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

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

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640:501    Theor Func Real Vari    T. Balaban    HLL 124    MW 6; 5:00-6:20

Basic real variable function theory, measure and integration theory pre-requisite to pure and applied analysis. Topics: Riemann and Lebesgue-Stieltjes integration; measure spaces, measurable functions and measure; Lebesgue measure and integration; convergence theorems for integrals; Lusin and Egorov theorems; product measures and Fubini-Tonelli theorem; Lp-spaces. Other topics and applications (such as Lebesgue's differentiation theorem, signed measures, absolute continuity and Radon-Nikodym theorem) as time permits.

Text: Wheeden and Zygmund, Measure and Integral
Pre-requisites: Undergraduate analysis at the level of Rudin's "Principles of Mathematical Analysis," chapters 1--9, including basic point set topology, metric space, continuity, convergence and uniform convergence of functions.

640:503    Theor Func Complex Variable    H. Sussmann    HLL 423    TTh 5; 2:50-4:10

Text Serge Lang, Complex Analysis, 4th edition.

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

640:509    Sel Topics in Analysis Nonlinear PDE's     A. Bahri    HLL 525    TTh 5; 3:20-4:40

640:510    Sel Topics in Analysis    M. Kruskal    T5 (3:20-4:40) in H705 & F2 (10:20-11:40) in H423

             THE 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 generalizes (encompasses) the usual real numbers,
Cantor's ordinal numbers (with their "natural" commutative
arithmetic), and a slew of infinite and infinitesimal numbers of
enormously varied sizes as well as all sorts of combinations of
them.  (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.

There is hope for significant and deep implications in standard
analysis and its applications to physics.  Some basic questions have
not yet been resolved and offer quick access to forefront research.

Prerequisite:  Merely some acquaintance with informal elementary
set theory

The course will be self-contained, since the approach adopted is
considerably simpler than that available in the literature.  However,
some relevant texts are:

Author              Title                     Publisher               Year

Donald Knuth     Surreal Numbers             Addison-Wesley Pub Co        1974
John H. Conway   On Numbers and Games        Academic Press               1976
Harry Gonshor    An Introduction to the      Cambridge University Press   1986
Theory of Surreal Numbers



640:517    Partial Diff Equations   Z. Han    HLL 425     MW 4; 1:40-3:00

This is the first half of a year-long introductory graduate course on PDE. PDE is an enormously vast field. PDEs arise from very diverse fields: from classical to modern physics, to more applied sciences such as material sciences, mathematical biology, and signal processing, etc, and from the more pure aspects of mathematics such as complex analysis and geometric analysis. This introductory course should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, complex analysis, differential geometry, and, of course, partial differential equations.

The theories developed for treating the most commonly encountered PDEs are very rich, and are hard to present in a compact, structured way, in contrast to some of the more neatly structured areas of analysis. For an introductory course, it is probably more important to examine some important examples to certain depth, to introduce the formulation, concepts, most useful methods and techniques through such examples, than to concentrating on presentation and proof of results in their most general form. The beginning weeks of the course aim to develop enough familarity and experience to the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation. Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing qualitative, characteristic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations. Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevskaya theorem, wellposedness. Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations. The purpose here is to motivate and introduce the discussion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.

Pre-requisites: a strong background on advanced calculus involving multivariables (esp. Green's Theorem and Divergence Theorem), the theory of ordinary differential equations (ODEs), and basic properties of Fourier transforms. We will also use some basic facts of Lp function spaces and the usual integral inequalities (mostly completeness and Holder inequalities in setting). These topics are covered in the first semester graduate real variable course (640:501). This course does not assume the students have had a PDE course at the undergraduate level.

Texts: The course material will be mostly drawn from "Partial Differential Equations" by Lawrence C. Evans, published by AMS, 2002; and "Partial Differential Equations: Methods and Applications, Second Edition" by Robert McOwen, Prentice Hall, 2002.
The former puts more emphasis on the theory, while the latter devotes some space to working out applications of the theory in some interesting cases, while leaving some full discussion of the theory to references. You may obtain one or both of the texts. I will put these two and some additional books on reserve in the math library:

• Jeffrey Rauch, Partial Differential Equations, Springer, 1997.
• G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1976.
• F. John, Partial Differential Equations, 4th ed., Springer-Verlag, 1982.

