Descriptions of fall 2002 courses in the Rutgers-New Brunswick Math Graduate Program

Descriptions of proposed fall 2002 courses in the Rutgers-New Brunswick Math Graduate Program

640:501 Theor Func Real Vari R. Wheeden HLL 525 MW 5; 2:50-4:10

The course will cover these topics from the book Measure and Integral by Wheeden and Zygmund:
Functions of bounded variation, Riemann-Stieltjes integrals, Lebesgue outermeasure and measure in n-dimensional Euclidean space, Lebesgue measurable functions including theorems of Lusin and Egorov, the Lebesgue integral in n-dimensional Euclidean space and its relation to Riemann and Riemann-Stieltjes integrals, Fubini and Tonelli theorems on repeated integration, Lebesgue's differentiation theorem, Vitali's lemma, differentiation of monotone functions on the line, absolutely continuous and singular functions on the line, and as much L^p and L² space theory as time permits, all in Euclidean space.

640:503 Theor Func Comp Vari O. Costin HLL 423 TTh 5; 2:50-4:10

The course covers: elementary properties of complex numbers, analytic functions, the Cauchy-Riemann equations, power series, Cauchy's Theorem, zeros and singularities of analytic functions, maximum modulus principle, conformal mapping, Schwarz's lemma, the residue theorem, Schwarz's reflection principle, the argument principle, Rouché's theorem, normal families, the Riemann mapping theorem, properties of meromorphic functions, the Phragmen-Lindelof principle and elementary properties of harmonic functions.
Text Serge Lang, Complex Analysis, 4th edition.
Prerequisite Advanced calculus.
Approximate syllabus:

The algebra of complex numbers and complex valued functions.
Elementary topology of the plane.
Complex differentiability; the Cauchy-Riemann equations; angles under holomorphic maps.
Power series, operations with power series.
Convergence criteria, radius of convergence, Abel's theorem.
Inverse mapping theorem, open mapping theorem, local maximum modulus principle.
Holomorphic functions on connected sets. Elementary analytic continuation.
Integrals over paths.
Primitive of a holomorphic function. The Cauchy-Goursat theorem.
Integrals along continuous curves, homotopy form of Cauchy's theorem.
Global primitives, definition of the logarithm.
Local Cauchy formula, Liouville's theorem, Cauchy estimates, Morera's theorem.
Winding number, global Cauchy theorem.
Uniform limits, isolated singularities.
Laurent series.
The residue formula.
Evaluation of definite integrals using the residue theorem.
More calculations with the residue theorem.
Conformal mapping, Schwarz lemma, automorphisms of the disc and upper half plane.
Other examples of conformal mappings. Level sets.
Fractional linear transformations.
Harmonic functions.
More properties of harmonic functions, the Poisson formula.
Normal families, formulation of the Riemann Mapping Theorem.
Weierstrass products. Functions of finite order. Minimum modulus principle.
Meromorphic functions, the Mittag-Lefler theorem
The Phragmen-Lindelof principle.
The D-bar operator.

640:507 Functional Analysis A. Soffer HLL 423 MW 6; 4:30-5:50

We begin with 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 and fixed point theorems.finally, the theory of bounded linear operators in Hilbert spaces.
Text Mostly Reed & Simon, volume I, Functional analysis
Prerequisites Real analysis, including basic measure theory.

640:509 Sel Topics in Analysis F. Treves HLL 425 TTh 5; 2:50-4:10

Algebraic Aspects of Completely Integrable Systems The course will introduce various completely integrable systems (main ones: the KdV, i.e., Korteveg-De Vries, Equations and the NLS, i.e., Nonlinear Schroedinger, equations) from the viewpoint of Differential ALgebra. The course will show how the hierarchies arise and how they can be viewed - as systems of generators of maximal commutative subalgebras of infinite dimensional (pseudo)nilpotent Lie algebras.
Prerequisites No prior knowledge of the equations will be assumed and there will be a minimum amount of Analysis (no solutions of the differential equations will be studied). No knowledge of Lie algebras (nilpotent or otherwise) will be assumed - beyond the Jacobi identity. Actually no deep algebraic results in Algebra are needed: there will be no Kac-Moody theory.
Everything will be done from scratch. However, many of the proofs will be omitted in the course and only provided in accompanying notes - sionce they are computational and basically uninteresting. Emphasis will be put on the conceptual aspects of the structures.

