Graduate Program

Senior and Graduate Course Offerings Fall 2001

For further information, contact the Department of Mathematics at 651 PGH, University of Houston, Houston, TX 77204-3476; Telephone (713) 743-3517 or e-mail to pamela@math.uh.edu.


MATH 4331: INTRODUCTION TO REAL ANALYSIS (Section 08332)
Time: 2:30-4 MW Rm. 348 PGH
Instructor: M. Friedberg
Prerequisites: Math 3334 (the catalogue lists Math 3333 or Math 3334 as the prerequisite, but we are in the process of changing that)
Text(s): Principles of Mathematical Analysis Walter Rudin, McGraw-Hill, 3rd Edition.
Description: This course will include an introduction to Topology of particular use in the field of analysis, a study of the convergence of sequences in metric spaces, continuity and its relation to topological properties and Riemann Stieltjes Integration.



MATH 4350: DIFFERENTIAL GEOMETRY (Section 10881)
Time: 4-5:30 MW Rm. 348 PGH
Instructor: A. Torok
Prerequisites: Math 2433, Math 2431 or equivalent.
Text(s): Differential Geometry of Curves and Surfaces, Manfredo P. do Carmo, Prentice Hall.
Description: This course will introduce the basic definitions of low-dimensional differential geometry. We will use these to describe curves and surfaces, exhibiting the interplay between local and global quantities. E.g.: the four-vertex and Fary-Milnor theorems for curves, the Gauss-Bonnet and Poincare index theorems for surfaces.



MATH 4362: ORDINARY DIFFERENTIAL EQUATIONS (Section 10880)
Time: 2:30-4 TTH Rm. 345 PGH
Instructor: P. Walker
Prerequisites: MATH 3333 or consent of instructor.
Text(s): None.
Description: An introduction to ordinary differential equations. Linear systems, Sturm-Liouville theory and related spectral theory for differential operators.



MATH 4364: NUMERICAL ANALYSIS (Section 08336)
Time: 4-5:30 TTH Rm. 127 SR
Instructor: E. Dean
Prerequisites: Math 2431 (Linear Algebra), Math 3431 (Differential Equations). Ability to do computer assignments in either FORTRAN or C (Cosc 1301 or 1304 or 2101.) This is the first semester of a two semester course.
Text(s): Numerical Analysis (latest 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 solving nonlinear equations, interpolation, numerical integration, direct methods for linear algebra, and initial value problems for ordinary differential equations. This is an introductory course and will be a mix of mathematics and computing.



MATH 4377: ADVANCED LINEAR ALGEBRA (Section 08338)
Time: 11:30-1 TTH Rm. 516 SR1
Instructor: J. Johnson
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: Topics to be covered in this course include linear equations, vector spaces, polynomials, linear transformations, and matrices.



MATH 4383: NUMBER THEORY (Section 08339)
Time: 10-11:30 MW Rm. 127 SR
Instructor: J. Hardy
Prerequisites: Math 3330.
Text(s): Elements of the Theory of Numbers, Dence and Dence, Academic Press.
Description: Divisibility, primes and their distribution, congruences, Fermat/Euler Theorems, Number-theoretic functions, primitive roots, Quadratic Reciprocity, Pythagorean Triples, and related topics. Note: Successful completion of Math 4383 and Math 4333 (Advanced Abstract Algebra, to be offered in Spring 2002) now satisfies the requirement of a 4000-level sequence for a bachelor's degree in mathematics.



MATH 4385: MATHEMATICAL STATISTICS (Section 10882)
Time: 1-2:30 MW Rm. 315 PGH
Instructor: C. Peters
Prerequisites: Math 3339 and Math 2431, or consent of instructor.
Text(s): Check with instructor.
Description: Linear models and multiple linear regression, regression diagnosis and model selection criteria, generalized linear models, time series (times permitting).



MATH 6198: TEACHING PRACTICUM (Section 11231)
Time: TO BE ARRANGED
Instructor: J. Hausen
Prerequisites: First year graduate assistantship.
Text(s): None.
Description: Course will meet two hours for the first half of the semester. Required of all first-year Teaching Fellows. Introduction to teaching and assisting at the University of Houston.



MATH 6298: INTRODUCTION TO COMPUTING RESOURCES (Section 08346)
Time: TO BE ARRANGED, Rm. 648 PGH
Instructor: A. Torok
Prerequisites: Graduate standing or consent of instructor
Text(s): Check with instructor.
Description: The purpose of this course is to familiarize students with the computer tools that are relevant for mathematical research in today's environment. It is intended primarily for graduate students and math majors, but it is useful for anybody interested in these topics. The topics we plan to discuss include the Unix and Linux operating systems, a multi-functional text editor (emacs), software for mathematical publications (TeX and its dialects), languages for formal and numerical computations (Maple, Mathemtica, Matlab), web-publishing (HTML) and Internet use (mail, electronic archives etc.). We will also mention a few principles of writing and presenting a mathematical paper.

