# Descriptions of proposed fall 2006 courses in the Rutgers-New Brunswick Math Graduate Program

640:501 `Theor Func Real Vari`
R. Wheeden
HLL 423 MTH 3; 12:00-1:20

The course will cover basic real variable theory in the context of
*n*-dimensional Euclidean space: functions of bounded variation, the
Riemann-Stieltjes integral, Lebesgue outermeasure and measure, Lebesgue
measurable functions and integration, convergence theorems for integrals,
Lusin and Egorov theorems, relations between Lebesgue and
Riemann-Stieltjes integrals, and Fubini and Tonelli theorems. As time
permits, additional topics such as the Lebesgue differentiation theorem,
absolute continuity and L^{p} spaces will be included.

**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`
S. Chanillo
HLL 124 TTh 5; 3:20-4:40

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:**

- 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.

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

**Prerequisite:** Advanced calculus.

640:507
`Functional Analysis` R. Nussbaum
HLL 423 MTh 3; 12:00-1:20

Historically, functional analysis arose from problems in analysis, e.g., questions about partial and ordinary differential equations, integral equations, approximation theory, convex sets, etc. We shall prove, in this course, a variety of "functional analysis theorems", but we shall attempt to illustrate all results with some analysis applications. We shall begin by by giving a number of examples of Banach spaces and Hilbert spaces -- Sobolev spaces, spaces of Holder continuous functions and L^p spaces. We shall then cover some basic results in Banach space theory: the Baire category theorem, the open and closed mapping theorems, the Hahn-Banach theorems about linear functionals (including geometric versions of the Hahn-Banach theorem), the weak and weak* topologies, reflexivity in Banach spaces and the Eberlein-Smulyan theorem.

The second half of the course will cover aspects of the the theory of linear operators in Banach and Hilbert spaces. After preliminaries about the spectrum and spectral radius of linear operators, we shall discuss compact linear operators, Fredholm determinants, trace class operators and self-adjoint operators. With luck, we may also have time to discuss some of the theory of so-called positive (in the sense of cone-preserving) linear operators and the linear Krein-Rutman theorem.

640:509
`Sel Topics in Analysis`
A. Bahri
HLL 425 MW 5; 3:20-4:40

**Deformation of Legendrian Curves and Contact Homology**

Let (M³,α) be a contact manifold and let v be a vector-field in a ker(α). Under the assumption

_{0}), α

_{0}standard).

The techniques are locally elementary.

640:515
`Ordinary Diff Equations` K. Mischaikow
HLL 124 MW 4; 1:40-3:00

This course provides an introduction to the qualitative theory for systems of differential equations. Topics include existence and uniqueness of solutions, Conley's decomposition theorem (attractors, Lyapunov functions), linear theory (stability, hyperbolicity, Floquet multipliers), local theory of equilibria (Hartman-Grobman theorem, stable and unstable manifolds), local and global bifurcations, and elementary properties of chaotic dynamics.

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

**Introduction to Geometric PDEs**

This is normally the first half of a year-long introductory graduate course on PDE. this semester, due to student input, the course will be re-oriented toward geometric analytic aspects of PDEs. A full course description is available at the course web page.

The aim of the course is to introduce ideas, methods, and techniques that are widely used in geometric analysis. We will achieve our goal through the examination of several important geometric examples, including various aspects of function theory on manifolds (harmonic and eigenfunctions, eigenvalue estimates, heat kernel estimates, etc), elementary aspects of minimal surface theory, harmonic maps, and Gauge theory. The techniques we will introduce include gradient estimates and its geometric applications, isoperimetric and Sobolev, Poincare type inequalities, barrier constructions on negatively curved manifolds, first and second variations of minimal surfaces and Dirichlet energies, stability and geometric applications, monotonicity properties and blow up at singularities, divergence structures in geometric PDEs.

We will draw material from several sources, including (the first four chapters of) Schoen and Yau, "Differential Geometry", Peter Li, "Lecture Notes on Geometric Analysis", J. Jost, "Nonlinear Methods in Riemannian and Kahler Geometry", T. Aubin, "Nonlinear Analysis on Manifolds, Monge-Ampere Equations", and some research papers. Students will be asked to present some parts of the course material.

The prerequisite for this course is a strong background in advanced calculus involving multivariables (esp. Green's Theorem and Divergence Theorem), analysis background equivalent to the trainings in the first semester graduate analysis courses, some experience with elementary PDEs and familiarity with some basic facts in Differential Geometry.