640:519 Sel Topics in Diff Equ S. Chanillo HLL 423 TTh 2; 9:50-11:10

Heat Equation Methods in Geometry Over the last 20 years or so the heat equation has been used a tool in the study of geometric problems. Examples of this are the Eels-Sampson heat flow for existence of harmonic maps, Hamilton's study of Ricci flow are the major works in this field. The course I will offer will study as a goal the theorem of Huisken from 1984 that a convex, closed hypersurface when deformed with a speed equal to its mean curvature shrinks in finite time to a hypersurface which is on re-scaling a round sphere. I will develop the evolution equations for various geometric quantities with arbitrary speed, not necessarily the speed given by mean curvature. Thus when a surface evolves one needs to know how the second fundamental form evolves, how the mean curvature evolves etc and then to these non-linear parabolic equations one applies the maximum principle or Nash-Moser techniques. Thus under mean curvature flow and the maximum principle one sees convexity is preserved. Another important question in this area is the availability of suitable Harnack inequalities, these are known to be available for evolving convex hypersurfaces by using ideas of Peter Li and Yau but in general this problem is open. I will also discuss flow by other speeds, by Gauss curvature where methods relating to Minkowski's problem are used and also flow by inverse mean curvature which has a direct application to the Hawking Mass for black holes.

640:521 Harmonic Analysis Y. Li HLL 423 MW 2; 9:50-11:10

Harmonic analysis has played an important role in partial differential equations, mathematical physics, probability, etc. We plan to cover the following material.

Fourier transform (L² theory and Plancherel theorem, Paley-Wiener theorem).
Maximal function, covering lemma, Lebesgue set, interpolation theorem.
Calderón-Zygmund decomposition, singular integrals.
Riesz transforms, Poisson integrals, spherical harmonics.
Littlewood-Paley theory and multipliers.
Riesz potentials, Sobolev spaces, Sobolev embedding theorems.
BMO and John-Nirenberg estimates.
Preliminaries of wavelets (if time permits).

For at least the first half of the course, we plan to follow closely the first four chapters of E. Stein, Singular integrals and differentiability properties of functions, Princeton, N.J., Princeton University Press, 1970. Some of the material will be taken from E. Stein, Harmonic analysis : real-variable methods, orthogonality, and oscillatory integrals, Princeton, N.J., Princeton University Press, 1993.

640:532 Differential Geometry X. Rong HLL 124 TTh 2; 9:50-11:10

Introduction to Riemannian Geometry This is an introductory course to the Riemannian geometry. However, we will start with the definition of a manifold and related notions (tangent space, differential forms, connection, etc). Hence, students who may not have the notion of a manifold could still take this course.
Then we will introduce the main subject: a Riemannian manifold and its structure. Topics that will be covered include:

Covariant derivatives of tensors.
Exponential maps and the Gauss lemma.
Geodesics and the completeness.
Jacobi fields and Cartan-Hadamard theorem.
The first and second variation formulae.
Injectivity radius and Klingenberg theorem.
Morse index and Bonnet-Myers theorem.
The comparison theory (Rauch, maybe Toponogov, Bishop-Gromov)
Morse index theorem and the connectedness principle of positive curvature.

This course is also a preparation for a succeeding course given by the instructor, Topics in Riemannian Geometry in the spring 2003 semester.

640:534 Sel Topics in Geometry P. Feehan HLL 525 TTh 3; 11:30-12:50

The goal of the course will be to discuss applications of differential geometry and non-linear PDEs to problems of interest in recent research in mathematical gauge theory, three and four-dimensional manifolds, the classification of smooth and symplectic manifolds, Gromov-Witten theory, Seiberg-Witten theory, and selected applications in mathematical physics. Sections 1-4 below will be covered as rapidly as we can or as needed, while section 5 will be the core of the course.
Depending on attendees, there may be a small amount of overlap between my Math 532 (Spring 2002) Differential Geometry course and the beginning of Math 534.

