Senior and Graduate Math Course Offerings Spring 2004 
MATH 4365: NUMERICAL ANALYSIS II (Section 10218)  
Time:  4:005:30 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 4397: Selected Topics in Mathematics  STOCHASTIC DIFFERENTIAL EQUATIONS (Section 12483 )  
Time:  1:002:30 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:  5:307:00 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 onedimensional FFT, the 2D Fourier transform and the Finite 2D FT. Line Integrals and Projections (an introduction to the Radon transform), Fourier Slice theorem, reconstruction algorithms for Xray computerized tomography, computer implementations, reconstruction from fan projections, 3D 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 onesemester 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:  J. Morgan 
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, SpringerVerlag (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:  10:0011:00 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, RadonNikodym theorem, product measures, absolute continuity and bounded variation, and the Riesz representation theorem. 
MATH 6323: COMPLEX ANALYSIS II (Section 12494)  
Time: 
11:0012:00 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: 
5:307:00 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 firstorder nonlinear equations, and gave an introduction to HamiltonJacobi 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 secondorder 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 57 of Evans's book. 
Remarks:  This is the second semester of a twosemester course. 
MATH 6343: TOPOLOGY II (Section 12427)  
Time: 
9:0010:00 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, PrenticeHall Publishers (required). 
Description:  This is the second semester of a twosemester
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:  10:0011:30 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:  E. Dean 
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 6374: NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS (Section 10301)  
Time:  11:301:00 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:  4:005:30 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:  2:304:00 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: CONTINUOUSTIME MODELS IN FINANCE (Section 12485 )  
Time:  4:005:30 pm, TTH, 347 PGH 
Instructor:  E. Kao 
Prerequisites:  MATH 6397 DiscreteTime Models in Finance. 
Text(s): 
Arbitrage Theory in Continuous Time, by Thomas Bjork, Oxford
University Press, 1998. ISBN 0198775180.
Financial Calculus: An Introduction to Derivative Pricing, by Martin Baxter and Andrew Rennie, Cambridge University Press, ISBN 0521552893, 1996. 
Description:  This is a continuation of the course enetitled "DiscreteTime Models in Finance." The course studies the roles played by continuoustime stochastic processes in pricing derivative securities. Topics incluse stochastic calculus, martingales, the BlackScholes model and its variants, pricing market securities, interest rate models, arbitrage, and hedging. 
MATH 6397: THEORETICAL COMPUTATIONAL NEUROSCIENCES II (Section 12486)  
Time:  4:005:30 pm, TTH, 350 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:  ? PGH 
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):  ? 
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. Some information about
this program is available on the web pages of
the instructor or Craig Larson .
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:  10:0011:30 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, SpringerVerlag, 1988. 
Description:  This course centers on:

Math 7351: GEOMETRY OF MANIFOLDS II (Section 12430)  
Time:  12:001:00 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:  1:002:30 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 mixedhybrid 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 diffisionconvection 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:  1:002:30 pm TTH, 315 PGH 
Instructor:  R. Hoppe 
Prerequisites:  Calculus, Lienar Algebra, and Numerical Analysis 
Text(s):  S.J. Wright, PrimalDual InteriorPoint Methods, Society for
Industrial and Applied Mathematics, Philadelphia, 1997.
Further references:

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 ,,oneshot techniques'', also called ,,allatonce 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 stateoftheart methods are variations of SQPtechniques (\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 primaldual approaches based either on interior point methods or active set strategies. 
*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.