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

Pre-requisites: Math 501, 502 and 503. Knowledge of implicit and inverse function theorems.
Text: There is no text book for the course.

640:523    Several Complex Variables     X. Huang    HLL 423     MW 2; 10:20-11:40

A function with n complex variables is said to be holomorphic if it can be locally expanded as power series in z. An even dimensional smooth manifold is called a complex manifold if the transition functions can be chosen as holomorphic functions. A CR manifold is a manifold that can be realized as the boundary of a certain complex manifold. Several Complex Variables is the subject to study the properties and structure of holomorphic functions, complex manifolds and CR manifolds.

Different from one complex variable, if n>1 one can never find a holomorphic function over the punctured ball that blows up at its center. This is the striking phenomenon that Hartogs discovered about 100 years ago, which opened up the first page of the subject. Then Poincaré, E. Cartan, Oka, etc, further explored this field and laid down its foundation. Nowadays as the subject is intensively interacting with other fields, providing important examples, methods and problems, the basic materials in Several Complex Variables have become mandatory for many investigations in pure mathematics.

This class tries to serve such a purpose, by presenting the following fundamental topics from Several Complex Variables.
(a) Holomorphic functions, plurisubharmonic functions, pseudoconvex domains and the Cauchy-Riemann structure on the boundary of complex manifolds
(b) Hörmander's L²-estimates for the ð-equation, sub-elliptic estimates, and the Levi problem
(c) Pseudo-differential operators and Hörmander's subelliptic estimates for the sum of vector fields squared.
(d) Cauchy-Riemann manifolds, Webster's pseudo-Hermitian Geometry; subelliptic and micro-local analysis on CR manifolds

Texts: The course materials will be largely taken from the following:
[1] L. Hormander, An introduction to complex analysis in several variables, Third edition, North-Holland, 1990.
[2] S. Chen and Mei-Chi Shaw, Partial differential Equations in Several Complex Variables, AMS/IP Studies in Advanced Mathematics, 19, American Math. Society, 2001.
Prerequisites: One complex variable and the basic Hilbert space theory from real analysis

640:532    Differential Geometry     X. Rong    HLL 124     TTh 2; 10:20-11:40

640:540    Intro Alg Topology (I)   S. Ferry    HLL 525 on T4 (1:40-3)   &   HLL 425 on W3 (12-1:20)

Text: Allen Hatcher's excellent new book Algebraic Topology, available for \$30 in paperback from Cambridge University Press, as well as online here

This course will be an introduction to the fundamental group, homology theory, and cohomology theory.
The plan is to cover chapters 1, 2, and the first part of chapter 3 of Hatcher's book. Topics include fundamental group, Van Kampen's Theorem, covering spaces, simplicial and singular homology, cohomology, Brouwer's fixed-point theorem, the Borsuk-Ulam theorem, and the Jordan-Brouwer separation theorem.

640:547    Topology of Manifolds    F. Luo    HLL 525 MTh 3; 12:00-1:20

Topology and Geometry of 3-Manifolds

This is an introduction to the topology and geometry of 3-manifolds. The theory was revolutionized in 1978 by W. Thurston who proposed the geometrization conjecture which supersedes the Poincare conjecture. The conjecture has been the main guideline for study of 3-manifolds during the past 25 years. Recent work of G. Perelman may have solved the geometrization conjecture using Hamiltons Ricci flow. The goal of the course is to introduce this rich theory from historical point of view with emphasizing on the topological and geometric motivations and examples.

We will introduce the basic problems in the field and the tools developed by topologists and geometers to study 3-manifolds. The course is intended to be self-contained. We will begin with the classification of surfaces. Students should know manifolds and basic algebraic topology.

We intend to cover the following topics.