Lie groups (introduction): Lie groups and Lie algebras
Connections on fiber bundles (core topics): Vector bundles, Principal bundles, Fiber bundles, Connections on principal bundles, Connections and metrics on vector bundles, Curvature, Spin and Spin^c manifolds, Dirac equation and spinors
Symplectic geometry (introduction to selected topics): Complex manifolds, Complex differential geometry, Kaehler metrics, Symplectic manifolds
Cohomology and characteristic classes (selected topics): De Rham cohomology, Poincaré duality, Euler and Thom classes, Characteristic classes
Applications (introduction to selected topics): Seiberg-Witten equations and Seiberg-Witten invariants, Anti-self-dual Yang-Mills equation and Donaldson invariants of 4-manifolds, Gromov-Witten invariants of symplectic manifolds, Floer homologies of 3-manifolds

Prerequisites This is a course which any mathematics graduate student (first-years included) should be able to take, but ought to contain something of interest to graduate students in all years. The level of analysis and algebra involved does not really go beyond what you would have seen in undergraduate courses. An exposure to the required first-year graduate courses (real and complex analysis, algebra) is more than adequate. If you have taken some of the optional graduate courses (such as Algebraic Topology, Algebraic Geometry, or Lie Groups), then that will help you appreciate how certain parts of the course fit in to the grand scheme of things, but it is not necessary to have taken them or be taking them concurrently. We will cover any topics from Topology or Lie Groups as needed.
To clear up any confusion, I will not assume that anyone has taken the course on "Differential Topology", which was advertised for Autumn 2001 but did not run. As this is a optional graduate course, there will be no exams or formal requirements; for those interested, I may suggest reading assignments from time to time.

Background references
Manifolds, tensor analysis, and applications, by R. Abraham, J. Marsden, and T. Ratiu
Nonlinear analysis on manifolds, by T. Aubin
Differential forms in algebraic topology, by R. Bott and L. Tu
Eigenvalues in Riemannian geometry, by I. Chavel
Geometry of four-manifolds, by S. Donaldson and P. Kronheimer
Modern geometry I, II, III, by B. Dubrovin, R. Fomenko, and S. Novikov
Seiberg-Witten theory, by D. Salamon (free pre-print book from author)
Principles of algebraic geometry, by P. Griffiths and J. Harris
Differential topology, by V. Guillemin and A. Pollack
Differential topology, by M. Hirsch
Dynamical systems, by M. Hirsch and S. Smale
Fiber bundles, by D. Husemoller
Differential geometry, by S. Kobayashi and K. Nomizu
Spin geometry, by B. Lawson and M-L. Michelsohn
Morse theory, by J. Milnor
Characteristic classes, by J. Milnor and J. Stasheff
Topology from a differential viewpoint, by J. Milnor
Notes on Seiberg-Witten theory, by J. Morgan
Foundations of global non-linear analysis, by R. Palais
A comprehensive introduction to differential geometry, by M. Spivak
Foundations of differentiable manifolds and Lie groups, F. Warner

640:535 Algebraic Geometry J. Tunnell HLL 525 MW 4; 1:10-2:30

This course will be an introduction to the study of algebraic varieties, that is the zero sets of polynomials in several variables. The subject of algebraic geometry has simultaneously the geometric flavor of surfaces, hypersurfaces, etc. and the algebraic structure of commutative algebra of rings of polynomial functions (just as linear algebra has a geometric and algebraic content). The emphasis of the course will be on examples of algebraic varieties and general attributes of varieties and morphisms as reflected in these examples. Algebraic varieties arise in many places in physics, topology, geometry, combinatorics and number theory and the examples studied in this course will be often be drawn from other areas of mathematics. I plan to concentrate on the geometrical aspects of the subject, where the classical beginnings lie, and to bring in the algebraic aspects as we accumulate examples. We will discuss morphisms, dimension, degree and smoothness of varieties. The course web site has more details.

640:551 Abstract Algebra R. Lyons HLL 425 WF 2; 9:50-11:10

Group theory: Sylow theorems, symmetric groups, group actions, free groups. Factorization in commutative rings. Categories. Modules over a principal ideal domain, applications to abelian groups and canonical forms of linear transformations.

