Senior and Graduate Math Course
Offerings Spring 2003
| MATH 4315: GRAPH THEORY WITH APPLICATIONS
(Section 09798) |
| Time: |
4:00-5:30 pm, TTH, Room 140-SR |
| Instructor: |
S. Fajtlowicz |
| Prerequisites: |
Discrete Mathematics. |
| Text(s): |
The course will be based on the instructor's notes. |
| Description: |
Planar graphs and the Four-Color Theorem. Trivalent
planar graphs with applications to fullerness - new forms of carbon. Algorithms
for Eulerian and Hamiltonian tours. Erdos' probabilistic method with applications
to Ramsey Theory and , if time permits, network flows algorithms with applications
to transportation and job assigning problems, or selected problems about
trees. |
| MATH 4332: INTRODUCTION TO REAL
ANALYSIS II (Section 11794) |
| Time: |
1:00-2:30 pm, MW, Room 132-SR |
| Instructor: |
V. Paulsen |
| Prerequisites: |
Math 4331. |
| Text(s): |
Principles of Mathematical Analysis, Walter Rudin,
McGraw-Hill, Latest Edition (required); Real Analysis with Real Applications,K.R.
Davidson and A.P. Donsig, Prentice Hall (ISBN 0-13-041647-9) (recommended).
Note: Selected topics from the recommended text Will be introduced throughout
the year. |
| Description: |
This course begins by repeating some of the topics from
MATH 3333 including properties of the real numbers and of continuous functions,
but done at a deeper level. The concepts of metric spaces, countability,
convergence of sequences and series of functions are all introduced. In
MATH 4332, we will study functions of several variables, polynomial approximation
and, time permitting, Fourier series.. |
| MATH 4333: ABSTRACT ALGEBRA (Section 09799) |
| Time: |
11:30-1:00 TTH, Room 345 PGH |
| Instructor: |
J. Hausen |
| Prerequisites: |
Math 3330 or equivalent. |
| Text(s): |
Rings, Fidel's, and Groups, An Introduction to Abstract Algebra, By R.B.J.T. Allenby, Routledge, Chapman and
Hall, Inc., New York, 1983. |
| Description: |
Topics from Ring Theory, Field Theory and Group Theory
including polynomial rings, quotient rings, field extensions and
finite fields. Structure theorems from group theory as time
permits. |
| MATH 4336: PARTIAL DIFFERENTIAL
EQUATIONS II (Section 11795) |
| Time: |
4:00-5:30 pm, MW, Room 345 PGH |
| Instructor: |
B. Keyfitz |
| Prerequisites: |
Ordinary Differential Equations (3331 or equivalent)
and Math 4335 or consent of instructor |
| Text(s): |
Partial Differential Equations: An Introduction, W.A. Strauss, Wiley, New York, latest edition (required), plus notes by
instructor. |
| Description: |
This is the second semester of a year-long first course in
partial differential equations. The objective of the course
is to explore the extensive connections between PDE and
mathematical analysis, as well as the role of PDE in applied
mathematics. The first semester focused on first-order
equations and the method of characteristics, and on linear
second-order equations in two independent variables:
types of equations, d'Alembert's method and separation of
variables.
In the second semester, we will study energy methods and
maximum principles for linear and nonlinear second-order
equations; boundary-value problems; hyperbolic systems;
Green's functions and Duhamel's principle; approximation
methods; and applications in fluid dynamics, continuum
mechanics, electromagnetics and ecology. |
| MATH 4351: DIFFERENTIAL GEOMETRY II
(Section 09800) |
| Time: |
12:00-1:00 MWF, Room 345 PGH |
| Instructor: |
M. Ru |
| Prerequisites: |
Math 4350. |
| Text(s): |
Elementary Differential Geometry by Andrew
Pressley, Springer-Verlag. |
| Description: |
This is the second part of the year-long course, which
introduces the theory of the geometry of curves
and surfaces in three-dimensional space using calculus techniques.
