Descriptions of Proposed Fall 2007 Courses
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640:501 Theor Func Real Vari J. Beck HLL 425 TTH 4; 1:40-3:00
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; Lp-spaces. Other topics and applications (such as Lebesgue's differentiation theorem, signed measures, absolute continuity and Radon-Nikodym theorem) as time permits.
Text: Wheeden and Zygmund, Measure and Integral
Pre-requisites: Undergraduate analysis at the level
of Rudin's "Principles of Mathematical Analysis," chapters 1--9,
including basic point set topology, metric space, continuity,
convergence and uniform convergence of functions.
640:503 Theor Func Complex Variable S. Greenfield HLL 525 TF 3; 12:00-1:20
Text: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) *by Robert E. Greene
The beginning of the study of one complex variable is certainly one of
the loveliest mathematical subjects. It's the magnificent result of
several centuries of investigation into what happens when R is
replaced by C in "calculus". Among the consequences were the
creation of numerous areas of modern pure and applied mathematics, and
the clarification of many foundational issues in analysis and
geometry. Gauss, Cauchy, Weierstrass, Riemann and others found this
all intensely absorbing and wonderfully rewarding. The theorems and
techniques developed in modern complex analysis are of great use in
all parts of mathematics.
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.
The course will include some appropriate review of relevant topics,
but this review will not be enough to educate the uninformed
student adequately. A previous "undergraduate" course in
complex analysis would also be useful though not necessary.
There are many excellent books about this subject. The official text
will be Function Theory of One Complex Variable, by Greene and
Krantz (American Math Society, 3rd edition, 2006). The
course will cover most of Chapters 1 through 5 of the text, parts of
Chapters 6 and 7, and possibly other topics. The titles of these
chapters follow.
640:507
Functional Analysis H. Sussmann
HLL 525 MW 6; 5:00-6:20
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.
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640:509
Selected Topics In Analysis A. Bahri
HLL 525 MW 4; 1:40-3:00
Let M3 be a three dimensional compact orientable manifold and let ® be a contact form on M. Let
v be a vector-¯eld in ker® which we assume to be non-singular (e.g ¼1(M) = 0), let ¯ = d®(v; :), L¯ =
fx 2 H1(S1;M); ¯x(x_ ) = 0g, C¯ = fx 2 L¯; ®x(x_ ) = positive constantg. Let J(x) = R1
0 ®x(x_ ) be the action
functional de¯ned on curves of C¯.
This variational problem has critical points (the periodic orbits of the Reeb vector-¯eld of ®, »). Its
intersection operator @ mixes them. However a careful analysis shows that @, restricted to the periodic orbits
and projected onto them (@1 = q ± @ ± p) satis¯es under reasonable conditions @1 ± @1 = 0.
A homology is thus de¯ned and it involves mainly the periodic orbits of ®.
This course is devoted to the de¯nition, presentation and ¯rst steps in the computation of this homology.
The relation(s) that this may have (identity?) with the Contact Homology de¯ned via pseudo-holomorphic
curves is clearly an interesting area of research to which this course should prepare.
The Contact Homology which we de¯ne has three advantages: compactness is reduced in this framework,
under our assumptions, to few cases which are well-de¯ned and restricted, which should help in any further
research; the value of the homology is understood for odd indexes, though not yet computed; it is fun to
deform immersed curves.
The general plan of the course is: 1-Set up of the variational problem (¯; J), 2- Critical points of J
on C¯, 3- Variational Flow and Asymptotes, 4- Unstable manifolds of periodic orbits of », 5-De¯nition of
Homology, 6- Compactness (under Hypothesis (A)), 7- Value of the Homology for odd generators (related to
the S1-invariant homology of the loop space of M),8- Discussion of the underlying assumptions (Hypothesis
(A).
640:515
Ordinary Diff Equations K. Mischaikow HLL 423 MTH2; 10:20-11:40
This course provides an introduction to the dynamics of differential equations and time permitting the following topics will be covered.
640:517
Partial Diff Equations Y. Li
HLL 425 TF 2; 10:20-11:40
This is the first half of a year-long introductory graduate course on PDE.
PDE is an enormously vast field. PDEs arise from very diverse fields: from
classical to modern physics, to more applied sciences such as material sciences, mathematical biology, and signal processing, etc, and from the more pure aspects of mathematics such as complex analysis and geometric analysis.
This introductory course should be useful for students with a variety of
research interests: physics and mathematical physics, applied analysis, numerical analysis, complex analysis, differential geometry, and, of course, partial differential equations.
For an introductory course, it is more important to examine some important
examples to certain depth, to introduce the formulation, concepts, most useful methods and techniques through such examples, than to concentrating on presentation and proof of results in their most general form.
