Descriptions of fall 2003 courses in the Rutgers-New Brunswick Math Graduate Program

Descriptions of proposed fall 2003 courses in the Rutgers-New Brunswick Math Graduate Program

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

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; signed measures, absolute continuity and Radon-Nikodym theorem; Lebesgue's differentiation theorem; $L^p$-spaces. Other topics and applications as time permits.

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 124    M3; 11:30-12:50 and W2; 9:50-11:10

Text Serge Lang, Complex Analysis, 4th edition.
Prerequisite Advanced calculus.

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.

640:507    Functional Analysis   R. Nussbaum    HLL 425    MW 6; 4:30-5:50

Text: Peter D. Lax, Functional Analysis (New, Wiley-Interscience, 2002)

We will begin with basic results about Banach Spaces: the Baire category theorem, the uniform boundedness theorem, the open mapping and closed graph theorems, and many variants of the Hahn-Banach theorem. We will discuss general bounded linear operators on Banach spaces and the theory of compact linear operators. Other topics will include weak and weak* topologies on Banach spaces, reflexive Banach spaces, Hilbert space and compact self-adjoint or normal operators on Hilbert space. Generally speaking, all topics will be illustrated by applications in analysis, e.g., to integral or differential equations. Some excursions to nonlinear techniques (e.g., the Schauder fixed point theorem) and their applications will be made.

640:509    Sel Topics in Analysis   S. Gindikin    HLL 525    TTh 5; 2:50-4:10

Radon Transform and its generalizations
The Radon transform is one of the most remarkable integral transforms of geometric analysis. In the 2-dimensional case it connects functions on the plane with their integrals along lines. In 1918 Radon discovered that this transform and its multidimensional generalizations have explicit inversion formulas.

The Radon transform has broad connections and applications inside mathematics: differential equations, harmonic analysis, analysis on homogeneous manifolds, representations of Lie groups, complex analysis. There are interesting discrete versions of the Radon transform on trees and buildings. On the other hand, the Radon transform is also the mathematical basis of tomography. I want to discuss in this course both sides of the Radon transform and its generalizations. We will not go into numerical problems of tomography but it is quite instructive to consider some purely mathematical problems which appear in modern tomography. The focus will be on specific examples and formulas.

We will need basic knowledge of mathematical analysis. I will distribute fragments of a book which will soon be published by American Mathematical Society.

640:517    Partial Diff Equations   Z. Han    HLL 425     MW 4; 1:10-2:30

This is the first half of a year-long graduate course on PDE. In this first course, the aim is to introduce some of the fundamental issues in the study of PDE's: the correct posting of boundary value problems or initial value problems for different kinds of equations, the main features for solutions of each kind of equations, the necessity and the correct use of a variety of function spaces, and the importance of a priori estimates for solutions. I plan to discuss these issues in the context of the prototype equations: the Laplace equation, the D'Alembert wave equation, and the heat equations. Explicit representations of solutions will be used for these prototype equations, but the aim of the course goes beyond this: after the introductory material, I will discuss how to extend these methods, or introduce new methods, to deal with these issues in more complex settings, where the coefficients may be variable or the equation may be nonlinear. Topics to be discussed include: the concept of characteristics, solutions of first order nonlinear differential equations, Fourier's method for discussing Cauchy problem of constant coefficient equations, the method of freezing coefficients to deal with perturbations of prototype equations, and introduction of non--perturbative methods, such as variational methods, or energy estimates for wave-type equations. I will place certain emphasis on illustrating the use of the methods in a variety of settings, leaving some detailed development of the theory and derivations of some estimates to the second semester.

The prerequisite for this course is 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. Towards the latter part of the course, more background is needed: $L^p$ function spaces and the usual integral inequalities. These topics are covered in the first semester graduate real variable course. For physics/engineering graduate students interested in this course, it is possible to follow this course, if your background in the prerequisite area is particularly strong and you are willing to accept a few facts from Lebesgue integration theory and to learn the use of integral inequalities.

The text proposed for this course is Partial Differential Equations by Lawrence C. Evans, published by AMS, 2002.

This course does not assume the students have had a PDE course at the undergraduate level. For students without any prior experience with PDEs, it will be helpful for you to do some advance reading of the first five chapters of the text Partial Differential Equations: An Introduction by Walter Strauss, publised by John Wiley & Sons (1992), ISBN# 0-471-54868-5, and use it concurrently with our required text as supplemental reading.

640:519    Sel Topics in Diff Equations    S. Chanillo    HLL 423    TF 2; 9:50-11:10
Pre-requisites: Math 501, 502 and 503. Knowledge of implicit and inverse function theorems.
Text: There is no text book for the course, but R. Osserman's book on Minimal Surfaces is a good source.

