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.
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).
This basic course will have a description soon.
640:507Functional 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.
I will give a balanced introductory course on pde (elliptic, parabolic, hyperbolic).
640:519Sel 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")
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.
640:546Topics Alg Topology P. Landweber HLL 525 TF 3 11:30-12:50
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
640:548Diff 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..
640:549Lie Groups S. Sahi HLL 525 MTh 2; 9:50-11:10
The course will discuss semisimple Lie groups and their representations.
640:551Abstract 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.
640:555:01Sel 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.
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.
This course will be an introduction to commutative algebra.
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.
640:573Spec Top Number Theory H. Iwaniec HLL 124 TF 3; 11:30-12:50
SPECTRAL THEORY OF AUTOMORPHIC FORMS
The course will include
- Selberg Trace Formula
- Phillips-Sarnak theory of deformation of groups
This course will examine the arithmetic of elliptic curves, in particular their zeta-functions and rational points.
642:527Methods 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.
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.
Functional analysis and spectral theory of Quantum mechanics (including the theory of QM)
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.
642:582Combinatorics J. Beck HLL 425 MTh 2; 9:50-11:10
Combinatorics I is a standard combinatorics grad. course discussing subjects like
- Pigeonhole and inclusion-exclusion principles
- Recurrence relations and generating functions
- Theory of graphs and hypergraph
- The Erdo"s type probabilistic method
642:587Sel 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.
642:593Math Fdns Ind Eng T. Butler HLL 124 MTh 2; 9:50-11:10
Advanced calculus for industrial engineering students
"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.
642:611:02Sel Top Appl Math CANCELLED O. Costin HLL 124 MW 6; 4:30-5:50
Advanced asymptotic methods in analysis.
642:661Topics 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.
Last Modified 9/15/2001.