The course will consists of weekly workshops accompanied by hands-on applications in the computer lab of the Math Department, followed by individual projects. These projects (e.g., typesetting a short mathematical paper, designing a web-page, writing programs in various languages) will give the students the opportunity to practice the notions they are being taught. The material used for this course will be either available on the web or handed out in class.



MATH 6302: MODERN ALGEBRA (Section 08347)
Time: 1-2:30 TTH Rm. 350 PGH
Instructor: J. Hausen
Prerequisites: Graduate standing or consent of instructor.
Text(s): Algebra, by Pierre Grillet, John Wiley & Sons, New York, 1999 (ISBN 9-471-25243-3).
Description: Group Theory, Ring Theory, and Module Theory with emphasis on universal properties and categorical aspects. Homework will be an integral part of the course. The continuation of the course, MATH6303, in the 2002 Spring Semester is expected to cover Category Theory and topics in Group, Ring and Module Theory at greater depth including tensor products and their applications.



MATH 6320: FUNCTIONS OF A REAL VARIABLE (Section 08357)
Time: 2:30-4 MW Rm. 127 SR
Instructor: D. Blecher
Prerequisites: An undergraduate real analysis sequence (Math 4331, 4332). A little topology and metric spaces would be useful.
Text(s): Lebesgue Integration on Euclidean Spaces, Frank Jones, Jones & Bartlett. Recommended reading: Real Analysis, H.L. Royden, (3rd Edition), Prentice Hall. Real and Complex Analysis, W. Rudin, McGraw Hill. Measure Theory, D. L. Cohn, Birkhauser.
Description: This is the first semester of a 2 semester sequence. This semester we will be developing the basic principles of measure and integration. This body of knowledge is essential to most parts of mathematics (in particular to analysis and probability) and falls within the category of "What every graduate student has to know". The one test and the final exam will be based on the notes given in class, and on the homework. After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter. The most important part of your task as a graduate student in this course is simply to reread the class notes making sure you understand everything. Please ask me about anything you don't follow. Less important is the assigned homework, a collection of problems, which are all worth trying, and some of which are important. Some of them are difficult, others are easy. Their purpose is to help you learn and internalize the material and techniques, or to touch on an aspect we don't have time in class for. You are encouraged to work with others, form study groups, and so on; however, do not simply copy homework unless you've tried hard to do the problem. Never copy homework you are required to turn in. Final grade is approximately 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. The syllabus for the first semester will cover some but not all of the following topics: Measures. Measurable functions. Integration. Convergence of sequences of functions. The Lp spaces. Signed and complex measures. Product measures and Fubini's theorem. Differentiation and integration.


MATH 6322: FUNCTIONS OF A COMPLEX VARIABLE (Section 10883)
Time: 2:30-4 TTH, 315 PGH
Instructor: Min Ru
Prerequisites: Math 3333.
Text(s): Introduction to complex analysis, Juniro Noguchi, AMS (Translations of mathematical monographs, Volume 168).
Description: This 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.


MATH 6326: PARTIAL DIFFERENTIAL EQUATIONS (Section 10885)
Time: 2:30-4 MW Rm. 345 PGH
Instructor: S. Canic
Prerequisites: Calculus I, II, III, Real Analysis.
Text(s): Partial Differential Equations, L.C. Evans, American Mathematics Society, Graduate Studies in Mathematics, Volume 19.
Description: Basic theory for linear PDEs; nonlinear first order PDEs; Holder spaces solutions; Sobolev spaces solutions.


MATH 6342: POINT SET TOPOLOGY (Section 11275)
Time: 4-5:30 TTH Rm. 309 PGH
Instructor: M. Friedberg
Prerequisites: Math 4331 and Math 4337 or consent of instructor.
Text(s): Topology, A First Course, J. R. Munkres, Second Edition, Prentice-Hall Publishers (required).
Description: This is the first semester of a two-semester introductory graduate course in Point Set Topology. We will consider the basic definitions of topology, topological spaces, bases for a topology and the basic properties of topological spaces: separation axioms, compactness, connectedness, local properties.