This is the way the course will be conducted.
The beginning weeks of the course aim to develop enough familarity and
experience to the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation.
Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods
are introduced for the purpose of establishing qualitative, characteristic properties of solutions to each class of equations. It is these properties that we
will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.
Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevskaya theorem, wellposedness.
Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations.
The purpose here is to motivate and introduce the discussion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.
Pre-requisites: a strong background on advanced calculus involving multivariables (esp. Green's Theorem and Divergence Theorem). We will also use some basic facts of Lp function spaces and the usual integral inequalities (mostly completeness and Holder inequalities in L² setting).
These topics are covered in the first semester graduate real variable
course (640:501).
Texts:
The course material will be mostly drawn from "Partial Differential
Equations" by Lawrence C. Evans, published by AMS, 2002; and "Partial Differential Equations: Methods and Applications, Second Edition" by Robert McOwen, Prentice Hall, 2002.
The former puts more emphasis on the theory, while the latter devotes some spaceto working out applications of the theory in some interesting cases, while leaving some full discussion of the theory to references.
You may obtain one or both of the texts. I will put these two and some additional books on reserve in the math library:
640:519
Sel Topics in Diff Equations
R. Nussbaum
HLL 423 TTH 4; 1:40-3:00
In the first half of this course we will follow some lecture notes of Oscar Lanford.
We shall develop the theory of invariant manifolds in Banch spaces--stable manifolds, unstable manifolds and the center manifold.
We shall then move on to generalizations of Poincare-Bendixson theory in dimensions greater than 2.
The ideas involved here have proved very useful in studying nonlinear differential-delay equations, and we shall give several applications to problems in this area.
640:534
Sel Topics in Geometry
X. Rong ARC 333 M 2; 10:20-11:40; HLL 525 TH 3; 12:00-2:00
This is an introduction course (self-contained) to the convergence
and collapsing theory in Riemannian geometry. The convergence theory
is formed by Cheeger-Gromov and the collapsing theory is founded
through the works of Cheeger-Gromov and Fukaya. The main contents
of this course include:
The prerequisite of this course is a basic knowledge related
to (Riemannian) manifolds (calculus on manifolds: such
as differentiable structure, tangent bundles, connection,
curvature, etc) and some basic knowledge in algebraic topology.
The general reference for this course are:
640:540
Text: Allen Hatcher's excellent new book
Algebraic Topology, available for $30 in paperback from
Cambridge University Press, as well as
online here
This course will be an introduction to the fundamental group, homology
theory, and cohomology theory.
640:551
Abstract Algebra
J. Lepowsky HLL 425 MW 4; 1:40-3:00
Main Texts: Jacobson, Basic Algebra, Volumes 1 and 2, second edition.
Note: These volumes seem to be out of print, but students can go to a
website such as addall.com to find used, and possibly new, copies of both volumes.
This is a standard course for beginners. We will consider a lot of examples.
640:555
Sel. Topics in Algebra S. Thomas
HLL 423 MW 4; 1:40-3:00
This 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, combinatorial group theorists worked directly with words. After
that came the realisation that more progress could often be made
if they pretended to doing something else. In the second wave,
combinatorial group theorists 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.
640:556
Representation theory is the study of linear group actions on vector
spaces, also called representations. The course will focus on groups
that are also compact manifolds, such as the unit circle or the group
of all unitary matrices of a given size. For such groups it is
natural to require that the group actions are differentiable maps, and
it turns out that representations of this type can be classified using
a combinatorial root system constructed from the group. The goal of
the course is to prove Weyl's character formula, which can be used to
decompose a representation as a direct sum of irreducible
representations.
Prerequisites: Linear algebra.
Text: Representations of Compact Lie Groups, GTM 98, by
Theodor Brocker and Tammo tom Dieck.
640:571
Number Theory
S. Miller
HLL 425 TTh 5; 3:20-4:40
This course is related, but complementary to, Iwaniec's courses on the spectral theory of automorphic forms. I will present a different viewpoint on the subject, through representation theory. The course will be self-contained and has no prerequisites, though preferably, students should have passed their written qualifying exams. The main goal is to discuss the GL(2) theory and generalize it to other Lie groups such as GL(n), and also to become acquainted with modern formulations over adele groups.
Topics to be covered include:
Text: Automorphic Forms and Representations, Dan Bump, Cambridge University Press. (I intend to also give frequent handouts of typed course notes.)
Prerequisites: none.