1. Basic theorems on 3-manifolds (the loop theorem, sphere theorem and Dehns lemma)
2. Haken's theory of normal surfaces and triangulations of 3-manifolds
3. Thurston's geometrization conjecture and the work of Perleman (brief introduction)
4. geometries in 3-dimension: constant curvature metrics and a few others
5. Seifert fibered 3-manifolds and its classification
6. volume of geometric structures
7. The work of Jones and Casson on finite type knot invariants and 3-manifold invariants
8. The conjecture of Kashaev-Murakarmi-Murakarmi relating quantum invariant with hyperbolic volume
If you have questions on the course, please let me know.
fluo@math.rutgers.edu

640:551    Abstract Algebra   D. Maclagan    HLL 525   MW 5; 3:20-4:40

Main Text: T. Hungerford, Algebra
Prerequisites: Any standard course in abstract algebra for undergraduate students.

This is a standard course for beginners. We will consider a lot of examples.
Group Theory: Basic concepts, isomorphism theorems, normal subgroups, Sylow theorems, direct products and free products of groups. Groups acting on sets: orbits, cosets, stabilizers. Alternating and Symmetric groups.
Basic Ring Theory: Fields, Principal Ideal Domains (PIDs), matrix rings, division algebras, field of fractions.
Modules over a PID: Fundamental Theorem for abelian groups, application to linear algebra: rational and Jordan canonical form.
Bilinear Forms: Alternating and symmetric forms, determinants. Spectral theorem for normal matrices, classification over R and C.
Modules: Artinian and Noetherian modules. Krull-Schmidt Theorem for modules of finite length. Simple modules and Schur's Lemma, semisimple modules.
Finite-dimensional algebras: Simple and semisimple algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.

640:555    Sel. Topics in Algebra    J. Lepowsky    HLL 425     MTh 3; 12:00-1:20

Vertex Operator Algebra Theory and Tensor Categories

The course will develop tensor product theory for representations of a vertex operator algebra, starting from the basics. This subject is related to developments in both mathematics and conformal field theory. We will discuss tensor category theory and also logarithmic conformal field theory. We will relate the mathematical and physical formulations of the theory, including the ideas of fusion rules and operator product expansions.

Students potentially interested in this course are encouraged to consult me, so that I can adjust the focus of the course according to people's interests.

Text: Lepowsky will distribute material, including background material and a new monograph by Y.-Z. Huang, J. Lepowsky and L. Zhang.
Please note: The Quantum Mathematics Seminar, which will meet on some Fridays at 1:00 (new meeting time, to fit with the new class schedule), 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.

640:558    Theory of Algebras    E. Taft

Hopf Algebras and Quantum Groups

Hopf algebras arose in topology and now play an important role in algebra and many other areas of mathematics. Basic examples include the universal enveloping algebra of a Lie algebra, the rational functions on an algebraic group, and certain deformations of these two examples which are now called quantum groups. Quantum groups have important connections to many fields in mathematics and physics.

We will study the algebraic structure of Hopf algebras. Quantum groups will be among the principal examples studied.

Prerequisites: Basic algebra, including linear algebra and tensor products.
Reference textbooks:

1. S. Dascalescu, C. Nastasescu, S. Raianu: Hopf Algebras: An Introduction, Marcel Dekker Inc., 2001.
2. C. Kassel: Quantum Groups, Springer-Verlag, 1995.
3. S. Montgomery: Hopf Algebras and Their Actions on Rings, Amer. Math. Soc., 1993.

640:559    Commutative Algebra    W. Vasconcelos    HLL 425     MTh 2; 10:20-11:40

Commutative Ring Theory
Text: D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry Springer-Verlag, 1996.

Commutative algebra is broadly concerned with solutions of structured sets of polynomial and analytic equations, and the study of pathways to methods and algorithms that facilitate the efficient processing in large scale computations with such data. This course will be an introduction to commutative algebra, with applications to algebraic gometry, combinatorics and computational algebra.