MATH 6360: APPLICABLE ANALYSIS (Section 08359)
Time: 12-1 MWF 345 PGH
Instructor: B. Keyfitz
Prerequisites: Math 4331-4332 and graduate standing.
Text(s): Introductory Real Analysis, N. Kolmogorov and S.V. Fomin, Dover, New York, 1975.
Description: This is the first semester of a two-semester course. In the first semester, we will cover Metric spaces: convergence, completeness and compactness. The Arzela-Ascoli theorem and applications to differential equations and the calculus of variations. The contraction mapping principle: inverse and implicit function theorems; applications in differential and integral equations.


MATH 6366: OPTIMIZATION THEORY (Section 10886)
Time: 4-5:30 MW 345 PGH
Instructor: G. Auchmuty
Prerequisites: Math 4332 or Math 6361 or consent of instructor. Also, some knowledge of multivariate analysis on Rn.
Text(s): Nonlinear Programming, Dimitri P. Bertsekas, Athena Scientific. (Order from athenasc@world.std.com or world.std.com)
Description: This course will treat the mathematical foundations of finite dimensional optimization theory and nonlinear programming. The prerequisites for the course is some knowledge of multivariate analysis on Rn;. Topics to be treated include
  • (i) Optimality conditions for unconstrained optimization, derivative tests,
  • (ii) theory of convex sets and functions,
  • (iii) optimality conditions for convex optimization,
  • (iv) theory of KKT conditions and multipliers,
  • (v) penality and augmented Lagrangian methods,
  • (vi) duality theory and saddle point analysis for convex optimization.
Throughout this course, specific examples of optimization problems will be treated. Also algorithms for the numerical approximation of solutions will be described and analyzed.


MATH 6370: NUMERICAL ANALYSIS (Section 08360)
Time: 5:30-7 TTH 348 PGH
Instructor: T. Pan
Prerequisites: Graduate standing or consent of instructor.
Text(s): Introduction to Numerical Analysis, J. Stoer and R. Bulirsch, Springer-Verlag, 2nd Edition, New York, 1993, ISBN 3-540-97878-X..
Description: We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. It will focus on approximation, numerical differentiation and integration, solving nonlinear equations, numerical linear algebra, and numerical solutions of ordinary differential equations.


MATH 6395: INTRODUCTION TO COMPLEX ANALYSIS AND GEOMETRY (Section 10917)
Time: 10-11:30 TTH 314 PGH
Instructor: S. Ji
Prerequisites: Graduate standing or consent of instructor.
Text(s): Principles of Algebraic Geometry, P. Griffiths and J. Harris.
Description: Holomorphic functions, complex manifolds, varieties, sheaves theory, vector bundles.


MATH 6397: MATHEMATICAL LOGIC WITH APPLICATIONS (Section 10887)
Time: 4-5:30 TTH 211 AH
Instructor: K. Kaiser
Prerequisites: Graduate standing in Mathematics or Computer Science.
Text(s): Logic for Applications, Anil Nerode and Richard A. Shore, 2nd Edition,, Graduate Texts in Computer Science,, Springer-Verlag. (ISBN 0-387-94893-7).
Description: Propositional Logic, Predicate Logic, Ultraproducts of Relational Systems with applications to Algebra and Analysis. The notion of "proof" will be based on tabeaux, versus the traditional method of rules and axioms. For the semantics, we cover some basic universal algebra and model theoretic constructions.


MATH 6397: AIR QUALITY MODELING I (Section 08366)
Time: 4:30-6:30 MW 134 SR.
Instructor: J. He
Prerequisites: Graduate standing.
Text(s): Fundamentals of Atmospheric Modeling, by M.Z. Jacobson, Cambridge Univ. Pr. 1998.
Description: The goal of this course is to introduce participants to the theories and applications of air quality modeling, specifically ozone formation and reactions in the atmosphere. Upon completion of the course, participants should understand the fundamental principles of ozone formation in the atmosphere and be able to understand and use numerical and computational science methods to study ozone science.


