Descriptions of fall 2004 courses in the Rutgers-New Brunswick Math Graduate Program

Descriptions of proposed fall 2004 courses in the Rutgers-New Brunswick Math Graduate Program

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640:501    Theor Func Real Vars    V. Retakh    HLL 423   TTh 6; 4:30-5:50

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 Complex Vars    S. Greenfield    HLL 525   MW4; 1:10-2:30

Disclaimer: The ideas expressed below are solely those of the wonderful professor who wrote them down.
Complex analysis is beautiful and useful. The course will be a rigorous introduction with examples and proofs foreshadowing modern connections of complex analysis with differential and algebraic geometry and partial differential equations.
Acquaintance with analytic arguments at the level of Rudin's Principles of Modern Analysis is necessary. Some knowledge of algebra and point-set topology is useful. A previous "undergraduate" course in complex analysis would also be useful though not necessary. Students who have taken a complex analysis course which carefully discussed a homotopy or homology version of Cauchy's Theorem may not need to take this course.

The text will be Complex Analysis in One Variable (second edition) by R. Narasimhan and Y. Nievergelt. The book is now available in the Rutgers University bookstore for about $68. Narasimhan's excellent text, with many interesting features, such as early presentation of results named for Picard and Runge, and the Corona Theorem, has in the second edition been supplemented with a useful and diverse collection of exercises by Nievergelt. I hope to cover most of Chapters 1 through 7. There will be written homework assignments and a written midterm and final exam.

640:508    Functional Analysis II    M. Vogelius    HLL 425   TTh 6; 4:30-5:50

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

Mathematical problems in the Navier Stokes equations

1) The Navier Stokes equations and the Euler equations

    a) formulation and local existence
    b) global existence theorems in 2 dimensions
    c) global regularity theorems in 2 dimensions and uniqueness

2) Leray's theory in 3 dimensions

    a) global existence
    b) generic regularity and fractal dimension of the singularity times
    c) the fractal dimension estimate of singularities in space-time
    d) Scheffer's theorem
    e) Caffarelli-Kohn-Nirnberg theorem

Book: Chapter III of Foundations of fluid dynamics, by G. Gallavotti, Springer Verlag, 2001.
The book is also downloadable from the web site at the page "Books"

640:517    Partial Diff Equations    A. Shadi Tahvildar-Zadeh    HLL 425     MW 4; 1:10-2:30

Introduction to Partial Differential Equations
This is the first half of a year-long graduate course on PDEs. In this first course, the aim is to introduce some of the fundamental issues in the study of PDE's: the correct posing 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, the heat equation and the Schroedinger equation. 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: Lp function spaces and the usual integral inequalities. These topics are covered in the first semester graduate real variable course (640:501). This course does not assume the students have had a PDE course at the undergraduate level.

My specialty is Partial Differential Equations (PDE) and especially those of hyperbolic type. These arise in mathematical physics as equations of motion in acoustics, electromagnetic theory, elasticity, fluid dynamics,and other areas. Consequently there are a wealth of applications and natural questions.

In broadest terms the subject of PDE addresses the question: what features are forced upon a function because it is the solution of a partial differential equation from some family? As very few problems are explicitly solvable, the analysis is almost always qualitative, the key ingredients being the derivation of estimates for solutions and the construction of approximate solutions. There is a vast variety of technique that has evolved for both purposes and a student usually approaches research by learning techniques developed for a small class of problems and then broadening with time.

The subject is appealing because:

  1. It uses the full arsenal of tools in analysis.
  2. Problems arise in a wide variety of settings: several complex variables, differential geometry, topology, mathematical physics, biology, engineering design, and continuum mechanics.
  3. Problems arise in context which makes them more interesting and suggest approaches. On the other hand, this demands that one appreciate the context. Interest from science and engineering means that there are opportunities for contacts with specialists in other areas.

640:519    Sel Topics in Diff Equations    Yanyan Li    HLL 423    TF 2; 9:50-11:10

The course will consist of two parts. In the first part, I will present a few useful methods in the study of some nonlinear partial differential equations. In the second part, I will present some results on the Yamabe problem and a fully nonlinear version of the Yamabe problem, and some results on the Monge-Ampere equations. For the second part, emphasis will be on the ideas, methods and open problems---- we do not carry out all the details.

