| MATH 4315: GRAPH THEORY WITH APPLICATIONS
(Section 10212) |
| Time: |
4:00-5:30 pm, TTH, 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 10213) |
| Time: |
1:00-2:00 pm, MWF, 315 PGH |
| Instructor: |
M. Freidberg |
| Prerequisites: |
Math 4331. or consent of instructor. |
| 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: |
Sequences and series of functions, Contraction Mapping Principle, Implicit and Inverse Function Theorems,
Lebesgue Theory for the Real Line.
|
| MATH 4365: NUMERICAL ANALYSIS II
(Section 10218) |
| Time: |
4:00-5:30 pm, MW, 309 PGH |
| Instructor: |
T. Pan |
| Prerequisites: |
Math 2331 (Linear Algebra), Math 3331 (Differential
Equations). Ability to do computer assignments in
one of FORTRAN, C, Pascal, Matlab, Maple, Mathematica.
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 10219) |
| Time: |
5:30-7:00 pm, TTH, 347 PGH |
| Instructor: |
Kaiser |
| 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: |
Syllabus: Solving Linear Equations, general theory of vector spaces and
linear maps, algebra of polynomials, determinants. |
| MATH 4378: ADVANCED LINEAR ALGEBRA II
(Section 10220) |
| Time: |
4:00-5:30 pm, TTH, 348 PGH |
| Instructor: |
M. Friedberg |
| Prerequisites: |
Math 4377 or consent of instructor. |
| Text(s): |
Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition,
Prentice-Hall. |
| Description: |
Determinants, Elementary Canonical Forms, Rational and Jordan Forms, Inner Product Spaces as time permits. |
| MATH 4397: Selected Topics in Mathematics - STOCHASTIC DIFFERENTIAL EQUATIONS
(Section 12483 ) |
| Time: |
1:00-2:30 pm, TTH, 301 AH |
| Instructor: |
M. Nicle |
| Prerequisites: |
Math 3334
(Recommended: Math 4331) |
| Text(s): |
Stochastic Differential Equations: An Introduction with
Applications by B Oksendal ISBN 3540047581.
Reference: Stochastic Calculus: A Practical Introduction by R Durrett ISBN 0849380715. |
| Description: |
This is an introduction at the advanced
undergraduate/beginning
graduate level to the theory and applications of stochastic differential
equations.
A knowledge of measure theory is strongly recommended but not required.
First we will review probability spaces, random variables and stochastic
processes.
Martingale theory, the martingale representation theorem and the
Ito integral will be introduced, followed by an introduction to diffusion
theory
and Brownian motion. Applications will include mathematical finance (arbitrage
and
option pricing) and stochastic control.
|
| MATH 4397: Selected Topics in Mathematics -
MATHEMATICS OF COMPUTERIZED TOMOGRAPHY
(Section 12484) |
| Time: |
5:30-7:00 pm, MW, 301 AH |
| Instructor: |
M. Papadakis |
| Prerequisites: |
Math 4355 or an equivalent course from ECE, or COSCI. |
| Text(s): |
Principles of Computerized Tomography , by A.C. Kak and M.Slaney, SIAM, Applied Mathematics # 33,
2001. |
| Description: |
Understanding sampling and one-dimensional FFT, the 2-D Fourier transform
and the Finite 2-D FT. Line Integrals and Projections (an introduction to
the Radon transform), Fourier Slice theorem, reconstruction algorithms for
X-ray computerized tomography, computer implementations, reconstruction
from fan projections, 3-D reconstructions (helical acquisition of data);
application of tomography, emission CT, MRI and PET.
|
| Math 5383: NUMBER THEORY (OnLine course) (Section 12753) |
| Time: |
On Line |
| 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 5397: ORDINARY DIFFERENTIAL EQUATIONS (OnLine course)
(Section 12754) |
| Time: |
On Line |
| Instructor: |
G. Etgen |
| Prerequisites: |
|
| Text(s): |
Linear Algebra and Differential Equations,
by Golubitsky and Dellnitz, Brooks/Cole.