MATH 6398: KNOWLEDGE BASED ALGORITHMS (Section 10918)
Time: 4-5:30 TTH 347 PGH
Instructor: S. Fajtlowicz
Prerequisites: Graduate standing in the NS&M College or consent of the instructor.
Text(s): Check with instructor.
Description: The purpose of this course is to discuss computer programs capable of making mathematical conjectures and scientific hypotheses. These ideas are based mostly on experiences with the computer program Graffiti whose conjectures inspired a number of papers, some by the most prominent mathematicians and more recently a few papers in chemistry. I will discuss also some other attempts to develop such programs including works of Nobel laureate Herbert Simon and Francis Crick and a critical approach of Roger Penrose in the context of what came to be known as Al debate. A version of Graffiti will be used by students to learn or to expand their knowledge of one of several possible subjects of their own choice. This will be done in so called Texas style - the method developed by UT Professor L.A. Moore. One significant difference will be that rather than to be led to discovery of known results, the students will work exclusively on conjectures of Red Burton - an educational version of Graffiti, often without getting any hints whether these conjectures are true or false. This will create a more realistic scenario for a graduate and undergraduate research experience. Active students will have the opportunity to discover new original results and possibly even new scientific hypotheses. I will also discuss various general models of computation including Universal Turing Machines and illustrate the domain-independence methods of Graffiti by showing that the program can make conjectures about arbitrary programs and I will show how this hypothetical knowledge can result in faster and more reliable computer programs. More information about the program can be found at http://cms.dt.uh.edu/faculty/delavinae/research/wowref.htm or http://www.math.uh.edu/~clarson/


MATH 7394: TOPICS IN MATHEMATICAL BIOLOGY (Section 10889)
Time: 10-11:30 TTH 203 PGH
Instructor: W. Fitzgibbon
Prerequisites: Consent of instructor.
Text(s): Mathematical Biology, J.D. Murray, Springer-Verlag (RECOMMENDED).
Description: This is envisioned to be a two semester lecture-seminar developed from topics in J.D. Murray's text, Mathematical Biology. No previous knowledge of biology will be presupposed. The instructor will provide a brief discussion of the underlying biological mechanisms required to understand the construction of a mathematical model. The models will be analyzed qualitatively with a viewpoint toward determining whether or not they accurately portray the processes they are intended to describe. The course will be concerned with continuous and deterministic models rather than discrete or stochastic models. Topics will be drawn from population ecology, reaction kinetics, neural models, biological oscillators, developmental biology, evolution, epidemiology, and other areas. The instructor will make allowances for disparity in the mathematical backgrounds of the students.


MATH 7394: PROBLEMS IN APPLIED MATHEMATICS (Section 10888)
Time: 11:30-1 TTH 347 PGH
Instructor: R. Glowinski
Prerequisites: Linear algebra; course on basic numerical analysis; basic course on partial differential equations.
Text(s): Check with instructor.
Description: The main goal of these lectures is to introduce the student to applied and computational mathematics by consideration of problems of practical interest. Our approach will be the following:
  • 1) Identification and formulation of problems of real life interest originating from science or engineering. Modeling aspects.
  • 2) Mathematical discussion of the problems.
  • 3) Computational discussion of the problems.
We shall take advantage of this approach to introduce the students to mathematical and computational tools which are sufficiently general to allow the solution of large classes of applied math problems. The course will be self-contained but senior level knowledge of linear algebra and differential equations will be most helpful.


MATH 7394: BASIC TOOLS FOR THE APPLIED MATHEMATICIAN (Section 12296)
Time: 4-5:30 TTH Rm TBA
Instructor: R. Sanders
Prerequisites: Second year Calculus. Elementary Matrix Theory. Graduate standing or consent of instructor.
Text(s): Lecture notes will be supplied by the instructor.
Description: Finite dimensional vector spaces, linear operators, inner products, eigenvalues, metric spaces and norms, continuity, differentiation, integration of continuous functions, sequences and limits, compactness, fixed-point theorems, applications to initial value problems.


MATH 7396: ITERATIVE METHODS FOR ELLIPTIC PROBLEMS (Section 10919)
Time: 1-2:30 MW 314 PGH
Instructor: Y. Kuznetsov
Prerequisites: Graduate standing.
Text(s): None
Description: Many important phenomena and processes in science and engineering are governed by elliptic partial differential equations. After discretization by finite element, finite volume or finite-difference methods these equations result in large scale systems of algebraic equations. The only realistic way to solve large scale algebraic systems on modern computers is to apply advanced iterative methods. In this course the most efficient and realizable iterative methods will be presented and discussed from both theoretical and applied aspects. Among others, Pint and Block Relaxation, Alternating Direction Implicit, Chebyshev, Generalized Minimal Residuals, Preconditioned Conjugate Gradient, and GMRES methods will be considered in the context of efficient solution of elliptic equations. We also consider the most advanced techniques based on domain decomposition, fictitious domain and algebraic multigrid methods. Throughout the course many practical examples from fluid mechanics and electro dynamics will illustrate the advantages and disadvantages of different iterative methods with emphasis on implementation aspects and applications to real life problems. In particular, several lectures will be devoted to efficient iterative solution of diffusion equations for strongly heterogeneous media.


*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.