Descriptions of fall 2001 courses in the Rutgers-New Brunswick Math Graduate Program

Descriptions of fall 2001 courses in the Rutgers-New Brunswick Math Graduate Program

Faculty members were asked to supply very informal descriptions of their courses. These descriptions were edited mildly. Some descriptions may not exist. Some descriptions which are given may change, hopefully not too much.


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

This course is the first half of the year-long graduate course in real analysis for all entering math graduate students. The topics are classical and will include the theory of Lebesgue measure and integral, Lp spaces, certain aspects of abstract measure theory, and some basic applications of this integration theory.
These standard topics can be presented in different ways. My intention is to emphasize the fundamental ideas and techniques, instead of generalities; to emphasize tools and techniques that have broad applications, instead of purely esoteric topics. Also, to make this course an enriching experience for students of different background, I will try to provide a variety of examples and assignment problems.
The technical prerequisite is a solid advanced calculus course (for instance, mastery of the topics and problems in Walter Rudin's Principles of Mathematical Analysis).

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640:503   Theor Func Comp Vari   X. Huang     HLL 423  MTh 2; 9:50-11:10

This basic course will have a description soon.

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640:507   Functional Analysis   S. Chanillo     HLL 423  MTh 3; 11:30-12:50

Hahn-Banach, uniform boundedness and open mapping theorem and then some more. How much spectral theory, how much of compact operators etc, I will have to ask people.

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640:517   Partial Diff Equations   Y. Li      HLL 124  MW 5; 2:50-4:10

I will give a balanced introductory course on pde (elliptic, parabolic, hyperbolic).

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640:519   Sel Topics Diff Equ   A. Cohen     HLL 425  TF 2; 9:50-11:10

This course is about the Korteweg-deVries Equation, one of several non-linear evolution equations with "solitons". The bulk of the course will consist of (the real-, complex-, and functional-analysis underlying) the inverse scattering construction of solutions with smooth initial data. The long-time asymptotics will show how every solution eventually looks like a superposition of solitons (plus decaying "noise")

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640:540   Intro Alg Topology   C. Woodward     HLL 425   M 3; 11:30-12:50 & W 2; 9:50-11:10

An introductory algebraic topology course. In the past year, it was based on Bredon's book, which also includes a fair amount of manifold theory, De Rham theory etc., and will likely be used again next year.

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640:546   Topics Alg Topology   P. Landweber     HLL 525  TF 3 11:30-12:50

Cohomology Operations. This will concern mainly the Steenrod algebra and its action on mod p cohomology, with an emphasis on algebraic aspects including rings of polynomial invariants in characteristic p. I expect to also cover the Adams operations in K-theory, and give applications. I may even discuss some aspects of operations in complex cobordism and their relation to formal groups.
Prerequisites: a year of algebraic topology and a year of abstract algebra

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640:548   Diff Topology CANCELLED  A. Kosinski      HLL 525  MW 6; 4:30-5:50

Basically, what I plan to do in 548 is the following:

  • tubular nbhds
  • operations on mflds: connected sum, attaching handles.
  • handle presentation and its consequences: Morse relations,
  • visualization of homology via intersection matrices etc..
I hope to use Milnor's Topology from the Differentiable Viewpoint and Differential Manifolds (Pure and Applied Mathematics, Vol 138)

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640:549   Lie Groups   S. Sahi      HLL 525  MTh 2; 9:50-11:10

The course will discuss semisimple Lie groups and their representations.

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

The main text will be Hungerford's Algebra. 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 (PID's), 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.

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640:555:01   Sel Topics Alg   J. Lepowsky     HLL 124   MW 4; 1:10-2:30

The course will be devoted to selected topics in vertex operator algebra theory and related subjects. I will choose the specific topics according to the needs and interests of the students. I intend for the course to be accessible to beginning students in this field, but also interesting for experienced students.

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640:555:02   Sel Topics Alg   S. Thomas     HLL 525  MW 4; 1:10-2:30

The course will be an introduction to Geometric Group Theory, including Gromov's theory of hyperbolic groups. There are no prerequisites, except for the most basic notions of group theory such as free groups, generators and relations, etc. Geometric group theory constitutes the third wave of combinatorial group theory. In the first wave, they messed about directly with words. After that came the realisation that more progress could be made if combinatorial group theorists pretended to doing something else. In the second wave, they pretended to be doing very low dimensional topology. In the third wave, they are pretending to do geometry; i.e. they are regarding finitely generated groups as metric spaces.

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640:559   Commutative Algebra   F. Knop      HLL 525  TTh 4; 1:10-2:30

This course will be an introduction to commutative algebra.