|
| Description: |
|
| Math 5397: REGRESSION AND LINEAR MODELS
(OnLine course) (Section 12896) |
| Time: |
On Line |
| Instructor: |
C. Peters |
| Prerequisites: |
Statistics or consent of instructor. |
| Text(s): |
Introduction to Linear Regression Analysis, 3rd Ed., by
Montgomery, Vining, and Peck, Wiley 2001. |
| Description: |
Simple and multiple linear regression, linear models with qualitative
variables, inferences about model parameters, regression diagnostics,
variable selection, other topics as time permits. The course will include
computing projects.
|
| Math 5397:
MATHEMATICAL MODELLING(OnLine course) (Section 12897 ) |
| Time: |
On Line |
| Instructor: |
SIBONA |
| Prerequisites: |
|
| Text(s): |
Mathematical Modeling, by F. R. Giordano, M.D. Weir and W.P. Fox, Thomson Brooks/Cole, 2003.
|
| Description: |
|
| MATH 6303: MODERN ALGEBRA II
(Section 10276) |
| Time: |
11:30 -1:00 pm, TTH, 309 PGH |
| Instructor: |
J. Johnson |
| Prerequisites: |
Math6302 or consent of instructor. |
| Text(s): |
Algebra, Thomas W. Hungerford, Springer-Verlag (required).
|
| Description: |
Topics from the theory of groups, rings,
fields with special emphasis on modules and universal constructions. |
| MATH 6321: FUNCTIONS OF A REAL
VARIABLE II (Section 10297) |
| Time: |
10:00-11:00 am, MWF, 121 SR |
| Instructor: |
V. Paulsen |
| Prerequisites: |
Math 6320 or consent of instructor. |
| Text(s): |
Foundations of Modern Analysis by Avner Friedman
|
| Description: |
This course is a continuation of 6320, beginning
where 6320 left off. We will cover a variety of topics in measure theory,
some deeper theorems in analysis and an introduction to functional
analysis.
The subjects covered include, Lebesgue's decomposition theorem,
Radon-Nikodym theorem, product
measures, absolute continuity and bounded variation, and the Riesz
representation
theorem. |
| MATH 6323: COMPLEX ANALYSIS II
(Section 12494) |
Time: |
11:00-12:00 am, MWF, 345 PGH |
| Instructor: |
S. Ji |
| Prerequisites: |
Math 6322 or equivalent. |
| Text(s): |
Introduction to complex analysis , Juniro Noguchi,
AMS (Translations of mathematical monographs, Volume 168). |
| Description: |
The course is an introduction to complex analysis. It covers the theory of holomorphic functions,
residue theorem, analytic continuation, Riemann surfaces, holomorphic mappings, and the theory of meromorphic functions.
This is the second semester of a two semester course. |
| MATH 6327: PARTIAL DIFFERENTIAL EQUATIONS II
(Section 12426) |
Time: |
5:30-7:00 pm, MW, 350 PGH |
| Instructor: |
B. Keyfitz |
| Prerequisites: |
6326, Fall 2003 or consent of instructor |
| Text(s): |
L. C. Evans, Partial Differential Equations, AMS. 1998. |
| Description: |
The fall semester was spent developing
representation formulas for some classical PDE: the
transport, potential, heat and wave equations. In
addition, we introduced the method of characteristics
for solving scalar first-order nonlinear equations,
and gave an introduction to Hamilton-Jacobi theory and
to a scalar conservation law. The fall ended with the
definition of Sobolev spaces and development of their
properties.
In the spring semester, we will complete the study of
Sobolev inequalities; prove existence of weak solutions
and develop regularity theorems for second-order elliptic
equations; study linear evolution equations and semigroup
theory; and, if time permits, cover some topics in
nonlinear equations. The basic material will come from
Chapters 5-7 of Evans's book. |
| Remarks: |
This is the second semester of a two-semester
course.
|
| MATH 6343: TOPOLOGY II
(Section 12427) |
Time: |
9:00-10:00 am, MWF, 350 PGH |
| Instructor: |
D. Blecher |
| Prerequisites: |
Math 6342 or consent of instructor. |
| Text(s): |
Topology, A First Course,
J. R. Munkres, Second Edition, Prentice-Hall Publishers (required). |
| Description: |
This is the second semester of a two-semester
introductory graduate course in topology. This is
a central and fundamental course and one which
graduate students usually enjoy very much! This
semester we will continue working through several of the
main topics in general topology, and perhaps a little algebraic
topology. We also will develop several examples that there
was no time for in the first semester. Thus in the
text (Munkres) we hope to
cover the main results up to Chapter 9, and a few important results
thereafter.