640:519
`Sel Topics in Diff Equations`
E. Teixeira
HLL 425 TF 2; 10:20-11:40

**An Introduction to Free Boundary Problems**

Free boundary problems appear naturally in the mathematical formulation of a great number of physical phenomena in technology and applied sciences as well as from areas of analysis, geometry, optimization and mathematical physics. Roughly speaking, a mathematical problem is said to have free boundary if one of the unknowns is a surface. A typical example is the evolving interphase between a solid and liquid phase: the ice-water problem.

In this course we shall present a series of ideas, methods, and techniques for treating basic issues of such problems: existence, regularity of solutions, regularity of the free boundary, etc. In particular, we shall describe some fundamental tools that make this possible. These serve for many other purposes and form a collection of useful mathematical instruments for studying several problems in applied analysis.

Our course will be organized as follows: We shall start by presenting some free boundary problems, with motivations and heuristic principles. Afterwards, we shall discuss some singular perturbation problems. Variational problems with free boundary and their relation to overdetermined problems will be presented in the sequel. Our ultimate goal is to develop the notion of viscosity solutions for free boundary problems.

This course will have an introductory character. We shall focus more on examples and applications, therefore, it should be suitable to a large range of graduate students. Indeed, this course is directed toward any student whose field is related to analysis and/or PDE. As prerequisite, we shall just assume some very basic knowledge in PDEs.

640:523
`Several Complex Variables`
S. Fu
HLL 423 W 2-3; 10:20-1:20

This will be a topics course on spectral theory of partial differential operators, especially the d-bar-Neumann Laplacian in several complex variables. It will be centered on Mark Kac's famous question: "Can one hear the shape of a drum?" and its various spin-offs. Listed below are tentative topics. (These are subject to changes depending on the audience.)

- Abstract spectral theory on Hilbert spaces.

a) Spectral theorems of self-adjoint operators.

b) Variational methods, Min-Max principle, quadratic forms. - Topics on spectral theory of the Dirichlet and Neumann Laplacians.

a) Counterexamples to Mark Kac's problem.

b) Isoperimetric inequalities.

c) Distribution of eigenvalues.

d) Inverse spectral problem. - Topics on spectral theory of Schrödinger operators.

a) Semi-classical analysis.

b) Diamagnetism and Paramagnetism. - Spectral theory of the d-bar-Neumann Laplacian.

a) Hörmander's L²-theory for the d-bar-operator.

b) Kohn's subelliptic estimates.

c) Compactness estimates.

d) Distribution of eigenvalues. - Morse theory

a) Witten's approach to the classical Morse inequality.

b) Demailly's holomorphic Morse inequality.

*Spectral theory and differential operators*by E. B. Davies, Cambridge University Press, 1996;

*An introduction to complex analysis in several complex variables*(Third edition) by L. Hörmander, Elsevier, 1991.

**Prerequisite for the course** is a solid knowledge of advance calculus.
Familiarity with real and complex analyses and
PDEs is a plus but is not required. There will be several homework
assignments but no exams.

640:532
`Differential Geometry`
C. Woodward
HLL 124 TF 2; 10:20-11:40

Introduction to smooth manifolds, connections, and Riemannian and symplectic geometry.

640:535
`Algebraic Geometry` A. Buch
HLL 525 MW 6; 5:00-6:20

The course will be an introduction to algebraic geometry, with the main emphasis on algebraic varieties over an algebraically closed field (e.g. the complex numbers). Varieties are algebraic analogues of manifolds, which locally look like geometric figures cut out by polynomial equations.

Topics will include products and morphisms of varieties, projective and complete varieties, dimension, non-singular varieties, rational maps, divisors, sheafs, and line bundles. We will take a closer look at algebraic curves, especially elliptic curves and consequences of the Riemann-Roch theorem. Along the way we will also introduce the more general notion of schemes, which makes it possible to work with varieties over an arbitrary commutative ring.

**Prerequisites:** Some familiarity with commutative algebra is
an advantage, for example Algebra II (Math 552), but is not required.

**Text:** I will use notes that I will distribute copies of. It will
still be very useful to own a copy of Hartshorne's book,
*Algebraic Geometry* (Springer GTM 52).

640:540
`Intro Alg Topology (I)` P. Landweber
HLL 525 TF3; 12:00-1:20
Please note: I expect the schedule to change!