640:573
Spec Top Number Theory H. Iwaniec
HLL 124 TF 3; 12:00-1:20
This course is for graduate students who are interested in number theory in a broad sense. I will present a variety of topics concerning diophantine equations, congruences and equations over finite fields. Some techniques from algebraic geometry will be applied, but the main focus will be on analytic methods. First of all I shall discuss in great detail the Circle method.
This depends heavily on estimates for exponential sums over finite field. Therefore I shall spend a considerable time to prove basic results, such as special cases of the
The Riemann Hypothesis for curves.
The most recent applications of the circle method are powered by exponential sums in many variables. In this case one needs the Riemann Hypothesis for varieties, which is far beyond the scope of this course. Nevertheless I shall try to give a comprehensive account of needed results.
Students are required to know only basic facts from arithmetic. Analytic techniques are less familiar so they will be developed from scratch. A large part of the course will follow the book by W.M. Schmidt, .Equations over Finite Fields. LNM 536.
642:527
Methods of Appl Math E. Speer
SEC 207 T 6; 5:00-6:20, SEC 117 TH 6; 5:00-6:20
This is a first semester graduate course appropriate for students in
mechanical and aerospace engineering, biomedical engineering, other
engineering, and physics. The topics to be covered are: power series
and the method of Frobenius for solving differential equations;
nonlinear differential equations and phase plane methods; perturbation
techniques; vector space of functions, Hilbert spaces and orthonormal
bases; Fourier seres and integrals; Sturm-Liouville theory; Fourier
and Laplace transforms; separation of variables for solving the linear
differential equations of physics, the heat, wave, and Laplace
equations.
More information is on the
course web page.
Text: M.Greenberg, Advanced Engineering Mathematics
(second edition); Prentice, 1998 (ISBN# 0-13-321431-1))
Students who are not prepared for this course should consider
taking 640:421.
642:550
Linear Alg & Applications R. Goodman
HLL 425 MW6; 5:00-6:20
Note: This course is intended for graduate students in
science, engineering and statistics.
This is an introductory course on vector spaces, linear
transformations, determinants, and canonical forms for matrices (Row
Echelon form and Jordan canonical form). Matrix factorization methods
(LU and QR factorizations, Singular Value Decomposition) will be
emphasized and applied to solve linear systems, find eigenvalues, and
diagonalize quadratic forms. These methods will be developed in class
and through homework assignments using MATLAB. Applications of linear
algebra will include Least Squares Approximations, Discrete Fourier
Transform, Differential Equations, Image Compression, and Data-base
searching.
Text: Gilbert Strang, Linear Algebra and its Applications,
4th edition, ISBN #0030105676, Brooks/Cole Publishing, 2007
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 423 T 5; 3:20-4:40 & F 4; 1:40-3:00
642:563
Statistical Mechanics
J. Lebowitz
HLL 525 MThF; 3:20-4:40
Statistical mechanics aims to relate the behavior of macroscopic objects to the dynamics of their constituent microscopic entities. Examples include the approach to equilibrium in isolated systems, properties of non-equilibrium stationary states of open systems, and the nature of phase transitions in equilibrium systems. Surprisingly, many aspects of these phenomena can be captured in greatly simplified models of the microscopic world, such as lattice gases evolving via simple local stochastic rules. These aspects emerge as collective properties of large aggregates which are independent of many details of the microscopic dynamics. In this course I will try to connect rigorous results on model systems of varying degrees of idealization with more heuristic arguments about the behavior of real macroscopic systems.
642:573
Numerical Analysis M. Vogelius
SEC 203 TTh 6; 5:00-6:20
This is the first part of a general survey of the basic topics in numerical
analysis -- the study and analysis of numerical algorithms for approximating
the solution of a variety of generic problems which occur in applications.
In the fall semester, we will consider the approximation of functions by
polynomials and piecewise polynomials, numerical integration, and the
numerical solution of initial value problems for ordinary differential
equations, and see how all these problems are related.
In the spring semester (642:574), we will study the numerical solution of
linear systems of equations, the approximation of matrix eigenvalues and
eigenvectors, the numerical solution of nonlinear systems of equations,
numerical techniques for unconstrained function minimization, finite
difference and finite element methods for two-point boundary value problems,
and finite difference methods for some model problems in partial
differential equations.
Despite the many solution techniques presented in elementary calculus and
differential equations courses, mathematical models used in applications
often do not have the simple forms required for using these methods.
Hence, a quantitative understanding of the models requires the use of
numerical approximation schemes. This course provides the mathematical
background for understanding how such schemes are derived and when they
are likely to work.
To illustrate the theory, in addition to the usual pencil and paper
problems, some short computer programs will be assigned. To minimize
the effort involved, however, the use of Matlab will be encouraged.
This program has many built in features which make programming easy, even
for those with very little prior programming experience.