Course Outline: The course will study Minimal surfaces. We will cover basic topics like the Weierstrass representation, Bernstein theorem and the Heinz curvature estimate. Then we will solve the Plateau problem in some form. We will also prove the monotonicity formula for minimal surfaces. Time permitting we will try to get into some geometric measure theory.

640:541    Intro Alg Topology (II)   P. Landweber    HLL 525    TTh 4; 1:10-2:30

This course will be a sequel to Math 540 being taught by Prof. Luo in Spring 2003, but can also be viewed as a mostly independent course on cohomology and homotopy theory for students who already have had an introduction to homology.

The text will be Allen Hatcher's excellent new book Algebraic Topology, available for $30 in paperback from Cambridge University Press, as well as online here
The plan is to start by covering some preliminary notions in Chapter 0 and reviewing the definition and main features of homology in Chapter 2. We will then cover Chapter 3 on cohomology in detail, including a selection of additional topics at the end of the chapter (such as H-spaces and Hopf algebras); there are features to cohomology that are not present in homology. We will end by covering basic results on homotopy groups in Chapter 4, aiming to reach the long exact sequence of homotopy groups for a fiber bundle, and to then study homotopy groups of classical groups as well as the cohomology of fiber bundles.

Note: The basics of cohomology theory can be found in Jim Vick's book Homology Theory. Hatcher's book gives many more examples, exercises, and additional topics on cohomology, and also gives a thorough treatment of homotopy theory with a large collection of additional topics.

640:547    Topology of Manifolds    F. Luo    HLL 525 MTh 3; 11:30-12:50

Topology and Geometry of 3-Manifolds

This is an introduction to topology and geometry of 3-manifolds with emphasis on the geometric aspect of the theory. More specifically, we will focus on the Ricci flow program of Richard Hamilton toward resolving the geometrization conjecture in dimension 3. Very recently (Nov. 2002), Grisha Perelman made a major progress on the Ricci flow program. He also claimed to have a proof of the geometrization conjecture (especially the 3-dimensional Poincare conjecture) in March 2003. Our goal in the course is to understand the basics of Ricci flow on surfaces and 3-manifolds.

Here is a sketch of what we are planning to cover.
(1) The work of Hamilton on the classification of 3-manifolds with positive Ricci curvature.
(2) The work of Hamilton on singularity formations in Ricci flow.
(3) An introduction to Perelman’s work toward a solution of the Geometrization conjecture.

This course is intended to be self-contained. The students should know basic topology (manifolds and covering spaces). Some knowledge of differential geometry will be helpful but not necessary. We will give a rapid introduction to Riemannian geometry at the beginning of the course with emphasis on tensor calculation.

The tentative meeting time will be MTH3. But that can be changed. For more information, please send an email to fluo@math.

640:550    Introduction to Lie Algebras   S. Sahi    HLL 425    MTh 2; 9:50-11:10

Text: Introduction to Lie algebras and representation theory by James Humphreys.
This course will be an introduction to the theory of semisimple Lie algebras.
We will carry out the (Cartan-Killing) classification of semisimple Lie algebras, and the (Cartan-Weyl) classification of their irreducible finite-dimensional representations.

The only prerequisite is a good understanding of linear algebra. Thus the course should be accessible to first and second year graduate students, and even to advanced undergraduate students. This course will be of interest to students planning to work in

    (a) Lie groups, differential geometry and topology, harmonic analysis.
    (b) Lie algebras, algebra, algebraic geometry, algebraic combinatorics, number theory.
    (c) Control theory, math. physics, statistical mechanics.

640:551    Abstract Algebra   V. Retakh    HLL 423    TTh 5; 2:50-4:10

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.

640:555:02    Sel. Topics in Algebra   L. Carbone    HLL 525    W2 W3; 9:50-12:50

Sphere Packings, Lattices and Group Actions
Texts: J.Conway and N.Sloane, Sphere Packings, Lattices and Groups, Springer 1992,
J.Milnor and D.Husemoller, Symmetric Bilinear Forms
R.Borcherds, Automorphic forms and Lie algebras, Current developments in mathematics, Harvard University 1996,
T. Hales, Cannonballs and honeycombs, Notices AMS 47 (2000), 440--449.