The first part of the course will cover: Leray-Schauder degree theory and some applications to nonlinear PDEs (existence of solutions, bifurcation of solutions, etc.), the Mountain Pass Lemma and some applications to semilinear elliptic equations, concentrated compactness and some applications.

The second part will cover:
I. Introduce the Yamabe problem ---- existence on a given compact Riemannian manifold of conformal metrics of constant scalar curvature. The problem was solved through the works of Yamabe (60), Trudinger (68), Aubin (76) and Schoen (84). We outline the proofs of these results, including the positive mass theorem of Schoen and Yau on which the proof of the last case relies.
II. Present some recent and ongoing joint work with Lei Zhang on compactness of all solutions to the Yamabe problem.
III. Present some recent and ongoing joint work with Aobing Li on a very general fully nonlinear version of the Yamabe problem.
IV. Present Pogorelov's interior second derivative estimates for solutions to the Monge-Ampere equations, the classical work of Caffarelli, Nirenberg and Spruck on the Dirichlet problem for the Monge-Ampere equations, some aspects of the regularity theory of Caffarelli on Monge-Ampere equations, and some recent and ongoing joint work with Caffarelli on multi-valued solutions to the Monge-Ampere equations.

640:533:01    Introduction to Differential Geometry    X. Huang     M5 in HLL 313 (2:50-4:10)   &   W5 in HLL 525 (2:50-4:10)

This is an introductory course to the basic topics in Differentiable Manifolds and Differential Geometry. It is designed for the first or second year graduate students who intend to major in any field of pure mathematics.

Textbooks: (a) L. Conlon, Differentiable Manifolds, Birkhauser, Advanced Texts
(b) S. S. Chern, Introduction to Differential Geometry, Chicago Lecture Notes Series

640:535:01    Algebraic Geometry    C. Weibel    HLL 425    TTh2; 9:50-11:10

Text: Hartshorne, Algebraic Geometry, Springer Graduate Texts in Math.~52, latest edition.
Prerequisite: Algebra II (Math 552). I'm not going to assume a lot of commutative ring theory (we'll just quote what we need and move on), but already knowing about localization at prime ideals is useful.

This will be an introduction to the subject of Algebraic Geometry.
The first part of the course will cover the general theory of varieties over an algebraically closed field (complex numbers are good). I'll do lots of examples using curves and surfaces, 'cause I can draw pictures. These examples will become useful grounding for part II.

The second part of the course will study Schemes as the algebraic analogue of manifolds. This allows us to talk about varieties over arbitrary fields, or even over the integers Z. Divisors and zero-cycles come in here.

The third part will cover curves, and the Riemann-Roch theorem. If there is sufficient interest, there will be a second semester.

640:540    Intro Alg Topology    S. Ferry    HLL 525    TTh 4; 1:10-2:30

Text: Algebraic topology by Allen Hatcher, Cambridge University Press, Cambridge, 2002.
"This book is one of the most interesting and accessible texts to come out in recent years. The subject is presented in three broad areas: elementary notions including fundamental group, homology and cohomology, and homotopy theory." -Math Review 2002k:55001

640:547    Topology of Manifolds    C. Woodward    HLL 525    MTh 2; 9:50-11:10

Symplectic Geometry


A course on manifolds oriented towards symplectic geometry and classical mechanics. The first part of the course will cover manifolds and calculus on them. The second part will cover symplectic geometry, including a start on Hamiltonian dynamics and Hamiltonian systems with symmetry. The pace of the course will depend on who is enrolled.

Text: Foundations of Mechanics by Abraham and Marsden, Perseus Publishing, 2nd Edition (1994).
Outline of Course (Same as Book)

  • Smooth Manifolds
  • Calculus on Manifolds
  • Symplectic Geometry
  • Hamiltonian Systems with Symmetry

640:550    Introduction to Lie Algebras   R. Goodman    HLL 425    MTh 3; 9:50-11:10

Text (required): Roe Goodman and Nolan R. Wallach, Representations and Invariants of the Classical Groups (3rd printing), ISBN 0-521-66348-2, Cambridge University Press, 2003.
Supplementary text (recommended): James E. Humphreys, Introduction to Lie Algebras and Representation Theory, ISBN 3-387-90052-7, Springer-Verlag, New York, 1987.