The first part of the course will treat basic notions and results---chain conditions, prime ideals, flatness, Krull dimension, Hilbert functions.

The other half of the course will study in more detail rings of polynomials and its geometry, and Gr\"{o}bner bases. It will open the door to computational methods in algebra (a few will be studied). Some other applications will deal with counting solutions of certain linear diophantine equations.

An useful reference will be David Eisenbud's {\em Commutative Algebra with a view toward Algebraic Geometry}, Springer. A graduate course in algebra (551 and/or 552) will suffice for prerequisite. For any additional clarification, I'm in Hill 228 or can be reached at vasconce@math.rutgers.edu.

640:569    Selected Topics in Logic   S. Thomas    HLL 525    TTh 6; 4:30-5:50

Set-theoretic forcing: an introduction to independence proofs
Text: Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, North Holland, Amsterdam.

This is an introductory course on proving independence results in set theory. (We say that a statement φ is independent of set theory iff φ can neither be proved nor disproved from the classical ZFC axioms of set theory. For example, it is well-known that the Continuum Hypothesis CH is independent of set theory.)
Initially we shall follow the lazy man's approach to obtaining independence results; namely, we shall study the consequences of the following two extra set-theoretic axioms.

• The Diamond Axiom (Diamond): a combinatorial principle which says intuitively that there exists a fortune-teller who correctly predicts the future often enough to be useful.
• Martin's Axiom (MA + ¬ CH): a powerful strengthening of the negation of the Continuum Hypothesis.
We shall see that there are many independent statements S such that ZFC + MA + ¬ CH implies S, while ZFC + Diamond implies ¬S. For example, this is true of the following statement.
• The Souslin Hypothesis: Suppose that the linear order { X, < } is complete and dense without endpoints. If there does not exist an uncountable family of pairwise disjoint nonempty open intervals of X, then { X, < } ~ { R, < }.
• However, there are many statements S whose independence cannot be established in this manner, such as:
• 2^{ℵn} = ℵn+1 iff n is prime.
• In order to prove the independence of such statements, we shall need to learn set-theoretic forcing. However, at this point in the course, those students who have successfully mastered the use of Martin's Axiom will realise that they already know how to force.

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

Prime Numbers

Questions about prime numbers lay in the heart of analytic number theory. Many subjects of the theory were created for solving problems of asymptotic distribution of primes, and later became useful for other topics. In particular the theory of the zeta and L-functions grew from these inspirations. In this course I will present a large panorama of problems, methods and results. Proofs will be given for the most fundamental results, while more advanced arguments will be surveyed in considerable details. The central areas of the course are:
1. Prime Number Theorem, elementary and analytic proofs
2. Primes in arithmetic progressions
3. Gaps between primes
4. The Grand Riemann Hypothesis and its substitudes
5. Density theorems for zeros off the critical line
6. The Pair Correlation Theory, and random matrix interpretations
This course is intended for beginning graduate students, however a quite advanced skill in complex and Fourier analysis is required. The lectures will be on Tuesdays and Fridays, yet the exact time and place will be fixed later.

642:527    Methods of Appl Math    D. Ocone    HLL 423    TTh 6; 4:30-5:50

This is a first semester graduate course appropriate for students in mechanical and aerospace engineering, biomedical engineering, other engineering, and physics. The topics to be covered are: power series and the method of Frobenius for solving differential equations; nonlinear differential equations and phase plane methods; perturbation techniques; vector space of functions, Hilbert spaces and orthonormal bases; Fourier seres and integrals; Sturm-Liouville theory; Fourier and Laplace transforms; separation of variables for solving the linear differential equations of physics, the heat, wave, and Laplace equations.

More information is on the course web page.

Text: M.Greenberg, Advanced Engineering Mathematics (second edition); Prentice, 1998 (ISBN# 0-13-321431-1))
Prerequisites: Topics the students should know, together with the courses in which they are taught at Rutgers, are: Introductory Linear Algebra (640:250); Multivariable Calculus (640:251); Elementary Differential Equations (640:244 or 640:252); Advanced Calculus for Engineering(Laplace transforms, sine and cosine series, introductory pde)(640:421).

