Senior and Graduate Math Course Offerings Spring 2004

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. 

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.

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.  

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
Text(s): Linear Algebra and Differential Equations, by Golubitsky and Dellnitz, Brooks/Cole.  

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
Text(s): Mathematical Modeling, by F. R. Giordano, M.D. Weir and W.P. Fox, Thomson Brooks/Cole, 2003.

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.  

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)

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. 


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)

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.  

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)

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 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.  

Time: 11:30-1:00 pm, TTH, 315 PGH  
Instructor: R. Glowinski
Prerequisites: Numerical analysis and an undergraduate PDE course.

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. 

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.  

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.

Time: 4:00-5:30 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: 10:00-11:30 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.

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.

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.

    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.  

    *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.