The final grade is aproximately based on a total score
of 400 points consisting of homework (100 points), a semester test
(100 points), and a final exam (200 points). The instructor may change this at his discretion. |
| MATH 6361: APPLICABLE ANALYSIS II
(Section 10299) |
| Time: |
10:00-11:30 am, TTH, 350 PGH |
| Instructor: |
J. Morgan |
| Prerequisites: |
Math 4331 and 4332. |
| Text(s): |
An Introduction to Variational Inequalities and Their
Applications , David Kinderlehrer and Guido Stampacchia, Academic Press.
|
| Description: |
Brief Description: Variational Inequalitities in finite dimensional
settings. Variational Inequalities in Hilbert space, the Lax Milgram
theorem, a theorem of Lions and Stampaccia, Applications to obstacle
problems and elliptic boundary value problems.
|
| MATH 6367: OPTIMIZATION II
(Section 12428) |
Time: |
5:30 -7:00 pm, TTH, 315 PGH |
| Instructor: |
PARKS |
| Prerequisites: |
Math 4331 and 4377 or consent of instructor. |
| Text(s): |
No textbook.
|
| Description: |
This is the second semester of a two semester course. The topics for
this second semester will include large scale optimization and time dependent
optimization. This course will be a mix of analysis and practicalities.
There will be no new textbook for the second semester. |
| MATH 6371: NUMERICAL ANALYSIS II
(Section 10300) |
| Time: |
4:00-5:30 pm, MW, 350 PGH |
| Instructor: |
J. He |
| Prerequisites: |
Graduate standing or consent of the instructor.
Students should have had a course on linear algebra and an introductory course
on analysis and
ODEs. This is the second semester of a two semester course. The first semester
is not a
prerequisite, but some familiarity with numerical solution of linear
system is assumed. |
| Text(s): |
Numerical Mathematics, Alfio Quarteroni, Riccardo Sacco, Fausto Saleri,
Springer Verlag, 2000, ISBN: 0387989595. |
| Description: |
This is the second semester of a two semester course. The focus in this
semester is
on approximation theory and numerical solution of ODEs. The applications of
approximation theory to interpolation, least-squares approximation,
numerical differentiation and Gaussian integration will be addressed. The
concepts of
consistency, convergence, stability for the numerical solution of ODEs will be
discussed. |
| MATH 6374: NUMERICAL PARTIAL
DIFFERENTIAL EQUATIONS (Section 10301) |
| Time: |
11:30-1:00 pm, TTH, 315 PGH |
| Instructor: |
R. Glowinski |
| Prerequisites: |
Numerical analysis and an undergraduate PDE course. |
| Text(s): |
|
| Description: |
|
| MATH 6378: BASIC SCIENTIFIC COMPUTING (Section 12429) |
| Time: |
4:00-5:30 TTH, 309 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 10302) |
| Time: |
2:30-4:00 pm, TTH, 309 PGH |
| Instructor: |
C. Peters |
| Prerequisites: |
Math 6382 or consent of instructor. |
| Text(s): |
No textbook required. |
| Description: |
Statistical
models, functionals and parameters, estimation theory, hypothesis testing,
likelihood methods. |
| MATH 6397: CONTINUOUS-TIME MODELS
IN FINANCE (Section 12485 ) |
| Time: |
4:00-5:30 pm, TTH, 347 PGH |
| Instructor: |
E. Kao |
| Prerequisites: |
MATH 6397 Discrete-Time Models in Finance. |
| Text(s): |
Arbitrage Theory in Continuous Time, by Thomas Bjork, Oxford
University Press, 1998. ISBN 0-19-877518-0.
