well-behaved fiber product. Combined with Grothendieck's philosophy that properties of schemes should be expressed as properties of morphisms between schemes, fiber products make the theory very flexible. The goal of the course is to cover the basic definitions and properties of schemes and morphisms, and to introduce and study the cohomology of sheaves, which provides a powerful tool for settling geometric questions. For example, one can use cohomological methods to give a simple proof of the classical Riemann-Roch theorem for curves.

Prerequisites: Math 535. Familiarity with commutative algebra is an advantage, but is not required.

Text: Hartshorne, Algebraic Geometry (Springer GTM 52).

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640:541     Intro Alg Topology II    S. Ferry     HLL 525    T3; 12:00-1:20; TH5 3:20-4:40

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".

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640:549    Lie Groups    S. Gindikin     HLL 423    MW 4; 1:40-3:00

This course is an elementary introduction to the theory of Lie groups and it addresses for the student from different areas of mathematics. Lie groups are an essential part of general mathematical knowledge and they apply in geometry, multidimensional complex analysis, differential equations, mathematical physics etc. The course does not use any special knowledge outside of calculus and linear algebra. An important part of the course is the consideration of classical Lie groups (linear, orthogonal, simplectic) and classical geometries.

We will use the textbook of W.Rossman, Lie Groups. An introduction through linear groups, Oxford University Press, 2002.

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640:552     Abstract Algebra II    V. Retakh     HLL 425   TF 2; 10:20-11:40

Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. Representative topics will be:

Galois Theory: Finite algebraic extensions, resolutions of equations by radicals (and without radicals)

Rings of Polynomials: Noetherian rings, Hilbert basis theorem, Noether normalization, Nullstellensatz

Basic Module Theory: Projective and injective modules, resolutions, baby homological algebra, Hilbert syzygy theorem

Representation Theory of Finite Groups: Maschke and Characters

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.

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640:555    Topics in Algebra    J. Lepowsky     HLL 425   MW5; 3:20-4:40

Vertex Operator Algebra Theory

This course will develop the axiomatic theory of vertex operator algebras from a contemporary point of view. Important examples and applications will be discussed.

Students in Yi-Zhi Huang's course this semester will be prepared for this course, which will be a continuation. Other students who might be interested in this course are welcome to consult me.

Please note: The Quantum Mathematics Seminar, which will meet on some Fridays at 1:00, will often be related to the subjects of the course. Students planning to take the course should also try to arrange to attend the seminar, although the seminar will not be required for the course.

Text: J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkh\"auser, Boston, 2003

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640:559    Commutative Algebra     D. Maclagan     HLL 525   MW 2; 10:20-11:40

Text: Eisenbud Commutative Algebra with a View Toward Algebraic Geometry, Springer, GTM 150, 1995.

Commutative algebra is the engine behind algebraic geometry and algebraic number theory. In addition, problems from other fields such as combinatorics or optimization can sometimes be phrased as commutative algebra problems.

This course will be an introduction to the basics of commutative algebra, including localization, primary decomposition, integrality, flatness, and dimension.

We will roughly follows Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. Computational aspects and examples relevant to algebraic geometry will be emphasized, but the only prerequisite is 551/552 or equivalent.

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640:569    Selected Topics Logic    G. Cherlin    HLL 423    MTH 2; 10:20-11:40

Tame Topology in a Logical Setting

Logicians refer to algebraic geometry as "the theory of algebraically closed fields" and refer correspondingly to real algebraic geometry (semi-algebraic geomety) as "the theory of real closed fields". Some generalizations of real algebraic geometry in the direction of analysis (exponential varieties, globally subanalytic functions, Khovanski's "Pfaffian functions") - fall under Grothendieck's "tame topology", or the logician's "o-minimal theories".

The course will deal with o-minimal theories.

As suggested by our introduction, these are geometrical theories in which all objects studied can be shown to be geometrically "tame". There are two aspects to the theory:

(1) the geometry of sets definable in o-minimal theories (ref: van den Dries, "Tame Topology") and
(2) a body of theorems showing that sets definable using analytically interesting functions (notably the exponential) fall into this framework.

We aim to treat one of the latter results, showing that the so-called "Pfaffian closure" of the real field has an o-minimal theory. This includes the deep result (Wilkie) that the theory of exponential varieties is an o-minimal theory.