Syllabus: This subject combines the study of sphere packings, lattices, finite groups, geometry of numbers and combinatorics. An outline of the syllabus is as follows:

  • Sphere packings in n-dimensional Euclidean space
  • The related kissing number, covering and quantizing problems
  • Lattices and quadratic forms
  • Classification of quadratic forms
  • The E_8 and Leech lattices
  • Reflection groups and root lattices
  • Connections with Kac-Moody Lie algebras and vertex operator algebras
  • Connections with finite simple groups

  • 640:559    Commutative Algebra   C. Weibel    HLL 425     T3; 11:30-12:50
    Reading course - Register for 640:615

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

    This course is an introduction to commutative rings, both from the algebraic and geometric viewpoint. We will only assume Algebra 551 and 552 as prerequisites; no prior knowledge of algebraic geometry is needed. Topics to be covered include: localization, the topological space Spec(R), primary decomposition, integral extensions, flatness, completions, dimension theory and Hilbert functions, differentials, regular sequences, regular rings.

    640:560    Homological Algebra   Y.-Z. Huang    HLL 423     MW 4; 1:10-2:30

    Theory and Applications of Tensor Categories

    The theory of tensor categories has recently become an important tool in the study of a number of mathematical and physical problems. The fundamental connections among quantum groups, knot and three-dimensional invariants and conformal field theories are best understood through ``modular'' tensor categories. Many classical algebraic notions, for examples, associative algebras and Frobenius algebras, have generalizations in the framework of tensor categories. More recently, tensor categories and these generalizations of classical algebraic notions have also been used to study boundary conformal field theories and D-branes in string theory.

    This course is an introduction to the theory of tensor categories and its applications in representation theories, quantum groups, knot invariants and conformal field theories. I will start with a review of basic notions in the theory of categories. The theories and applications of monoidal categories, tensor categories, symmetry tensor categories, braided tensor categories and modular tensor categories will be discussed next. Then I plan to discuss examples of modular tensor categories constructed from representations of quantum groups and vertex operator algebras.

    Prerequisites: I will assume that the students have some basic knowledge in algebra, as covered in the first-year graduate courses.
    This course was originally numbered 555:01 (Sel. Topics in Algebra). It was renumbered 560 because the registrar cannot get good programmers.

    Texts: (i) Categories for the Working Mathematician by Saunders Mac Lane (Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1971);
    (ii) Lectures on Tensor Categories and Modular Functors by Bojko Bakalov and Alexander Kirillov, Jr. (University Lecture Series, Vol. 21, American Mathematical Society, Providence, 2001);
    (iii) Some papers to be distributed.

    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. The only prerequisites are some basic set theory, including cardinals, ordinals and transfinite induction. 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, < } ~ \langle { R, < }.
    However, there are many statements S whose independence cannot be established using \Diamond and MA + -CH. For example, while the following statement is independent of ZFC, it is implied by both ZFC + \Diamond and ZFC + MA + - CH.
    Additivity of Measure: If \kappa < 2^{\aleph_{0}} and A_{\alpha} \subseteq R is a nullset for each \alpha < \kappa, then \bigcup_{\alpha < \kappa}A_{\alpha} is also a nullset.

    In order to prove the independence of such statements, we shall need to learn set-theoretic forcing. But 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.

    640:571    Number Theory   S. Miller    HLL 124    TF 3; 11:30-12:50
    Prerequisites: the standard undergraduate curriculum suffices.

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

    This is a first-semester graduate course appropriate for students of

    1. mechanical and aerospace engineering,
    2. biomedical engineering or other engineering areas,
    3. materials science, or
    4. physics.
    We begin with power series expansion, the method of Frobenius, and Bessel functions, and go on to nonlinear differential equations, phase plane methods, and an introduction to perturbation techniques. We then study vector spaces of functions, including the L2 inner product, orthogonal bases, Sturm-Liouville theory, Fourier series and integrals, and the Fourier and Laplace transform. These ideas are applied using the method of separation of variables to solve partial differential equations, including the heat equation, the wave equation, and the Laplace equation. The course focuses mainly on applied techniques and conceptual understanding, rather than on theorems and rigorous proofs.

    642:550    Linear Alg & Applications   J. Tunnell    HLL 423    MW6; 4:30-5:50
    Text: Gilbert Strang, Linear Algebra and its Applications, 3 rd ed., ISBN #0-15-551005-3, Saunders Publishing/Harcourt Brace Jovanovich

    This course is primarily 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.