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640:561   Mathematical Logic   G. Cherlin     HLL 423  MTh 4; 1:10-2:30

This is an introductory course in mathematical logic aimed at graduate students in mathematics rather than prospective logicians; so for example it will discuss the Banach/Tarski paradox.

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640:573   Spec Top Number Theory   H. Iwaniec     HLL 124  TF 3; 11:30-12:50

SPECTRAL THEORY OF AUTOMORPHIC FORMS
The course will include

  1. Selberg Trace Formula
  2. Phillips-Sarnak theory of deformation of groups
  3. L-functions

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640:574   Spec Top Number Theory   J. Tunnell     HLL 425  MW 5 2:50-4:10

This course will examine the arithmetic of elliptic curves, in particular their zeta-functions and rational points.

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642:527   Methods of Appl Math   G. Goldin      HLL 124  TTh 6; 4:30-5:50

A first-semester graduate course, appropriate for students of mechanical and aerospace engineering or other engineering areas, materials science, or physics. Topics include ordinary and partial differential equations, perturbation techniques, vector spaces of functions, Fourier series and integrals, Laplace and Fourier transforms, and applications.

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642:550   Linear Alg & Applications   R. Goodman     HLL 423  MW 7; 6:10-7:30

This course aimed at graduate students in engineering (usually from the Electrical and Computer Engineering program). Matrix factorization methods (LU and QR factorizations, Singular Value Decomposition) will be used to solve linear systems, find eigenvalues, and diagonalize quadratic forms. The theory and application of these methods will be developed in class and through homework assignments using MATLAB. The course will also treat vector spaces, linear transformations, determinants, and Jordan canonical form. Applications selected from the following topics: Graphs, Least Squares Approximations, Discrete Fourier Transform, Difference and Differential Equations, Markov chains, and Image Processing.

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642:561   Intro Math Physics   A. Soffer      HLL 423  TF 3; 11:30-12:50

Functional analysis and spectral theory of Quantum mechanics (including the theory of QM)

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642:577   Math Topics Syst Theory   H. Sussmann     HLL 423  TTh 6; 4:30-5:50

We will discuss optimal control and related mathematical questions: (a) Hamiltonian formalisms in classical mechanics, calculus of variations, and control, (b) Lie brackets and intrinsic versions of the necessary conditions in optimal control, illustrated with such things as the explanation of how when you park a car you are really computing a third-order Lie bracket (c.f. E. Nelson's book), (c) nonsmooth analysis (generalized gradients, etc.) and its connection to optimal control, (d) lots of examples of classical problems about minimizing curves.

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642:582   Combinatorics   J. Beck      HLL 425  MTh 2; 9:50-11:10

Combinatorics I is a standard combinatorics grad. course discussing subjects like

  1. Pigeonhole and inclusion-exclusion principles
  2. Recurrence relations and generating functions
  3. Theory of graphs and hypergraph
  4. The Erdo"s type probabilistic method

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642:587   Sel Top Discret Math   J. Kahn      HLL 525   TF 2; 9:50-11:10

587 will most likely (not a promise) be probabilistic methods (applications thereof to problems in discrete math and related parts of CS, geometry, physics), much of the material will come from Alon-Spencer.

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642:593   Math Fdns Ind Eng   T. Butler      HLL 124   MTh 2; 9:50-11:10

Advanced calculus for industrial engineering students

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642:611:01   Sel Top Appl Math   D. Zeilberger     ARC (not Hll!) 118   MTh 2; 9:50-11:10

Experimental Mathematics With MAPLE
"Proving is dead, long live Programming."
While the truth value of the above statement is not (yet) 1, it is strictly larger than 0. Mathematics, until recently, was at the stage that Physics was in Antiquity and the Middle Ages, i.e. a `deductive', not an empirical, science. Thanks to the computer, and especially thanks to computer algebra, mathematics is nowadays becoming an increasingly experimental science. We will learn how to perform `mathematical experiments' that will enable us to conjecture, and often prove, interesting results. In the process, we will enhance our computer literacy, and sharpen our programming skills. This course is indispensable for surviving professionally in this new millennium. No prerequisites are required besides an open mind. In particular, no previous computer programming experience is assumed. The students will become Maple whizzes.

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642:611:02   Sel Top Appl Math CANCELLED  O. Costin      HLL 124  MW 6; 4:30-5:50

Advanced asymptotic methods in analysis.

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642:661   Topics Math Physics   D. Ruelle & G. Gallavotti      HLL 425   MW 4; 1:10-2:30

We will study the theory and applications of dynamical systems and chaos. The first part of the course will emphasize the conceptual background from which one can evaluate the success of particular applications; the second will study some examples from the viewpoint of both theory and simulation.

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Last Modified 9/15/2001.

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