This course will be an introduction to Lie algebras in the context of linear algebraic groups, with emphasis on the classical complex matrix groups (the general and special linear group, orthogonal group, and symplectic group). It will cover material from Chapters 1-5 of the Goodman-Wallach book, with additional topics from Humphrey's book.

Topics will include basic notions of linear algebraic groups, their representations, and their Lie algebras. The Lie algebras of the classical groups will be studied using root systems and Weyl groups relative to a maximal torus. These notions will then be extended to general semisimple Lie algebras, as in Humphrey's book. The complete reducibility of finite-dimensional representations will be proved and the Cartan-Weyl highest weight theory of irreducible finite-dimensional representations will be developed. For the classical simple Lie algebras explicit models for the irreducible representations will be constructed. The classification of finite-dimensional simple Lie algebras using Cartan matrices and Dynkin diagrams will also be discussed, as time permits.

640:551    Abstract Algebra   S. Sahi    HLL 124    MW 5; 2:50-4:10

This is a standard course for first-year graduate students.
Group Theory:Sylow theorems. Representations of finite groups. Groups acting on sets: orbits, cosets, stabilizers. Alternating and Symmetric groups, and their representations.
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.
Modules: Artinian and Noetherian modules. Krull-Schmidt Theorem for modules of finite length. Simple modules and Schur's Lemma, semisimple modules.
Finite-dimensional algebras: Simple and semisimple algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.

640:555    Sel Topics in Algebra    Haisheng Li    HLL 425    M 5&6; 2:50-5:50

Vertex Operator Algebras
Vertex operator algebras, introduced in the mid 80's, are a new and fundamental class of algebraic structures, and they have found applications in numerous fields. The mathematical counterparts of chiral algebras in physics, vertex operator algebras provide the mathematical foundation of conformal quantum field theory. The theory of vertex operator algebras has been developing rapidly as a unique and important branch of mathematics.

In this course we shall mostly follow the book listed below step by step for the fundamental theory of vertex operator algebras. We shall discuss motivation, basic axiomatic properties, general construction theorems and important examples. This book was designed to be a textbook for graduate students, and it is highly self-contained, with minimal prerequisites. Any graduate or advanced undergraduate student familiar with linear algebra will be able to take this course without much difficulty, though some familiarity with Lie algebras and associative algebras would be helpful.

In addition to the book, we shall discuss some other significant topics, as time permits.

Textbook: Introduction to Vertex Operator Algebras and Their Representations,
James Lepowsky and Haisheng Li, Progress in Mathematics 227, Birkhauser, 2004.

640:559    Commutative Algebra    W. Vasconcelos    HLL 425    MTh 2; 9:50-11:10

This course will be an introduction to commutative algebra, with applications to algebraic gometry, combinatorics and computational algebra.
The first part of the course will treat basic notions and results - chain conditions, prime ideals, flatness, Krull dimension, Hilbert functions.
The other half of the course will study in more detail rings of polynomials and its geometry, and Gröaut;bner bases. It will open the door to computational methods in algebra (a few will be studied). Some other applications will deal with counting solutions of certain linear diophantine equations.
Participants will be encouraged to give talks of selected topics.

An useful reference will be David Eisenbud's {Commutative Algebra with a view toward Algebraic Geometry, Springer.
A graduate course in algebra will suffice for prerequisite. For any additional clarification, I'm in Hill 228 or at

640:573:01    Special Topics in Number Theory    H. Iwaniec    HLL 124    TF3; 11:30-12:50

Analytic Methods in Modular Forms

Modular forms are important in analytic number theory, first of all because they offer the most powerful tools to work with classical problems. They are also quite attractive in their own structures. Our objectives in this course will be to show both sides of the theory. First we shall develop the foundation of the real-analytic forms (the Maass forms) paying particular attention to methaplectic forms, because these are rarely presented in details in the literature, and they are instrumental for many works concerning quadratic forms. The featured results in this area are the Shimura correspondence and the Waldspurger theorem. Having applications in mind, our approach will be somewhat different from the original works, that is to say we shall try to carry constructions more explicitly.

One cannot succeed without the spectral theory of automorphic forms, which itself would take a one semester course. We shall appeal to this without complete arguments, however only after substantial surveys and sketches of proofs.

The theory of Hecke operators is also indispensable for arithmetical applications of modular forms. Here we do give complete proofs of basic results.