Topics in the second semester include:
Gaussian curvature and the Gaussian map, geodesics, minimal surfaces,
Gauss' Theorem Egrigium. Depending on the interests of the
audience, some advanced topics will also be discussed.
|
| MATH 4362: ORDINARY DIFFERENTIAL EQUATIONS
(Section 11821) |
| Time: |
1:00-2:30 pm, TTH, Room 203 AH |
| Instructor: |
P. Walker |
| Prerequisites: |
MATH 3331 and 3334. |
| Text(s): |
An Introduction to Ordinary Differential Equations, by
Earl A. Coddington, Dover Paperback, ISBN 0486659429. |
| Description: |
Existence, uniqueness, and continuity of solutions
of single equations and systems of equations; other topics at the discretion of the instructor. |
| MATH 4365: NUMERICAL ANALYSIS II
(Section 09801) |
| Time: |
5:30-7:00 pm, MW, Room. 350 PGH |
| Instructor: |
T. Pan |
| Prerequisites: |
Math 2331 (Linear Algebra), Math 3331 (Differential
Equations). Ability to do computer assignments in
either FORTRAN or C. The first semester is not a
prerequisite. |
| Text(s): |
Numerical Analysis (Seventh edition), R.L. Burden
and J.D. Faires. |
| Description: |
We will develop and analyze numerical methods for approximating the
solutions of common mathematical problems. The emphasis this semester
will be on the iterative methods for solving linear systems, approximation
theory, numerical solutions of nonlinear equations, iterative methods for
approximating eigenvalues, and elementary methods for ordinary differential
equations with boundary conditions and partial differential equations.
This is an introductory course and will be a mix of mathematics and
computing. |
| MATH 4377: ADVANCED LINEAR ALGEBRA
(Section 09802) |
| Time: |
1:00-2:30 pm, TTH, Room 309 PGH |
| Instructor: |
M. Friedberg |
| Prerequisites: |
Math 2431 and minimum 3 hours of 3000 level math. |
| Text(s): |
Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition,
Prentice-Hall. |
| Description: |
Fields, vector spaces, subspaces, linear transformations, matrices and applications to simultaneous
equations. |
| MATH 4378: ADVANCED LINEAR ALGEBRA II
(Section 09803) |
| Time: |
2:30-4:00 pm, TTH, Room 309 PGH |
| Instructor: |
J. Johnson |
| Prerequisites: |
Math 4377 |
| Text(s): |
Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition,
Prentice-Hall. |
| Description: |
Topics to be covered in this course include linear equations,
vector spaces, polynomials, linear transformations, and matrices. |
| MATH 4397: Selected Topics in Mathematics - NONLINEAR DYNAMICS
II (Section 11854) |
| Time: |
11:00-12:00 am, MWF, Room 134 SR. |
| Instructor: |
K. Josic |
| Prerequisites: |
MATH 4340 |
| Text(s): |
Nonlinear Dynamics and Chaos by Steven H. Strogatz, Perseus Books (2000). |
| Description: |
Nonlinear terms enter into equations for many systems.
The dynamics of such systems varies from relatively trivial, to extremely
complex. Still many - often surprisingly general - principles can be used
to describe nonlinear dynamics and chaos. This course will cover the
theoretical foundations, numerical techniques, and some current
applications of nonlinear dynamics. |
| MATH 6303: MODERN ALGEBRA II (Section 09858) |
| Time: |
1:00-2:30 pm, TTH, Room 315 PGH |
| Instructor: |
K. Kaiser |
| Prerequisites: |
Math6302 or consent of instructor. |
| Text(s): |
Algebra, Thomas W. Hungerford, Springer-Verlag (required). I will also circulate my own class notes. |
| Description: |
During the first semester we will cover the basic theory
of groups, rings and fields with strong emphasis on principal ideal domains. We will discuss the most important algebraic constructions from a universal
algebraic as well as from a categorical point of view.
The second semester will be mainly on modules over principal ideal domains,
Sylow theory and field extensions. |
| MATH 6321: FUNCTIONS OF A REAL
VARIABLE II (Section 09878) |
| Time: |
2:30-4:00 pm, TTH, Room 202 AH |
| Instructor: |
A. Torok |
| Prerequisites: |
Math 6320 or consent of instructor. |
| Text(s): |
We will use mainly Real Analysis, H.L.
Royden, (3rd Edition), Prentice Hall.