Students who are not prepared for this course should consider taking 640:421.

642:550    Linear Alg & Applications   R. Goodman    HLL 423    MW6; 5:00-6:20

Note: This course is intended for graduate students in science, engineering and statistics.

This is an introductory course on vector spaces, linear transformations, determinants, and canonical forms for matrices (Row Echelon 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.

Text: Gilbert Strang, Linear Algebra and its Applications, 4th edition, ISBN #0030105676, Brooks/Cole Publishing, 2005
Grading: Written mid-term exam, homework, MATLAB projects, and a written final exam.
Lecturer: Prof. Roe Goodman, Hill 428, 445-3071

Prerequisites: Familiarity with matrices, vectors, and mathematical reasoning at the level of advanced undergraduate applied mathematics courses.

642:563    Statistical Mechanics     J. Lebowitz    T6 (5:00-6:20) in H705 & Th2 (10:20-11:40) in H423 & F6 (5:00-6:20) in H423

"Statistical Mechanics and Cooperative Phenomena"

Statistical mechanics 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 system 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.

The key ingredient in the statistical mechanical explanation of the emergence of macroscopic behavior from microscopic dynamics is the great disparity between the microscopic and macroscopic temporal and spatial scales. This gives rise to autonomous macro behavior in accord with the "law of large numbers". One can also go beyond that to obtain fluctuations and large deviations. This is beautifully captured, for equilibrium systems, by the elegant formalism of Gibbsian ensembles. There is no such general formalism for nonequilibrium systems and the subject is still in a state of exciting development.

This course will treat the subject from both the mathematical and physical point of view. Topics will include:

1. Microscopic and macroscopic descriptions of systems containing many elementary units.
2. Gibbs ensembles, free energy and correlation functions.
3. High temperature and low density expansions.
4. Phase transitions, low temperature behavior.
5. Nonequilibrium phenomena.
6. Applications of statistical mechanical ideas and methods to biological, sociological and economic systems.

A surprisingly large amount of the theory can be discussed in terms of idealized models; Ising spin systems, lattice gases, hard sphere fluids, etc. which require only little knowledge of physics or chemistry.

Prerequisite: A general mathematical background equivalent to that of a second year graduate student in math or a knowledge of stat. mech. obtained from a physics, chemistry or engineering course in the subject.

Time period will be determined by students' schedule. If you have questions please see me in room 612 of Hill Center; phone 5-3117; e-mail: lebowitz@math.

642:573    Numerical Analysis   Richard S. Falk    HLL 425     TTh 6; 5:00-6:20

This is the first part of a general survey of 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. In the fall semester, we will consider the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations, and see how all these problems are related.

In the spring semester (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.

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.

For more information, see the Course Web Page or Prof. Falk in Hill 722 (445-2367).

642:582    Combinatorics I    J. Beck    HLL 525    TF 2; 10:20-11:40

Combinatorics I
Basic Course Information: This course is the first part of a two semester advanced introduction to combinatorial theory.
We are going to discuss some classical results with applications in other branches of mathematics. The main source of the course is
• B. Bollob\'as: Combinatorics, Cambridge Univ. Press and
• L. Lov'asz: Combinatorial problems and exercises, North Holland. A brief list of what we are going to cover is the following:
1. Inclusion-exclusion principle, number-theoretic sieves
2. Extremal Set Theory: Sperner systems, Erd\H os-Ko-Rado theorem, factorizing complete hypergraphs: Baranyai's theorem
3. Linear algebra technique: Frankl-Wilson theorem, applications in Ramsey theory and geometry, Kahn-Kalai theorem disproving the Borsuk conjecture, eigenvalues
4. Probabilistic method: applications in graph theory, geometry, and number theory
5. Analytic method: Roth's theorems on arithmetic progressions, geometric discrepancy
6. Tic-Tac-Toe like games
Students are expected to solve homeworks.
Prerequisites: For Mathematics Ph.D. students: no formal prerequisites. For graduate students from other departments and undergraduates: permission of instructor. At various points in the course, we will need a variety of standard undergraduate material: linear algebra (350), advanced calculus (411), complex analysis (403) and elementary probability theory (477).