640:560 Homological Algebra C. Weibel HLL 124 TTh 4; 1:10-2:30

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

640:566 Axiomatic Set Theory S. Thomas HLL 124 TTh 5; 2:50-4:10

Choice versus Determinacy: an introduction to classical descriptive set theory The Axiom of Choice (AC) implies the existence of various pathologies on the real line R such as a non-measurable set of reals, an uncountable set of reals with no perfect subset, a 2-coloring of the 2-subsets of R with no uncountable monochromatic subset, etc. In this course, we shall explore the question of whether such pathologies can arise within the "definable sets" of reals; namely, the Borel sets, the analytic sets, ... , etc. While these questions cannot be fully settled using the usual axioms of set theory ZFC, we shall find that they all have negative answers if we assume the Axiom of Projective Determinacy (PD): an extra set-theoretic axiom which posits the existence of winning strategies for a broad class of infinite 2-player games.
The course will cover the following topics:

The axiom of choice
We shall study AC and some of its pathological consequences.
Basic descriptive set theory
We shall study the hierarchies of the Borel sets and projective sets, and analyze the structure of the sets in each of these hierarchies.
Determinacy
We shall study infinite 2-player games played on the real line R. We shall see that the existence of winning strategies for suitably defined games implies the non-existence of set-theoretic pathologies within the projective hierarchy.

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

The sieve ideas were first used by Erathosthenes for the purpose of creating tables of prime numbers. This simple algorithm (inclusion-exclusion steps ) was further refined by Fibonacci, Legendre, Euler, Lebesgue and many others in the ninteenth century. However the real evolution began in 1916 from the works of Viggo Brun who created quite sophisticated combinatorial methods. The original goal was to prove the Goldbach conjecture which asserts that every even number larger than two is a sum of two primes. This remains still an open problem in spite of great effort by many prominent mathematicians (the one million dollar prize didn't help). A new impetus was given by Selberg in 1948 with very elegant and powerful techniques based on ideas from the Prime Number Theory. Sadly enough Selberg also revealed that no sieve method alone was capable to produce genuine prime numbers, but only almost primes. Nevertheless the sieve methods are still intensively studied, because of their importance for many other applications (including cryptography). Only recently new axioms were developed within which the prime numbers could be captured.
In this course I will present completely the theory of sieves up to date, and I will give fundamental applications. These tools and results will be useful in future courses on analytic number theory. Here are some of the topics to be presented in the Fall:

Eratosthenes sieve
Combinatorial methods
Selberg's upper-bound sieve
Weighted sieves
The linear sieve
Bilinear forms for the remainder
Bombieri's asymptotic sieve
The parity problem
The Large Sieve
Sieving primes
Mollification as sieve concepts.

Among numerous applications I shall show that certain polynomials in two variables represent primes infinitely often.
The course is addressed to anyone who loves numbers. No special knowledge of advanced mathematics will be required. Occasionally for applications we shall borrow hard products from analytic number theory. Complete notes will be distributed before lectures.

642:527 Methods of Appl Math G. Goldin HLL 705 MW 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 L² 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.

642:550 Linear Alg & Applications R. Goodman HLL 423 M 7; 6:10-7:30 & W 4; 1:10-2:30

An introductory course on vector spaces, linear transformations, determinants, and canonical forms for matrices (Row Echelon form, Schur triangular form and Jordan canonical form). Matrix factorization methods (LU and QR factorizations, Singular Value Decomposition) will be emphasized and applied to solve linear systems, find eigenvalues, and diagonalize quadratic forms. These methods will be developed in class and through homework assignments using MATLAB. Applications of linear algebra will include Least Squares Approximations, Discrete Fourier Transform, Differential Equations, Image Compression, and Data-base searching.
Grading: Written mid-term exam, five MATLAB projects, and a written final exam.
Prerequisites: Familiarity with matrices, vectors, and mathematical reasoning at the level of advanced undergraduate applied mathematics courses.
Note: This course is primarily intended for graduate students in science, engineering and statistics.

642:575 Num Solutions of PDE M. Vogelius HLL 525 TTh 5; 2:50-4:10

This course is intended as an introduction to the subject. The idea is to provide the basic theory (convergence, stability, ...) but at the same time give the participants some practical experience with the performance of different approaches (using Matlab, ...). The course will concentrate on linear problems of elliptic and parabolic character. It will provide a treatment of

Finite difference methods (discrete maximum principle, ...)
Variational methods (inf-sup condition, relation to approximation theory, ....)
Boundary integral methods (singular integrals, Nystrom's method, ....)