If time permits we shall finish the course by applications to various questions of equidistribution, for example the equidistribution of closed geodesics and the roots of quadratic congruences.

Prerequisites: This course is addressed to any student with some experience in complex analysis and interest in arithmetic. No special knowledge of modular forms is required. We shall not follow closely any book, nevertheless some text books will be recommended to look at as supplement for improving the skill (optional). Notes will be distributed frequently before lectures when the subject matter is hard to find in standard literature.

640:574:01    Topics in Number Theory    J. Tunnell    HLL 425    MW5; 2:50-4:10

Elliptic Curves
An elliptic curve over a field K is an algebraic curve defined by an equation of the form y² = x³ + Ax + B (A, B in K), where the cubic on the right has distinct roots. Of paramount importance in the theory is that the set of solutions (x,y) to these equations with coordinates in a field L containing K form a group (three points on the intersection of a line with the graph of the curve sum to 0 in this group).

We will study this group for K the complex numbers, finite fields and number fields. The group structure on the set of points on a elliptic curve allows an array of technical tools such as f-adic representations, Galois cohomology, group schemes, and Selmer groups to be utilized to analyze problems. The abstract tools used to study elliptic curves must ultimately be brought to bear on specific number theoretic problems. Which integers are the areas of right triangles with integer sides? When is an integer the sum of two rational cubes? Each of these questions leads to open problems in rational points on elliptic curves.

We will emphasize examples as a means of exploring the many open problems and conjectures. Much of the theory will be illustrated by special curves such as y² = X³ - Dx or y² = x³ + D which are related to the classical problems above. These elliptic curves exhibit many of the phenomena present in general elliptic curves which arise in many areas of mathematics. Topics will include the following:

  1. The group law for adding points on an elliptic curve
  2. Elliptic curves over complex, real, finite and p-adic fields
  3. The group of rational solutions to E : y² = X³ + Ax + B form a finitely generated abelian group (the Mordell- Weil group of E)
  4. Effective bounds on the rank of the Mordell- Weil group
  5. Conjectural effective algorithms to find all rational solutions to y² = X³ + Ax + B
  6. L-series of elliptic curves and relations to modular curves
  7. Applications to classical Diophantine problems
Prerequisites: We will assume no prior knowledge of elliptic curves and will assume no more than an introductory graduate number theory course.
Text: The text for this course will be Arithmetic of Elliptic Curves by J .H. Silverman.
This text is a standard reference in the field. Other texts will be on reserve and notes will be distributed for certain topics.
Course Format: There will be periodic problem assignments and term projects involving elliptic curves.
More Information: Contact J. Tunnell in Hill 546, email to tunnell@math or consult the course web page

642:527    Methods of Appl Math    Z.-C. Han    HLL 124     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.

642:582:01    Combinatorics    J. Kahn    HLL 525    TF2; 9:50-11:10

This is the first part of a two-semester course surveying basic topics in combinatorics, including:

  1. Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)
  2. Matching theory, polyhedral issues if time allows
  3. Partially ordered sets and lattices, Möbius functions
  4. Theory of finite sets, hypergraphs, combinatorial discrepancy, Ramsey theory, correlation inequalities
  5. Probabilistic methods
  6. Algebraic methods
  7. Matroids, block designs, finite geometry, coding theory: probably small amounts of some of these as time allows

Prerequisites: The course is mostly self-contained, though some previous combinatorics, linear algebra, rudimentary probability are all occasionally helpful. Check with me if in doubt.
Text: van Lint and Wilson, A Course in Combinatorics. (Optional. We won't really follow it, but it's a nice book and has significant overlap

642:587:01    Sel Topics Discrete Mathematics    M. Saks    HLL 423    W4; 1:10-2:30 & F5; 2:50-4:10

Topological Methods in Discrete Mathematics

In recent years, algebraic topology has been applied to a number of diverse problems in combinatorics, discrete geometry and theoretical computer science. This course will survey some of these applications. Students are not required to know more than basic point-set topology; we will develop other things as needed.
The material for the course will be taken from several sources. The two main references will be:
  1. J. Matousek, Using the Borsuk-Ulam Theorem, Springer, 2002. (This will be available in the bookstore.)
  2. A. Bjorner, Topological Methods, in Handbook of Combinatorics V.II, R. Graham, M. Grotschel and L. Lovasz, eds., 1995.
There will be approximately 4 problem sets during the term, and no exams.
Prerequisites: The minimal background for this course is undergraduate linear algebra (350) and advanced calculus (411), and a graduate level course in discrete mathematics, e.g., 581 or 582.
Permission of instructor is required for students other than Mathematics Ph.D. students.

