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

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

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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 Lp 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:

  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.

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
(A)       β = dα(v,-)
is a contact form, we have defined a homology and we have computed it in some cases. This contact homology which is defined using Legendrian curves of β is invariant under deformation of the contact form. The aim of this course is:
  • to describe the homology and compute it in standard cases (e.g., (S³, α0), α0 standard).
  • Discuss (A) and weaken it.

    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.)

    1. Abstract spectral theory on Hilbert spaces.
      a) Spectral theorems of self-adjoint operators.
      b) Variational methods, Min-Max principle, quadratic forms.
    2. 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.
    3. Topics on spectral theory of Schrödinger operators.
      a) Semi-classical analysis.
      b) Diamagnetism and Paramagnetism.
    4. 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.
    5. Morse theory
      a) Witten's approach to the classical Morse inequality.
      b) Demailly's holomorphic Morse inequality.
    We will not follow any particular texts. References are:
    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:

    1. The Brown-Mazur proof of the topological Schoenfliess theorem.
    2. Basic piecewise linear topology: general position, regular neighborhoods, etc.
    3. Basic differential topology: tangent bundles, transversality, tubular neighborhoods, etc.
    4. The proof of the Poincare Conjecture in dimensions greater than 4.
    5. Whitehead torsion and the s-cobordism theorem.
    6. Introduction to characteristic classes and cobordism.
    7. Construction of exotic spheres in high dimensions.
    If there is time, I will introduce the basic ideas and techniques of controlled topology.


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

    This course will be an introduction to the theory of semisimple Lie algebras.
    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 :

    1. basic properties of groups, rings and modules
    2. categories and functors
    3. groups acting on sets and vector spaces
    4. rings, ideals, chain conditions
    5. 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:

    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.


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

    Large Cardinals and Forcing
    Set-theoretic forcing can be used to prove the independence of many mathematical statements, such as the Continuum Hypothesis or the Souslin Hypothesis. However, there exist many interesting problems which cannot be tackled using only this technique. For example, is it possible to extend Lebesgue measure on 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 Gonshor    An Introduction to the      Cambridge University Press   1986
                     Theory 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.
    1. Geometry of the hyperbolic plane
    2. Differential and integral operators
    3. Discrete groups
    4. Automorphic forms
    5. Spectral theorem
    6. Trace formula
    7. Selberg zeta-function
    8. Weyl's law
    9. Small eigenvalues
    10. Kloosterman sums formula
    11. Hecke operators on congruence groups
    12. 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, 4th 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
    This course is an introduction to basic quantum mechanics and its mathematical analysis.
    Quantum mechanics was first developed when experiments indicated that particles behave as waves and waves behave well ... as particles.
    The resulting theory is fundamental to our understanding and description of the physical reality. Quantum theory had profound implications to virtually all sciences, basic and applied; it opened new directions for research in many mathematical fields, from algebra to analysis. It poses a challenge to our understanding of basic notions like information, randomness, computation and recently led to the new field of quantum computation encryption and teleportation.

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

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


    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.

    For more information, see the Course Web Page or Prof. Falk in Hill 722 (732-445-2390 x2367).


    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
    Correlation inequalities are statements about positive and negative reinforcement among random variables (as in the well-known Harris and FKG Inequalities). Such considerations are fundamental for various aspects of discrete probability, and also for some classical combinatorial problems. Nonetheless many simple, beautiful and seemingly basic questions remain open. We will survey the state of this area. Emphasis will be on combinatorial aspects, but we will also take the opportunity to explore a few related topics in probability.

    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

    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     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

    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.
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