The course will be an introduction to algebraic topology, based on
Allen Hatcher's excellent recent book *Algebraic Topology*. We will cover
the fundamental group and covering spaces (Chapter 1), homology of
spaces (Chapter 2), and perhaps make a start at cohomology (Chapter 3).
There will be some preliminaries about basic notions of homotopy
(Chapter 0) at the start, to set the stage. There will be an emphasis
on working problems, since many fine problems are offered.

Students should ideally have some familiarity with topological spaces,
as covered in the undergraduate topology course Math 441, which uses
the 2000 edition of Jim Munkres' book *Topology*. It is also important to
know the essentials of group theory, especially abelian groups.

It is expected that the remaining topics in the book on cohomology theory (Chapter 3), especially Poincaré duality, and homotopy theory (Chapter 4) will be covered in the sequel Math 541 in Spring 2007.

Anyone wanting some preliminary exposure to the ideas of algebraic
topology, in relation to other areas of mathematics and low dimensions
(where one can draw pictures) might take a look at Bill Fulton's 1995
Springer book, *Algebraic Topology*.

**Text:** Allen Hatcher's book *Algebraic Topology* is available
for $30 in paperback from Cambridge University Press.

Let me add that Hatcher's book is available free on
his web page,
http://www.math.cornell.edu/~hatcher.

640:547 `Topology of Manifolds`
S. Ferry
HLL 525 MTh 3; 12:00-1:20

This course will be an introduction to the topology of (mostly) high-dimensional manifolds in the topological, differentiable, and piecewise linear categories.Topics to be covered include:

- The Brown-Mazur proof of the topological Schoenfliess theorem.
- Basic piecewise linear topology: general position, regular neighborhoods, etc.
- Basic differential topology: tangent bundles, transversality, tubular neighborhoods, etc.
- The proof of the Poincare Conjecture in dimensions greater than 4.
- Whitehead torsion and the s-cobordism theorem.
- Introduction to characteristic classes and cobordism.
- Construction of exotic spheres in high dimensions.

640:550
`Introduction to Lie Algebras` R. Wilson
HLL 423 TF 2; 9:50-11:10

Text:

*Introduction to Lie algebras and representation theory*by James Humphreys.

640:551
`Abstract Algebra` J. Tunnell
HLL 423 MW 5; 3:20-4:40

This is an introduction to the mathematics of groups, rings and modules. The concept of groups acting on vector spaces will be used as a unifying idea which illustrates the interplay of these topics. Examples that provide concrete interpretation of the theory will be discussed.

**Topics will include the following :**

- basic properties of groups, rings and modules
- categories and functors
- groups acting on sets and vector spaces
- rings, ideals, chain conditions
- modules over principal ideal domains, including abelian groups

**Text:**T. Hungerford,

*Algebra*, Springer GTM 73

**Prerequisites:**A standard undergraduate knowledge of algebra is required. It will be assumed that students understand the concepts of group, ring, vector space and linear algebra.

640:555
`Sel. Topics in Algebra` YZ. Huang
HLL 425 MTh 3; 12:00-1:20

**Infinite-dimensional Lie Algebras**

In this course, I will give an introduction to infinite-dimensional Lie algebras and vertex operator algebras.

For the part on
infinite-dimensional Lie algebras, I will discuss mostly
infinite-dimensional Heisenberg algebras, affine Lie algebras the
Virasoro algebras and their representations. Vertex operator algebras
and their representations will be introduced as natural structures on
representations of these infinite-dimensional Lie algebras. Various
formulations of Vertex operator algebras, modules and intertwining
operators, including the component formulation, the formal variable
formulation, the complex variable formulation, the conformal geometric
formulation and the *D*-module formulation, will be discussed.

**Prerequisites:** I will assume that the students have
some basic knowledge in algebra and complex variables, as covered in
the first-year graduate courses.

640:558
`Theory of Algebras` E. Taft
HLL 423 TTh 6; 5:00-6:20

**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:**

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

640:561
`Mathematical Logic` S. Thomas
HLL 525 TTh 6; 5:00-6:20

**Large Cardinals and Forcing**

**R**to a countably additive measure μ which is defined on

*every*subset of

**R**? (Of course, such a measure cannot be translation invariant.) It turns out that the existence of such a measure μ is equiconsistent with the existence of a suitable ``large cardinal''. (Roughly speaking, a large cardinal is a cardinal which is so large that it proves the consistency of its own nonexistence.)