Financial Calculus: An Introduction to Derivative Pricing, by Martin
Baxter and Andrew Rennie, Cambridge University Press, ISBN 0-521-552893, 1996. |
| Description: |
This is a continuation of the course enetitled "Discrete-Time Models in
Finance." The course studies the roles played by continuous-time stochastic
processes in pricing derivative securities. Topics incluse stochastic
calculus, martingales, the Black-Scholes model and its variants, pricing market
securities, interest rate models, arbitrage, and hedging. |
| MATH 6397:
THEORETICAL COMPUTATIONAL NEUROSCIENCES II
(Section 12486) |
| Time: |
4:00-5:30 pm, MW, 347 PGH |
| Instructor: |
Kresimir Josic |
| Prerequisites: |
One semester of Differential Equations. Student who did not take the first semester of this course
should talk to the instructor. |
| Text(s): |
None. It will use notes which will be posted on the web. |
| Description: |
This is the second semester of a two semester introductory
course to mathematical and computational neuroscience. Basic knowledge of
the biophysics of single
neurons is assumed. The goal of the course is to study how single neurons
behave in small to intermediate
sized networks using analytical and numerical techniques.
|
| MATH 6397:
Tutorials with Graffiti
(Section 13757) |
| Time: |
The course will be conducted by a discussion list |
| Instructor: |
Siemion Fajtlowicz |
| Prerequisites: |
Graduate standing in the College of Natural Sciences and Mathematics or consent of the instructor.
No previous knowledge of any of the subjects listed above is a prerequisite. |
| Text(s): |
No textbook is required. |
| Description: |
The purpose of this course is to learn the basics or to expand one's knowledge of selected mathematical
topics by working exclusively on conjectures of the computer program Graffiti.
A version of Graffiti will be used individually by participants to learn or to
expand their knowledge of one of several subjects of their own choice, including:
graphs theory, number theory, eigenvalues, benzenoids, diamondoids, and possibly hypergraphs.
One significant difference between the Texas ( the method developed by the UT Professor R. L. Moore)
style, and what we refer to as the Red Burton style, is that rather leading the participants to the
rediscovery of known results, the students will work exclusively on conjectures of selected versions
of Graffiti, without getting any hints whether these conjectures are true or false. This will create
a more realistic setting for acquisition of research experience. As in the past, active participants
will have an opportunity to discover new original results.
That does not mean that the course will be more difficult than other math classes. The only prerequisites
are graduate standing in the College of Natural Sciences and Mathematics or consent of the instructor.
No previous knowledge of any of the subjects listed above is a prerequisite; one advantage of running
Graffiti individually, is that the difficulty of conjectures can be tailored to a preferred level of
users, presumably making the class actually easier. The course will be conducted by email and a discussion list.
|
| MATH 7325: BIFURCATION THEORY II
(Section 12495) |
| Time: |
10:00-11:30 am, TTH, 347 PGH |
| Instructor: |
M. Golubitsky |
| Prerequisites: |
Some familiarity with ODE's, linear algebra, and undergraduate group theory will be useful.
Math 7324 is not a prerequisite - but some acquaintance with Hopf bifurcation will be assumed. This course should be accessible to
graduate students in science and engineering department, as well as in mathematics.
|
| Text(s): |
Required Text: M. Golubitsky and I. Stewart. The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical
Space, Birkhauser, Basel. Softcover edition, Due November 2003.
Recommended Text: M. Golubitsky, D.G. Schaeffer and I.N. Stewart. Singularities and Groups in Bifurcation Theory, Vol. II,
Springer-Verlag, 1988.
|
| Description: |
This course centers on:
- Equivariant bifurcation theory with preliminaries on representation theory.
- Periodic solutions with spatio-temporal symmetries.
- The dynamics of coupled cell systems with comments on animal gaits.
- Pattern formation.