Other possible text to be used are: Measure and Integration,
M.E.Munroe; Lebesgue Integration on Euclidean Space, Frank
Jones; Foundations of Modern Analysis, Avner Friedman; Real Analysis:
Modern Techniques and their Applications, G.B. Folland.
|
| Description: |
This is the continuation of Math 6320. The topics to be discussed
in Math 6320-6321 include: Lebesgue measure and integration, differentiation of real functions, functions of bounded variation, absolute
continuity, the classical L^p spaces, general measure theory, and
elementary topics in functional analysis. |
| MATH 6325: DIFFERENTIAL
EQUATIONS II (Section 11798) |
Time: |
10:00-11:30 am, TTh, Room 345 PGH |
| Instructor: |
M. Golubitsky |
| Prerequisites: |
A course in ordinary differential equations
which includes some of the basic theory (such as MATH 6324) and an
(undergraduate) course in linear algebra. |
| Text(s): |
J. Guckenheimer and P. Holmes, Nonlinear
Oscillations, Dynamical Systems, and Bifurcations of Vector
Fields, Appl. Math. Sci. 42, Springer-Verlag. |
| Description: |
This course will stress the local bifurcation
theory of dynamical systems through codimension two, including
Liapunov-Schmidt and center manifold reductions, normal form
theory, steady-state bifurcation, Hopf bifurcation, Takens-Bogdanov
bifurcations and other codimension two mode interactions.
Some aspects of chaotic dynamics, including Melnikov's method and Smale
horseshoes, will be presented. Emphasis will be on the mathematical
ideas (rather than on formal proofs) and how to apply these ideas. |
| MATH 6361: APPLICABLE ANALYSIS II
(Section 09881) |
| Time: |
4:00-5:30 pm, MW, Room 309 PGH |
| Instructor: |
G. Auchmuty |
| Prerequisites: |
MATH 4332 [Math 6360 is not required. |
| Text(s): |
Linear Operator Theory in Science and Engineering,
Naylor and Sell, Springer-Verlag. |
| Description: |
Topics to be covered will include:
(i) An introduction to spaces.
(ii) Theory of continuous linear operators on H.
(iii) Solvability of linear operator equations; including linear integral equations.
(iv) Spectral theorem for compact self-adjoint operators.
(v) Introduction to boundary value problems for ordinary differential equations and Green's functions
.
The 1-d Fourier transform and its properties.
|
| MATH 6371: NUMERICAL ANALYSIS II
(Section 11818) |
| Time: |
4:00-5:30 pm, TTH, Room 309 PGH |
| Instructor: |
E. Dean |
| Prerequisites: |
Graduate standing or consent of the instructor.
This is the second
semester of a two semester course. The first semester is not a
prerequisite. |
| Text(s): |
Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch
(Springer-Verlag, (2nd or 3rd) ed.) |
| Description: |
We will develop and analyze numerical methods for approximating the
solutions of common mathematical problems. The emphasis this semester
will be on iterative methods for numerical linear algebra, including
large systems of linear equations and eigenvalue problems, numerical
methods for ordinary differential equations, and a brief introduction
to numerical methods for partial differential equations. This course will
prepare students for other graduate level courses in numerical mathematics. |
| MATH 6374: NUMERICAL PARTIAL
DIFFERENTIAL EQUATIONS (Section 11796) |
| Time: |
5:30-7:00 pm, MW, Room 301 AH |
| Instructor: |
J. He |
| Prerequisites: |
Numerical analysis and an undergraduate PDE course. |
| Text(s): |
A First Course in the Numerical Analysis of Differential Equations
by Arieh Iserles, Cambridge University Press, 1996.
ISBN: 0521556554 |
| Description: |
This course presents the methods and underlying theory used in the
numerical solution of partial differential equations.
It emphasizes finite difference methods and spectral
methods for simple examples of elliptic, parabolic
and hyperbolic equations; finite element and finite volume are
discussed at a level that is understandable to beginning graduate
students in engineering and in the sciences.
In detail, topics covered include:
1) review of numerical solution of ordinary differential equations
by
multistep and Runge-Kutta methods;
2) finite difference and finite elements techniques for the
elliptic equations:
multivariate differences, Galerkin approximation, maximum
principle,
shape function, matrix structure;
3) algorithms to solve large sparse algebraic systems: sparse
Gaussian
elimination, classical and conjugate gradient iterations,
multigrid iterations;
4) and methods for parabolic and hyperbolic differential equations
and
techniques of their analysis: explicit and implicit schemes,
Lax-Richtmyer stability, von Neumann analysis,
method of lines approach and relation to stiff ODEs,
dissipation and dispersion error, operator splitting,
convection-diffusion and reaction-diffusion equations.