As time allows, we may treat the following topics from the general theory (here the real field may be replaced by other real closed fields):

- Miller's dichotomy: if there is a definable superpolynomial function, then exp is definable;
- Structure of groups definable in o-minimal theories (simple or definably compact); - complex analysis in o-minimal structures.

There are no prerequisites from logic. Some terminology from logic will be convenient (mainly to give the term "definable" a precise and useful meaning) but can be introduced as needed.

This particular material is not covered in van den Dries' book Tame topology, but on the other hand that book gives a thorough account of the fundamental results of the general theory, and may be useful as a general reference.

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640:573    Spec Topics in Number Theory     H. Iwaniec     HLL 124   TF 3; 12:00-1:20

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.

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640:574    Topics Number Theory     S. Miller    

CANCELED    This course is in some sense a continuation of Iwaniec's fall semester course on the Spectral Theory of Automorphic Forms, but does not require it or any number theory course as a prerequisites.

Preferably, students should have passed their written qualifying exams. The course treats automorphic forms from the beginning using the representation-theoretic point of view introduced by Gelfand. [Iwaniec's courses and the current seminar on Goldfeld's book will be helpful in that they provide many examples of automorphic forms, but are not necessary for the course.

The main goal is to discuss the SL(2) theory and generalize it to other Lie groups such as GL(n), and also to become acquainted with modern formulations over adele groups.

Topics to be covered include:

Representation Theory of SL(2,R)

Spectral decomposition of L2(Gamma\G)

Maass raising and lowering operators

L-functions and Gamma factor computations

Piatetski-Shapiro's Fourier expansion on GL(n)

Adeles, Ideles, Tate's thesis

Jacquet-Langlands Theory

Automorphic Distributions

Text: Automorphic Forms and Representations, Dan Bump, Cambridge University Press.

Prerequisites: none.

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640:640    Experimental Mathematics    D. Zeilberger    ARC 118     MTH3; 12:00-1:20;

* Dr. Zeilberger's Office: Hill Center 704 (Phone: (732) 445-1326)
* E-mail: zeilberg at math dot rutgers dot edu
* Classroom: ARC 119 (IML room inside computer lab)

TEXT: A=B by M. Petkovsek, H. Wilf, and D. Zeilberger (a free download) and handouts.

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 Wilf-Zeilberger theory and the so-called Holonomic ansatz.

There are no prerequisites, and no previous programming knowledge is assumed. The final projects for this class may lead to journal publications.

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640:651    Category Theory     R. Bumby    

CANCELED   

Category theory was developed to exploit the idea that a mathematical subject tends to be organized around the properties of the structure-preserving mappings connecting the objects of the subject. This allowed a unified approach to common constructions within a subject and a formal study of connections between different subjects.

This will be an introductory course devoted to the "ideas and methods which can now be used by Mathematicians working in a variety of other fields of Mathematical research", as stated in the Preface to the textbook.

Textbook: Saunders Mac Lane, "Categories for the Working Mathematician", Springer-Verlag

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642:528    Methods of Appl Math II    G. Speer     HLL 423   TTh 6; 5:00-6:20

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

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642:561    Intro Mathematical Physics     A. Soffer     HLL 423   TTh 4; 1:40-3:00

This course is an introduction to basic quantum mechanics and its mathematical analysis.
Quantum mechanics was first developed when experiments indicated that particles behave as waves and waves behave well ... as particles.
The resulting theory is fundamental to our understanding and description of the physical reality. Quantum theory had profound implications to virtually all sciences, basic and applied; it opened new directions for research in many mathematical fields, from algebra to analysis. It poses a challenge to our understanding of basic notions like information, randomness, computation and recently led to the new field of quantum computation encryption and teleportation.

Topics include: The physical basis of Q.M., basic postulates, Hilbert spaces and linear operators, square well potentials, point and continuous spectrum, hydrogen atom, harmonic oscillator, path integrals, gauge invariance, self-adjointness, symmetries, 1 qubit computer, 2 qubit systems, Approximation methods: bound states, scattering states.

Prerequisites: Real analysis, Linear algebra
Books:
Quantum Mechanics I - A. Galindo, P. Pascual
Functional Analysis - Reed Simon I (recommended)
Hilbert space operators in Q. physics - Blank, Exner, Havlicek (recommended)
Quantum Mechanics - Schwabl (recommended)

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642:574    Numerical Analysis    R. Falk     HLL 425   TTh 6; 5:00-6:20

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.