642:591:01    Topics Probability & Ergodic Theory    R. Gundy    HLL 423    MW 5; 2:50-4:10

Martingales in Ergodic Theory

This course will focus on the concept of a martingale and present some of its various applications in fields ranging from statistics to harmonic analysis.
Preliminary outline:

  1. Basic concepts. Probability spaces and filtrations. Dyadic filtrations on the unit interval and Rn. Conditional expectations: the elementary and general case.
  2. Convergence theorems. The Calderon-Zygmund decomposition of Rn and its extension. Stopping times. Super and submartingales. (The meaning of life.)
  3. Examples of discrete martingales. The Wald inequality (statistics). Black-Scholes option pricing in discrete time. The existence of the conjugate function of R1. (Harmonic analysis.) A Markov process on the unit interval and statistical mechanics.
  4. The B-D-G inequalities and the ``good lambda'' technique.
  5. Martingales in continuous time. The Wiener and Ito integrals. Gaussian Markov processes and their structure. Harmonic functions of Brownian motion and Hp-spaces.
Suggested texts: There are two texts that will provide useful support for the course.
  • ``Probability with Martingales,'' by David Williamson, Cambridge Univ. Press.
    This is an eccentric account of probability that has received reviews ranging from rave to scorn. I recommend it because the point of view is consistent with this course.
  • ``Probability Theory and Examples,'' by Richard Durrett, Duxbury Press, 2nd ed.
    This is a standard text with a much wider scope than the Williamson book. There is a supplement with answers to the exercises. Together, the two books cancel whatever style shortcomings each may have.
There will be a set of notes provided for this course.

642:593:01    Math Foundations Ind Eng    T. Butler    HLL 423    MTh 2; 9:50-11:10

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.

642:611   Sel Topics Applied Math     D. Zeilberger    ARC PC IML-116     MTh 3; 11:30-12:50

Experimental Mathematics

TEXT: The Maple Book by Frank Garvan (Chapman and Hall) and handouts.
* Teacher: Dr. Doron ZEILBERGER ("Dr. Z")   E-mail: zeilberg at math dot rutgers dot edu
* Classroom: "smart" classroom ARC bldg (inside computer lab).
* Dr. Zeilberger's Office: Hill Center 704 ( Phone: (732) 445-1326)

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in this direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they'll decide to do research in.

We will first learn Maple, and how to program in it. Then we will learn how to design and conduct mathematical experiments, that often lead to completely rigorous proofs.

642:612   Sel Topics Applied Math     Joel Lebowitz    Hill Center     times TBA

Statistical Mechanical Models of Complex Systems

Lattice systems also known (by probabilists) as interacting particle systems are used to model both static and dynamical phenomena in a great variety of physical systems composed of a large number of identical (or similar) components. These range from phase diagrams of alloys to phase transitions in epidemics. The utility of such models rests on the fact that their (relatively) simple stochastic microscopic dynamics gives rise to collective behavior similar to that observed in many physical and social systems.

While these models have been traditionally studied on regular lattices there is now much interest in extending them to more general graphs or networks. The topology of these networks, such as the (probabilistic) connections between different sites, contains relevant information about the systems being considered, e.g., the structure of social networks is important for following the evolution of certain epidemics.

The course will focus on the emergence of complex collective behavior, strikingly demonstrated by the existence of sharp phase transitions in the latter, from the simpler microscopic dynamics. We will discuss rigorous mathematical results (when available) and approximate physical ones. The latter will be based on suitable heuristics combined with computer simulations.

The course will be informal and interactive. Some familiarity with statistical mechanics and/or probability theory, dynamical systems theory is desirable but not essential. If you are interested and have any questions, please contact me. (

642:621   Financial Mathematics     T. Petrie    HLL 423     TF3; 11:30-12:50

Financial Mathematics

See the Course Web Page for updated information
This course is an introduction to the field of financial mathematics. One of the chief aims is to provide the theoretical framework in which to value securities especially derivative securities. (Basic securities are stocks and bonds. A derivative security is any security whose value is derived from a basic security. Examples are stock indices, options and futures.) Another aim is to provide the mathematical tools and develop mathematical models which will lead to valuation of securities and to trading and hedging strategies. Another aim of the course is to provide the framework to evaluate and manage risk in holding securities.

