# 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 L^{2} 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**

*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**

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 (L
^{2}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).

*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**

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.

*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**

**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.

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 L^{2} 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, ....)

**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 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.

**Complex systems: physical reality and mathematical models**

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

- 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.