642:587    Sel Topics in Combinatorics   R. Radoicic    HLL 425     MW 5; 3:20-4:40

TBA
See bulletin board posting, or here.

642:591    Topics Probability & Ergodic Theory   J. Kahn    HLL 525     MW 4; 1:40-3:00

Probabilistic Methods in Combinatorics

We will discuss applications of probabilistic ideas to problems in combinatorics and related areas (e.g. geometry, graph theory, complexity theory). We will also at least touch on topics, such as percolation and mixing rates for Markov chains, which are interesting from both combinatorics/TCS and purely probabilistic viewpoints.

Text: Alon-Spencer, The Probabilistic Method (optional, but useful).
Prerequisites: I will try to make the course self-contained except for basic combinatorics and very basic probability. See me if in doubt.

642:593    Math Fdns Ind Eng   T. Butler

Math Foundations for Industrial Engineering
CANCELLED in 2005 (The course will run in 2006.)
This course is offered specifically for graduate students in Industrial Engineering.

Proof Structure for the Development of Concepts Based on the Real Numbers

1. Axioms for the Real Numbers
2. Logical Principles
The Continuity Axiom
1. The supremum concept and useful implications
2. Convergence of sequences and series
Development of the Calculus of Functions of One Variable
1. Continuous functions and basic properties
2. Differentiable functions and basic properties (the Mean value Theorem and Taylor's Theorem)
3. The Riemann Integral and its basic properties
4. The Fundamental Theorem of Calculus and implications
5. Uniform convergence of sequences of functions
Text: Bartle and Sherbert, Introduction to Real Analysis, 3rd Edition, Wiley & sons, 1992.

642:621   Financial Math     D. Ocone    HLL 124     TTh 4; 1:40-3:00

Financial Mathematics

This course is an introduction to modern mathematical analysis of financial markets and financial instruments. The finance concepts, such as financial derivatives and no arbitrage, and the basic probabilistic ideas for their analysis will be introduced first and briefly for discrete time models. After this introduction, the course will move to continuous time models. It will cover Brownian motion, martingales, stochastic calculus, diffusions and their related partial differential equations, and apply these to modeling financial markets and to the valuation of derivatives. Major goals are the Black-Scholes option pricing formula, risk neutral pricing, hedging, and the study of American and exotic options.

More information may be found at the course web page.

642:661    Topics Math Physics    G. Gallavotti and D. Ruelle     T2 (10:20-11:40) in HLL 423 & Th3 (10:20-11:40) in HLL 425

"Classical Mechanics and Non-equilibrium Statistical Mechanics".
Professors G. Gallivotti and D. Ruelle will teach this class joinntly, each giving part of the lectures.
Part I (Gallivotti): Stability theory of Hamiltonian equations

1) Canonical transformations and variational principles
2) Integrable systems and perturbation theory
3) Convergent and divergent series; and diagrammatic expansions
4) Applications to celestial mechanics (3 body problem)
Textbook: downloadable review article on Classical Mechanics (www.math.rutgers.edu/~giovanni)
Part II (Ruelle): NONEQUILIBRIUM STATISTICAL MECHANICS
(AN INTRODUCTION TO THE PHYSICS AND MATHEMATICS).
1) A broad overview. [Equilibrium, entropy and its increase, diversity of nonequilibrium, macroscopic evolution equations, diversity of mathematical idealizations, life, thermostats].
2) Infinite quantum spin systems. [Model and assumptions, NESS, entropy production and its positivity, the question of asymptotic abelianness in time].
3) Finite classical systems with isokinetic thermostats. [Deterministic thermostat, chaos, NESS=SRB, entropy production=phase space contraction, linear response formula, fluctuation theorems].
4) Appendix. [Hyperbolic dynamics and SRB states].