PREREQUISITES: Advanced calculus, linear algebra,
and familiarity with differential equations.
642:582
Combinatorics I J. Kahn
HLL 423 TF 2; 10:20-11:40
This is the first part of a two-semester course surveying basic topics in com-
binatorics. The second semester will be taught by Van Vu. Topics in the
first semester will be some subset of:
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 signi¯cant overlap with
the course. It and other relevant books will be on reserve.)
1
642:587
Sel Topics in Combinatorics M. Saks
HLL 425 MTH 2; 10:20-11:40
The basic problem to be investigated is: given a finite metric space, represent
each point of the space by a low dimensional vector so that
the distances between points in the space are well approximated
by the distance between the vectors in some appropriate norm
(usually L_2, L_1 or L_{infinity}). Typically there is a tradeoff
between the dimension required and the closeness of approximation.
This general problem gives rise to very interesting mathematical
questions, as well as surprising applications to combinatorial
optimization (e.g., multicommodity flow, graph bandwidth).
(See, e.g., this article)
642:593
Math Fdns Ind Eng T. Butler
HLL 525 MTh 2; 10:20-11:40
This course is offered specifically for graduate students in Industrial
Engineering.
Proof Structure for the Development of Concepts Based on the Real Numbers
642:613
Mathematical Foundations of Systems Biology E. Sontag
HLL 260 W; 10:00-1:00
There are a very large number of possible topics to choose from, and the
syllabus will evolve based on student's interest and input. Some of the
possible topics include the dynamics of cell signaling networks including
memories, switches, and adaptation, oscillators, chemotaxis, pattern
formation, neural transmission, synthetic biology, reverse engineering of gene
and protein networks, Markov chains for population models, epidemiology, and
the mathematics behind phylogenetic trees, sequence alignment methods, and
shotgun DNA sequencing.
In addition to mathematics students, the level of the course will be
appropriate for graduate students from BioMaPS, various Engineering
departments, chemistry, life sciences, pharmacy, physics, statistics, and
computer science.
Prerequisites: working familiarity with linear algebra, differential
equations, and basic probability, at the level of an advanced undergraduate or
beginning graduate student.
More information may be found at the
course web page.
642:621 Financial Math P. Feehan
HLL 705 T 7& 8; 6:40-9:30
This course is an introduction to modern mathematical analysis of
financial markets and financial instruments. The finance concepts,
such as financial derivatives and no arbitrage, and the basic
probabilistic ideas for their analysis will be introduced first and
briefly for discrete time models. After this introduction, the course
will move to continuous time models. It will cover Brownian motion,
martingales, stochastic calculus, diffusions and their related partial
differential equations, and apply these to modeling financial markets
and to the valuation of derivatives. Major goals are the Black-Scholes
option pricing formula, risk neutral pricing, hedging, and the study
of American and exotic options.
More information may be found at the
course web page.
642:661
Topics Math Physics
G. Gallavotti
TBA
The lectures wll be extracted from published literature available on my
website at http://www.math.rutgers.edu/~giovanni
Publisher: American Mathematical Society; 3rd edition (March 29, 2006)
ISBN-13: 978-0821839621
1: Fundamental Concepts; 2: Complex Lines Integrals; 3: Applications
of the Cauchy Integral; 4: Meromorphic functions and Residues; 5: The
Zeros of a Holomorphic Function; 6: Holomorphic Functions as Geometric
Mappings.
If time permiting, we plan to have a reading/workshop session
at the end of this course (two or more weeks) on the topic
"Collapsed $3$-manifolds with sectional curvature bounded from below."
The plan is to cover chapters 1, 2, and the first part of chapter 3 of
Hatcher's book. Topics include fundamental group, Van Kampen's Theorem,
covering spaces, simplicial and singular homology, cohomology,
Brouwer's fixed-point theorem, the Borsuk-Ulam theorem, and the
Jordan-Brouwer separation theorem.
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.
Categories and functors: Introduction
Prerequisites:
Topics the students should know, together with the courses
in which they are taught at Rutgers, are: Introductory Linear
Algebra (640:250); Multivariable Calculus (640:251);
Elementary Differential Equations (640:244 or 640:252);
Advanced Calculus for Engineering(Laplace transforms,
sine and cosine series, introductory pde)(640:421).
Grading: Written mid-term exam, homework, MATLAB projects, and a
written final exam.
Lecturer:
Prof. Roe Goodman, Hill 428, 445-3071
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 Continuity Axiom
Development of the Calculus of Functions of One Variable
Text: Bartle and Sherbert, Introduction to Real Analysis,
3rd Edition, Wiley & sons, 1992.
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Last Modified 9/11/2007.