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

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

    642:561    Intro. Math. Physics   M. Kiessling    HLL 425    MTh 7; 6:10-7:30

    Introduction to Mathematical Physics
    Text: "Classical Mathematical Physics: Dynamical Systems and Field Theory"
    Author: Walter Thirring     Translator: (from German) by Evans M. Harrell II,
    List Price: $77.95 (on 3-31-2003.)    Hardcover: 543 pages
    Publisher: Springer Verlag; 3rd edition (October 17, 1997)     ISBN: 0387948430
    The course introduces the student to a modern mathematical treatment, using the tools of differential geometry, of the classical physical theories of space, time, matter, gravity and electromagnetism, going all the way to the beginnings of relativistic quantum theory. Topics:
    1. The Newtonian universe (Galileian space and time, point particles, Newton's laws of motion, Newton's law of gravitational forces, Coulomb's law of electrical forces, Lorentz' law of magnetic forces, the two-body (Kepler) problem, the N-body problem; Lagrange formalism, Hamiltonian formulation, symplectic geometry, probability and statistical physics (brief); Hamilton-Jacobi formalism),
    2. The Einsteinian universe (Minkowski's spacetime, Maxwell's electromagnetic field equations, electromagnetic waves, relativistic energy and momentum; Lorentzian manifolds, Einstein's gravitational field equations, geodesics, black holes, gravitational waves; the Cauchy problem, the problem of self-interactions, relativistic Hamilton-Jacobi theory),
    3. The quantum world (Limits of validity of the classical theories and the beginnings of quantum theory).

    642:563    Statistical Mechanics    J. Lebowitz    T6; 4:30-5:50 in H425 & W4;1:10-2:30 in H612 (sometimes M6; 4:30-5:50 in H525)

    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   M. Vogelius    HLL 425    TTh 4; 1:10-2:30

    This course has an eponymous title
    Text: K. Atkinson, An Introduction to Numerical Analysis

    642:582    Combinatorics I    M. Saks    HLL 525    TF 2; 9:50-11:10

    Combinatorics I
    Basic Course Information: This course is the first part of a two semester advanced introduction to combinatorial theory.Topics will include: hypergraphs, probabilistic, linear algebraic and polyhedral methods, enumeration, partially ordered sets and lattices, Ramsey theory, and combinatorial designs. For more information, see the course web page.

    There is no textbook that covers all the material of the course and in the past we've had no text for the course. A number of books will be placed on reserve at the library. This year I'm considering using the following text (I'm awaiting receipt of a copy of the book from the publisher.)

    S. Jukna, Extremal Combinatorics with applications to computer science.

    The grade in the course is based on problem sets. There will be 5 to 7 problem sets, which will normally be assigned every other week. There are no exams.

    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    Algebraic Methods in Combinatorics   J. Beck    HLL 525     W4; 1:10-2:30 & F5; 2:50-4:10

    Discrepancy Theory
    See bulletin board posting

    642:593    Math Fdns Ind Eng   T. Butler    HLL 423    MTh 2; 9:50-11:10

    Advanced calculus for industrial engineering students

    642:611   Sel Top Appl Math     P. Feehan    HLL 124     TTh 4; 1:10-2:30

    Financial Mathematics

    Mathematical finance is an emerging discipline wherein mathematical tools are used to model financial markets and solve problems in finance. Finance as a sub-field of economics concerns itself with the valuation of assets and financial instruments as well as the allocation of resources. History and experience have produced fundamental theories about the way economies function and the way we value assets. Mathematics comes into play because it allows theoreticians to model the relationship between variables and represent randomness in a manner that can lead to useful analysis. Mathematics, then, becomes a tool chest from which researchers can draw to solve problems, provide insights and make the intractable model tractable.

    Our course will primarily consist of an introduction to some of the most important mathematical tools in use today by practitioners of mathematical finance, stochastic partial differential equations, which is used to price options, derivatives, and other complex financial securities, and extreme value theory, used in risk analysis. After an introduction, as needed, to the background theory of probability and stochastic processes, we shall focus on diffusion processes, such as a Brownian motion, and jump diffusion processes. In discussing applications, such as the Black-Scholes option pricing formula, we shall give an introduction to derivative securities.

    We shall also discuss the application of stochastic processes to risk analysis. Empirical observation suggests that large market movements occur more often than models based on a convenient normal probability distribution would indicate - so, in reality, probability distributions accurately modeling the behavior of the market should have ?fat tails?. Hence, we shall discuss extreme value theory and its application to risk analysis. Time permitting, techniques for measuring and managing the risk of trading and investment positions will be discussed, including the portfolio risk management technique of Value-at-Risk, stress testing, and credit risk modeling. Important applications include risk analysis for credit derivatives and collateral debt obligations.

    Principal Texts:Stochastic Differential Equations, Øksendal
    Brownian Motion and Stochastic Calculus, Karatzas and Shreve
    Derivatives, Wilmott

    Our target audience is second graduate students or higher in all areas of mathematics, as well as graduate students in statistics, economics, and business, so we shall do our best to cater for a diverse audience. For all graduate students, familiarity with basic probability theory at the level of an applied text such as Papoulis is important background knowledge. Students in mathematics and statistics should ideally have an enthusiasm for learning how mathematics is used in finance, while economics and business students should have an interest in analytical, quantitative methods.