The course will begin with a discussion of the financial aspects of the course. This involves definitions of various securities and the details and operation of the markets in which they trade. Here are some examples: Forward and Future Contracts. Future Markets and Hedging, Options, Swaps, Leaps.

The central mathematical feature of this field concerns modeling the individual markets mathematically. Serious work in this area requires applied propability, statistics, some computer skills and stochastic partial differential equations. This material will be developed in the course. For background, participants should have some probability background like a good undergraduate course in probability and or statistics and some ability with a computer such as connecting to the internet and downloading files. Excel is an important computer tool which is often used in practical applications.

Any model in this area begins with some probability assumptions. The valuations which occur may come from probabilitic arguments or equilibria arguments. Both approaches will be discussed. A famous example of this is the derivation of the Black-Scholes formula for the price of a European Option. We will treat the model for this and the derivation in detail from both points of view. In the same spirit we will also treat other securities mentioned above in the same light. A basic tool in managing risk is the Capital Asset Pricing Model. This too will be treated.

For further information including registration information, please contact Professor Petrie at:
References: Options, Futures and Other Derivative Securities by J. Hull ; An introduction to probability and its applications vols. I and II by W. Feller; Arbitrage Pricing-notes by Musiela and Rutkowski. The Econometrics of Financial Markets by Campbell et. al.

642:661:02    Topics Math Physics    S. Goldstein    HLL 124     Tues 1&7 (8:10-9:30 AM & 6:10-7:30 PM)

Bohmian Mechanics

Quantum theory is the most successful physical theory yet devised.
It is also bizarre, bordering on incoherent.
It is widely claimed that this quantum weirdness is inevitable.

This course will be concerned with Bohmian mechanics, a formulation of nonrelativistic quantum mechanics that is a striking counterexample to such claims. Bohmian mechanics is the deterministic theory of particles in motion which naturally emerges from orthodox quantum theory when we demand conceptual clarity and physical precision. This theory resolves all paradoxes associated with quantum mechanics. Put simply, Bohmian mechanics is quantum mechanics made coherent.

Possible topics:

  1. Review of orthodox quantum theory
  2. Conceptual difficulties of orthodox quantum theory
  3. Bohmian mechanics
  4. Global existence and uniqueness for the Bohmian dynamics
  5. The empirical equivalence between Bohmian mechanics and orthodox quantum theory
  6. Bohmian mechanics for configuration spaces having nontrivial topology; Bohmian mechanics for systems of identical particles
  7. Bohmian scattering theory
  8. The classical limit of Bohmian mechanics
  9. Nonlocality
  10. Relativistic extensions of Bohmian mechanics
  11. Bohmian quantum field theory
  12. Quantum cosmology and the meaning of the wave function
Prerequisites: Linear algebra and advanced calculus. There are no physics prerequisites, but prior exposure to standard quantum theory would be helpful. Some knowledge of probability theory would also be good.

642:662    Topics Math Physics    M. Kruskal    HLL 632     MW6; 4:30-5:50

Asymptotic techniques, especially in Fluid Dynamics

This course should be of particular interest to applied mathematicians, theoretical or mathematically inclined engineers, and theoretical or mathematical physicists, since so many problems they deal with can best be treated by these techniques. The approach will be largely formal and heuristic, but rigorous when possible and appropriate.

Topics such as the following will be discussed:

  1. The use of small or large parameters to . . .
    • - Solve algebraic and transcendental equations
    • - Evaluate integrals and summations
    • - Solve ordinary and partial differential equations, difference equations, etc
  2. General boundary-layer theory
  3. Matched asymptotic expansions
  4. WKB and turning point theory
  5. Multiscale expansions
  6. PLK method
  7. Maximal balance and other principles of asymptotology
  8. Asymptotics beyond all orders (exponential asymptotics)
Prerequisites: Some background in ordinary and partial differential equations and complex analysis

Text: Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw Hill, 1978

Martin Kruskal
Hill 632      445-5788 (messages at -3921)

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