In this course, we shall study the Large Cardinal Hierarchy, beginning with weakly compact cardinals and then passing through Ramsey cardinals and measurable cardinals to the the pinnacle of supercompact cardinals ... and then we'll collapse them to small cardinals such as the continuum. As we'll see, if this collapse is done correctly, the resulting small cardinal will retain some residue of its former glory and this will lead to many interesting independence results.

**Prerequisites:**
This course is intended for students who already know basic
set-theoretic forcing such as c.c.c forcing, Martin's Axiom and
Diamond. We shall not be using any textbook.

640:569
`Sel Topics in Logic` M. Kruskal
by arrangement

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 GonshorAn Introduction to theCambridge University Press 1986Theory of Surreal Numbers

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

**Spectral Theory of Automorphic Forms**

This is a graduate course for both beginning and advanced students. No special knowledge of the subject matter is required, except for a basic knowledge of complex analysis and differential calculus. The main goal is to present the theory of automorphic forms in analytic aspects, with emphasis for applications to number theory.

- Geometry of the hyperbolic plane
- Differential and integral operators
- Discrete groups
- Automorphic forms
- Spectral theorem
- Trace formula
- Selberg zeta-function
- Weyl's law
- Small eigenvalues
- Kloosterman sums formula
- Hecke operators on congruence groups
- Automorphic L-functions

**Text:**I will use various original sources and my recent book

*Spectral Methods of Automorphic Forms*, AMS Graduate Studies in Mathematics, vol. 53.

642:527
`Methods of Appl Math` D. Ocone
HLL 124 TTh 6; 5:00-6:20

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. Bumby
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,
4^{th} edition, ISBN #0030105676, Brooks/Cole Publishing, 2006

**Grading:** Written mid-term exam, homework, MATLAB projects, and a
written final exam.

**Lecturer:** TBA

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

642:561
`Intro. Math. Physics` A. Soffer
HLL 425 MTh 7; 6:40-8:00

**Introduction to Quantum Mechanics**

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)

642:563
`Statistical Mechanics
` J. Lebowitz
time TBA

**COMPLEX SYSTEMS: PHYSICAL REALITY AND MATHEMATICAL MODELS**

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 direct way, from the properties of its constituents.

In this course we will investigate interesting features of such complex systems or phenomena. We will utilize both deterministic equations describing the macro behavior, as well as probabilistic 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. We will investigate how one may 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.

**Prerequisites:** The course will be informal and interactive.
Some familiarity with statistical mechanics and/or probability theory,
is important.

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.

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.

642:577
`Sel Topics System Theory` E. Sontag
HLL 423 TF 3; 12:00-1:20

**Introduction to Control Systems Theory**

See the course web page

Note: Course time will change!

642:582
`Combinatorics I` V. Vu
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.

**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` J. Kahn
HLL 425 MW 2; 10:20-11:40

**Correlation Inequalities**

**Text:** None; there will be books on reserve.

**Prerequisites:** I will try to make the course self-contained.
See me if in doubt.

642:593
`Math Fdns Ind Eng` T. Butler
HLL 124 MTh 2; 10:20-11:40

**Math Foundations for Industrial Engineering**

This course is offered specifically for graduate students in Industrial Engineering.

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

- Axioms for the Real Numbers
- Logical Principles

- The supremum concept and useful implications
- Convergence of sequences and series

- Continuous functions and basic properties
- Differentiable functions and basic properties (the Mean value Theorem and Taylor's Theorem)
- The Riemann Integral and its basic properties
- The Fundamental Theorem of Calculus and implications
- Uniform convergence of sequences of functions

**Text:**Bartle and Sherbert,

*Introduction to Real Analysis*, 3rd Edition, Wiley & sons, 1992.

642:621 `Financial Math`
P. Feehan
HLL 705 M 7-8; 6:40-9:30

**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.

**Text:** S. E. Shreve, "Stochastic Calculus for Finance II:
Continuous-Time Models", Springer Verlag, 2004

Sample syllabus

642:661
`Topics Math Physics`
G. Gallavotti and D. Ruelle
W3 (12:00-1:20) in HLL 425 & F4 (1:40-3:00) in HLL 425

**Renormalization Group, Ideas in Equilibrium Statistical Mechanics,**

Entropy Production and Fluctuations in Nonequilibrium Systems

Entropy Production and Fluctuations in Nonequilibrium Systems

The course will be adopted to the interests of the students. A preparation in statistical mechanics or dynamical systems is necessary.

Students with questions, please contact Joel Lebowitz, lebowitz@math.rutgers.edu , phone: 732-445-2390 x3923. Back to course listings Last Modified 4/20/2006.