- Structurally stable dynamics in symmetric differential equations.
|
| Math 7351: GEOMETRY OF MANIFOLDS II (Section 12430) |
| Time: |
12:00-1:00 pm, MWF, 345 PGH |
| Instructor: |
M. Field |
| Prerequisites: |
7350 Geometry of manifolds I or consent of instructor. |
| Text(s): |
Topology from the Differentiable Viewpoint
by John Milnor. Paperback, 1997. Available from here at the
princely price of $14.95 (plus shipping). |
| Description: |
Transversality and integration.
|
| Math 7396: NUMERICAL METHODS FOR ELLIPTIC EQUATIONS
(Section 12487) |
| Time: |
1:00-2:30 pm, MW, 345 PGH |
| Instructor: |
Y. Kuznetsov |
| Prerequisites: |
Graduate Courses on PDEs and Numerical Analysis |
| Text(s): |
None |
| Description: |
In this course we discuss new advanced discretization
methods and iterative solvers for elliptic partial differential
equations.The basic part of the course is devoted to the mixed
and mixed-hybrid finite element methods on arbitrary polygonal
and polyhedral meshes which may contain nonconvex cells and
locally refined cells.New multilevel preconditioners for
finite element systems on unstructured meshes is another impot-
ant part of the course.The methods are demonstrated on diffusion
and diffision-convection problems relevant to nowadays indust-
rial and environmental applications.We also consider applications
of mixed finite element methods to the Stokes problem and to the
linear elasticity equations.
|
| MATH 7396: NUMERICAL SOLUTION OF OPTIMIZATION PROBLEMS WITH PDE
CONSTRAINTS
(Section 12488) |
| Time: |
1:00-2:30 pm TTH, 315 PGH |
| Instructor: |
R. Hoppe |
| Prerequisites: |
Calculus, Lienar Algebra, and Numerical Analysis |
| Text(s): |
S.J. Wright, Primal-Dual Interior-Point Methods, Society for
Industrial and Applied Mathematics, Philadelphia, 1997.
Further references:
D. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods.
Athena Scientific , Belmont, MA, 1996
D. Bertsekas, Nonlinear Programing: 2nd Edition. Athena Scientific ,
Belmont, MA, 1999
J. Nocedal, S.J. Wright, Numerical Optimization . Springer,
Berlin-Heidelberg-New York, 1999
|
| Description: |
Optimization problems with PDE constraints typically arise in
structural optimization (shape and topology optimization), in the
optimal control of PDEs (distributed and/or boundary control), and
in inverse problems associated with PDEs (parameter
identification). There are two fundamental approaches: The first
one relies on an appropriate discretization of the optimization
problem leading to a large scale constrained finite dimensional
optimization problem for which efficient numerical solution
techniques have to be provided. The second approach does it the
other way around and begins with an optimization in function space
followed by a discretization and numerical solution of the
optimality conditions.
We will focus on ,,one-shot techniques'', also called
,,all-at-once methods'', where in contrast to traditional
approaches the numerical solution of the state equations is an
integral part of the optimization routine. Most of the
state-of-the-art methods are variations of SQP-techniques
(\underline{S}equential \underline{Q}uadratic
\underline{P}rogramming) which are iterative methods where each
iteration requires the solution of a constrained quadratic
optimization problem. In particular, we will address primal-dual
approaches based either on interior point methods or active set
strategies. |
| MATH 7397: FINANCIAL-ENERGY TIME SERIES ANALYSIS (Section
12489) |
| Time: |
10:00-11:30 am, TTH, 345 PGH |
| Instructor: |
E. Kao |
| Prerequisites: |
MATH 6383 Mathematical Statistics |
| Text(s): |
Analysis of Financial Time Series, by Ruey S. Tsay, Wiley, ISBN
0-471-415448, 2002. |
| Description: |
This is a data analysis course with a focus on financial and energy time series
for applications in pricing contingency claims, value-at-risk (VAR), and
portfolio optimization. Topics include autoregressive and moving averages
(ARMA) models, conditional heterscedastic (GARCH) models, nonlinear and
multivariate time series, estimation and analysis of jump diffusion models.
The computation software chosen for data analysis is S-Plus. Students enrolled
in the course are expected to have some proficiency in computer usage and a
strong interest in developing expertise in computational statistics as it is
applied to financial and energy data analysis. |
听听
*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.