The point of departure is mathematical but the exposition strives
to maintain a balance among theoretical, algorithmic and applied aspects of
the subject.
|
| MATH 6378: BASIC SCIENTIFIC COMPUTING (Section 11788) |
| Time: |
5:30-7:00 pm, TTH, Room 350 PGH |
| Instructor: |
R. Sanders |
| Prerequisites: |
Elementary Numerical Analysis. Knowledge of
C and/or Fortran. Graduate standing or consent of instructor. |
| Text(s): |
High Performance Computing, O'Reilly, Kevin Dowd & Cbarles Severance, the 2nd edition. |
| Description: |
Fundamental techniques in high performance scientific
computation. Hardware architecture and floating point performance. Pointers
and dynamic memory allocation. Data structures and storage
techniques related to numerical algorithms. Parallel programming
techniques. Code design. Applications to numerical algorithms
for the solution of systems of equations, differential equations
and optimization. Data visualization. This course also provides an introduction
to computer programming issues and techniques related to large scale
numerical computation. |
| MATH 6383: PROBABILITY MODELS
AND MATHEMATICAL STATISTICS II(Section 11800) |
| Time: |
2:30-4:00 pm, TTH, Room |
| Instructor: |
E. Kao |
| Prerequisites: |
Math 6382 or equivalent. |
| Text(s): |
Statistical Inference, by George Casella and Roger Berger,
Second Edition, Duxbury Press, 2002 |
| Description: |
This course is an introduction to statistical inference. In this
course we will cover random samples, principles of data reduction, point
estimation, hypothesis testing, interval estimation, asymptotic methods, analysis of variance and regression. Emphases will be placed on
conceptual development of basic ideas. Whenever applicable, we will also
address issues relating to computational statistics and topics involving
financial econometrics. |
| MATH 6395: INTRODUCTION TO COMPLEX
ANALYSIS AND GEOMETRY II (Section 11799) |
| Time: |
11:30-1:00, TTH, Room 315 PGH |
| Instructor: |
S. Ji |
| Prerequisites: |
Graduate standing or consent of instructor. |
| Text(s): |
none |
| Description: |
We shall introduce geometric measure theory (currents,
distributions, etc.), the L^2 vanishing theorem, and their applications in
complex analysis and geometry. |
| MATH 6395: WAVELET ANALYSIS II
(Section 11793) |
| Time: |
4:00-5:30 pm, TTH, Room350 PGH |
| Instructor: |
M. Papadakis |
| Prerequisites: |
MATH 6320, MATH 6395 (Wavelet analysis I ). |
| Text(s): |
E. Hernandez and G. Weiss, A first course on wavelets, CRC
Press, and P. Wojtaszyck's: A mathematical introduction to wavelets,
Cambridge University press. |
| Description: |
Orthogonality conditions, Characterizations of
scaling functions, low pass filters and wavelets.
The construction of Daubechies' compactly supported wavelets, transfer
operators, time-domain fractal constructions
of scaling functions. Translations, dilations and multiresolution in one
and multidimensions (definitions).
Wavelet and scaling sets. Biorthogonal multiresolution analysis,
multiscaling functions and multiwavelets. Multiresolution analysis
on intervals. Uncertainty principles, Weyl-Heisenberg frames, Frames with
erasures in finite dimensional spaces.
Non-separable multiresolution analysis. Wigner-Ville transforms, the
ambiguity function and time-frequency atoms,
Wavelet packets. Multiplexing and demultiplexing filter banks and
applications of frame designs in cellular technology. |
| MATH 6395: PDES AND APPLICATIONS
(Section 13313) |
| Time: |
2:30-4:00 pm, MW, Room 106 AH |
| Instructor: |
C. Suncica |
| Prerequisites: |
Multivariable Calculus, Real and Complex Analysis.
|
| Text(s): |
Textbook: None required. (Texbooks that will be partially used
are: Strauss's PDEs, R. LeVeques's "Conservation Laws",
Renardy and Rogers' "PDEs", Research Papers) |
| Description: |
Review of basic linear PDEs. Introduction to fundamentals
of fluid mechanics (basic equations of motion: continuity, momentum, energy,
vorticity). Incompressible/compressible flow examples. Analysis of
quasilinear PDEs with the focus on hyperbolic conservation laws.
Basic numerical methods.
Special topics in modeling, analysis and numerical simulation arising
in the study of blood flow through compliant blood vessels. |
| Math 6397: NUMBER THEORY (OnLine course) (Section 13423) |
| Time: |
?, Rm. ? |
| Instructor: |
M. Ru |
| Prerequisites: |
None |
| Text(s): |
Discovering Number Theory, by Jeffrey J. Holt and John
W. Jones, W.H. Freeman and Company, New York, 2001. |
| Description: |
Number theory is a subject that has interested people for thousand of
years. This course is a one-semester long graduate course on number
theory. Topics to be covered include divisibility and factorization,
linear Diophantine equations, congruences, applications of
congruences, solving linear congruences, primes of special forms, the
Chinese Remainder Theorem, multiplicative orders, the Euler function,
primitive roots, quadratic congruences, representation problems and
continued fractions. There are no specific prerequisites beyond basic
algebra and some ability in reading and writing mathematical proofs.