    For more information, see course web page

    642:611:02   Sel Top Appl Math     N. Komarova    HLL 525     T3; 11:30-12:50 & Th5; 2:50-4:10

    Mathematical Foundations for Biology
    Prerequisites: calculus, some undergrad exposure to ODE's, linear algebra, and basic probability

    This course is primarily intended for students in the BioMaPS program, as well as other students from Engineering, Computer Science, Statistics, and other departments who are interested in biological applications but whose undergraduate background did not include basic differential equations and stochastic processes.
    Tentative syllabus:
    I. Ordinary Differential Equations

    1. review of separation of variables and 2nd order homogeneous linear systems
    2. 2D systems: phase plane, nullclines, steady states, stability, parameter dependence and the concept of bifurcation
    3. periodic orbits, limit cycles, Poincare'-Bendixon
    4. n-dim linear time-invariant systems: matrix exponential, variation of parameters
    5. Laplace transforms: definition, basic properties, application to 2nd-order forced systems, computing inverses
    II. Fourier Series and Transforms
    1. Basic properties (linearity, convolutions), computation of expansions of simple examples.
    2. FT as continuous limits of FT, relate to Laplace transform, mention briefly basic properties, and inverse transform.
    3. FFT: numerical practice using applets or MATLAB code.
    III. Diffusion PDE's
    1. Introduction, separation of variables solutions in one-dimension (boundary value problems; use of FS for initial conditions)
    2. Gaussian kernels on infinite domain; speed of diffusion.
    IV. Probability
    1. review: events, independence, conditional prob, Bayes' theorem.
    2. densities and distributions, including Poisson an exponential (and how they relate), binomial, normal, etc
    3. from binomial to normal
    4. random walks in 1 dimension; relate to diffusion PDE
    5. finite Markov chains: equilibrium distributions, Perron-Frobenious
    6. continuous-time, finite-space processes - derive ODE for probabilities
    7. birth and death (queuing) as an example of a countable continuous-time process
    8. linear stochastically forced systems dx = Ax dt + dw: intuition as limit of discrete-time continuous-state process, calculation of spectrum of x.

    642:613    Sel Top Physiol & Medicine    E. Sontag    HLL 423    W 2; 9:50-11:10 & F 5; 2:50-4:10

    Mathematical Physiology
    Please visit the course page for updated information
    Text: James P. Keener and James Sneyd, Mathematical Physiology, Springer-Verlag New York, 1998. (Hardcover, 766pp, ISBN: 0387983813)
    This course will cover materials from the textbook, selected according to instructor and students' interests. The main prerequisite is familiarity with (ordinary) differential equations and general mathematical maturity, but the course will emphasize intuition and modeling, not mathematical rigor per se.
    The book's Table of Contents is as follows:
    • Part I, Cellular Physiology.
      • Biochemical Reactions
      • Cellular Homeostasis
      • Membrane Ion Channels
      • Excitability
      • Calcium Dynamics
      • Bursting Electrical
      • Activity Intercellular Communication
      • Passive Electrical Flow in Neurons
      • Nonlinear Wave Propagation
      • Wave Propagation in Higher Dimensions
      • Cardiac Propagation
      • Calcium Waves
      • Regulation of Cell Function
    • Part II Systems Physiology
      • Cardiac Rhythmicity
      • The Circulatory System
      • Blood
      • Respiration
      • Muscle
      • Hormone Physiology
      • Renal Physiology
      • The Gastrointestinal System
      • The Retina and Vision
      • The Inner Ear

    642:661:02    Topics Math Physics    G. Gallavotti and D. Ruelle    HLL 124    TF2; 9:50-11:10

    "Hyperbolic dynamical systems, chaotic motions and stochastic processes".
    Professors G. Gallivotti and D. Ruelle will teach this class joinntly, each giving part of the lectures.
    1) Quasi periodic motions: lack of chaotic motions. Their relations with the theory of continued fractions (text: Kintchin's book "Continued fractions")
    2) Rotation number and applications
    3) Hyperbolic maps of the $2$--torus.
    4) Chains of coupled chaotic maps
    Topics 2-4 will be extracted from a book that is already available on internet and that can be found and downloaded from my internet site at Rutgers:
    G.Gallavotti, F. Bonetto, G. Gentile: Aspects of the ergodic, qualitative and statistical properties of motion

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