Some familiarity with partial differential equations and elementary numerical techniques is desirable. The precise level of the course may to some extent be adjusted to reflect the background of the participants. Depending on the interest of the participants I could devote special attention to one of following topics: a) the equations of elasticity, b) Stokes' problem, c) eigenvalue problems d) preconditioning (solution of the discrete problems) or e) aposteriori error estimation and adaptivity.

References
K. Atkinson, The numerical solution of intergal equations of the second kind, Cambridge Univ. Press, 1997
S. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer Verlag, 1994
P.G. Ciarlet, The finite element method for elliptic problems, North Holland, 1978
C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge Univ. Press, 1987
R. Kress, Linear integral equations, Springer Verlag, 1999
J.C. Strikwerda, Finite difference schemes and partial differential equations, Wadsworth & Brooks/Cole, 1989

642:582 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 Algebraic Methods in Combinatorics J. Kahn HLL 425 TF 3; 11:30-12:50

This course will survey applications of ideas from algebra (mostly linear) to problems in discrete mathematics and related areas. Areas of application include extremal problems for finite sets and the n-cube; theoretical computer science; discrete geometry; graph theory; probability; additive number theory and group theory; etc. Various open problems will be discussed.
Prerequisites: I will try to make the course self-contained except for basic combinatorics and linear algebra. A course in each of these would be helpful. See me if in doubt.
Text: Babai-Frankl, Linear Algebra Methods in Combinatorics. (This is actually only a manuscript. It's not mandatory: we won't really follow it, but will overlap it to some extent; on the other hand, it has lots of nice material and is relatively cheap.)

642:591 Topics in Probability and Ergodic Theory J. Beck HLL 425 MW 5; 2:50-4:10

We are going to discuss some classical results of Probability Theory like Strong Laws of Large Numbers, the Law of the Iterated Logarithm, Central Limit Theorems, Arc-Sine Law, and so on.
We motivate Ergodic Theory by some classical results of H. Weyl in Uniform Distribution . Then we prove general ergodic theorems (Neumann, Birkhoff, etc.), and show applications in the theory of continued fractions.
Prerequisites Nothing really.
Text We are not going to follow any particular book, instead we will pick some of the best parts from several books.

642:611:01 Sel Top Appl Math M. Kruskal To be given as a reading course. Please contact the instructor.

Arithmetic and Analysis of Surreal Numbers The surreal number system is a relatively new and fascinatingly rich creation by J. H. Conway. In a compellingly natural way, it simultaneously encompasses the usual real numbers, Cantor's ordinal numbers, and a slew of infinite and infinitesimal numbers of enormously varied sizes as well as all sorts of combinations thereof. (Though superficially similar to the hyperreal number systems of nonstandard analysis, it is actually quite different).
The numbers, as well as the operations and relations on them, are defined very simply and explicitly, and their elementary arithmetic properties have strikingly simple proofs with virtually no special cases, so that even restricted to the real numbers the treatment is a great improvement on the somewhat intricate classical development of the real numbers.
The course will be self-contained, since the approach adopted is considerably simpler than that available in the literature.
Prerequisites: None, except for a modicum of mathematical maturity and some familiarity with elementary set theory.

640:661:01 Topics Math Physics J. Lebowitz HLL 124 M 4; 1:10-2:30 & W 2; 9:50-11:10 & F 4; 1:10-2:30 (sometimes!)

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.

642:661:02 Topics Math Physics G. Gallavotti To be given as a reading course. Please contact the instructor.

General properties of Anosov systems
Existence of stable and unstable manifolds
Construction of stable unstable manifolds
Markov partitions and Gibbs states
Thermodynamic formalism

Based on

Chapter 5 of Foundations of fluid mechanics (by G. Gallavotti)
Chapters 5, 6 and 9 of the book Qualitative statistical aspects of ergodic theory by Gallavotti, Bonetto, and Gentile.

The first book is published by Springer Verlag. The second is a draft : BOTH can be (legally) downloaded from http://www.math.rutgers.edu/~giovanni at the books page.

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