The method of presentation in this
course is quite different. Rather than simply presenting
the material, students first work to discover many of the important
concepts and theorems themselves. After reading a brief
introductory material on a particular subject, students work through
electronic materials that contain additional background, exercises,
and Research Questions, using either mathematica, maple, or HTML with
Java applets. The research questions are typically more open ended
and require students to respond with a conjecture and proof.
We then present the theory of the material which the students
have worked on, along with the proofs. The homework problems contain
both computational problems and questions requiring proofs. It is
hoped that students, through this course, not only learn the material,
learn how to write the proofs, but also gain valuable insight into
some of the realities of mathematical research by developing the
subject matter on their own.
|
| Math 6397: STATISTICS (OnLine course) (Section 13424) |
| Time: |
?, Rm. ? |
| Instructor: |
C. Peters |
| Prerequisites: |
Math 1432: Calculus II or consent of instructor. |
| Text(s): |
Applied Statistics with Microsoft Exceel, by Gerald Keller, Duxbury 2001.
ISBN: 0534382029 |
| Description: |
Fundamentals of probability and statistics. Descriptive and inferential
methods of statistics.
|
| MATH 7321: FUNCTIONAL ANALYSIS II
(Section 11797) |
| Time: |
12:00-1:00 pm, MWF, Room 128 SR |
| Instructor: |
D. Blecher |
| Prerequisites: |
Graduate standing. |
| Text(s): |
There will be a xeroxed set of lecture notes available. There are several good books on the market, such as Pedersen's
Analysis Now or Conway's Courses in Functional Analysis. |
| Description: |
The first semester will be a leisurely and general presentation,
starting from scratch, of the basic facts in Linear Analysis,
Banach spaces and Hilbert space. The second semester will be a
more technical development of the theory of linear operators
on Hilbert space. We will also cover topics which the students
request. We will prop ably only make it to the middle of section
III in the first semester. |
| MATH 7394: SPLITTING METHODS IN COMPUTATIONAL MECHANICS AND PHYSICS (Section
11711) |
| Time: |
2:30-4:00 pm, TTH, Room 314 PGH |
| Instructor: |
R. Glowinski |
| Prerequisites: |
|
| Text(s): |
None, since the course is self-contained. However, a preliminary
reading of Neumann Control of Unstable Parabolic Systems: Numerical
Approach , by R. Glowinski and J.W. He, published in the Journal of
Optimization Theory and Control, 96, pp. 1-55, can help. |
| Description: |
|
| MATH 7394: REACTION DIFFUSION
SYSTEMS II (Section 11820) |
| Time: |
10:00-11:30 am, TTH, Room 350 PGH |
| Instructor: |
W. Fitzgibbon |
| Prerequisites: |
The Preceding course, Reaction Diffusion Equations or
consent of instructor. |
| Text(s): |
Notes. |
| Description: |
This will be a seminar with a lecture format. The focus will be systems as opposed to
scalar reaction diffused advection equations. Applications as well as theory will be discussed.
|
| MATH 7396: MULTIGRID METHODS (Section 11710) |
| Time: |
1:00-2:30 pm, MW, Room 345 PGH |
| Instructor: |
R. Hoppe |
| Prerequisites: |
Graduate Standing |
| Text(s): |
1. J. K. Brainble, Multigrid Methods, Longman, Harlow, 1993.
2.Wolfgang Hackbusch; Iterative Solution of Large Sparse Systems of Equations.
Appl. Math. Sciences, Vol. 95, Springer, New York, 1993
ISBN 0-387-94064-2.
|
| Description: |
Multigrid methods and related multilevel approaches are the most
efficient solution techniques for the numerical solution of PDEs, integral
equations, and other kind of operator equations. The course starts from an
introduction to the basic principles and then proceeds to a detailed analysis of the
convergence behavior of various multigrid schemes. Both linear and nonlinear
problems will be addressed.
|
*NOTE: Teaching fellows are required to register for three regularly scheduled math courses for a total of 9 hours. Ph.D
students who have passed their prelim exam are required to register for one regularly scheduled math course and 6 hours of
dissertation.