Graduate Program

Senior and Graduate Course Offerings Fall 2002

For further information, contact the Department of Mathematics at 651 PGH, University of Houston, Houston, TX 77204-3008; Telephone (713) 743-3517 or e-mail to pamela@math.uh.edu.
 
 
MATH 4315: GRAPH THEORY WITH APPLICATIONS (Section 11148)
Time: 4-5:30pm TTH, Rm.347-PGH
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 4331: INTRODUCTION TO REAL ANALYSIS (Section 09446) 
Time: 1-2:30pm MW, Rm. 302-AH
Instructor: V. Paulsen
Prerequisites: Math 3333 
Text(s): Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, Latest Edition (required); Real Analysis with Reas Applications,K.R. Davidson and A.P. Donsig, Prentice Hall (ISBN 0-13-041647-9) (recommended). Note: Selected topics from the recommended text wil 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 4335: PARTIAL DIFFERENTIAL EQUATIONS (Section 11149)
Time: 4-5:30pm MW, Rm. 345-PGH
Instructor: B. Keyfitz
Prerequisites: Advanced Calculus, Ordinary Differential Equations (MATH 3331 or equivalent).
Text(s): Partial Differential Equations: An Introduction,  W.A. Strauss, Wiley, New York, latest edition (required), plus notes by instructor.
Description: This is the first 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. Along with techniques for obtaining solution formulas, such as the method of characteristics and separation of variables, the course will discuss the relationship between algebraic properties of the differential operator and analytic behavior of solutions (that is, reasons for the distinction between elliptic, parabolic and hyperbolic equations), the use of PDE in modeling continuum phenomena, and the distinction between linear and nonlinear equations. Topics:
    I. First-order equations:
                                                i. linear,
                                               ii. quasilinear,
                                               iii. fully nonlinear,
                                               iv. applications to turbulence and traffic flow. 
  II. Classification, normal forms and examples: 
                                                i. definitions: order, linearity, type, systems, 
                                               ii. initial and bouondary value problems, 
                                               iii. classification of second-order linear equations, 
                                               iv. solution methods, examples and applications.
 III. Theory and methods for linear equations.
 IV. Nonlinear Equations.

 
 
MATH 4337:: POINT SET THEORY (Session 11150)
Time: 4-5:30pm  TTH, Rm 309-PGH
Instructor: M. Friedberg
Prerequisites: MATH 3333 or MATH 3334 or Consent of Instructor.
Text(s): Elementary Topology, Dennis Roseman, Prentice-Hall.
Description:  An introduction to concepts of Point Set Topology: metric space topologies, general topologies, continuity, compactness, connectedness, and other topological properties. 

 
 
MATH 4340: NONLINEAR DYNAMICS AND CHAOS (Section 11151)
Time: 12-1pm MWF, Rm. 348-PGH
Instructor: M. Field
Prerequisites: Calculus sequences, I, II, III. A first course in differential equations (for example MATH 3331) would be helpful for the first semester, essential for the second. 
Text(s): References: Chaos: An Introduction to Dynamical Systems, K.T. Alligood, T.D. Sauer and J. Yorke, Springer-Verlag, 1996; An Introduction to Chaotic Dynamical Systems, R.L. Devaney, Addison Wesley, Second Edition; and A First Course in Discrete Dynamical Systems, Richard A. Holmgren, Springer-Verlag (Universitext). 
Description:  The course is intended to be an introduction to nonlinear analysis and chaotic dynamics. In the fall semester, much of the course will be devoted to studying one-dimensional iterated mappings - examples of discrete dynamical systems. We will spend a few class periods in the mathematics computer lab. In the second semester the emphasis will be on differential equations. 

 
 
MATH 4350: DIFFERENTIAL GEOMETRY (Section 09447)
Time: 12-1pm MWF, Rm. 345-PGH (NOTE TIME AND ROOM CHANGE!)
Instructor: M. Ru
Prerequisites: Math 2433 (Calculus of Functions of Several  Variables) and  Math 2431 (Linear Algebra).
Text(s): Elementary Differential Geometry  by Andrew Pressley,  Springer-Verlag.
Description: This course will introduce the theory of the geometry of courves and surfaces in three-dimensional space using calculus techniques. Topics include: curves in the plane and in space, global properties of curves, surfaces in three dimensions, the first fundamental form, curvature of surfaces, Gaussian curvature and the Gaussean map, geodesics, minimal surfaces, Gauss' Theorem Egrigium, the Gauss-Bonnet theorem. 

 
 
MATH 4364: NUMERICAL ANALYSIS (Section 09449)
Time: 5:30-7pm MW, Rm. 28-H
Instructor: T. Pan
Prerequisites: Math 2431 (Linear Algebra), Math 3331 (Differential Equations). Ability to do computer assignments in either FORTRAN , C , PASCAL, Matlab, etc.This is the first semester of a two semester course.
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 solving nonlinear equations, interpolation, numerical integration, initial value problems for ordinary differential equations, and direct methods for solving inear systems of algebraic equations. This is an introductory course and will be a mix of mathematics and computing.

 
 
MATH 4377: ADVANCED LINEAR ALGEBRA (Section 09450)
Time: 2:30-4pm TTH, Rm. 309 PGH
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 09451)
Time: 10-11:30 TTH,  Rm. 127 SR
Instructor: J. Hardy
Prerequisites: Math 3330. or equivalent. 
Text(s): TBA
Description: Topics will include ivisibility, primes and their distribution, congruences, Fermat's and Euler's Theorems, Number-theoretic functions, primitive roots, Quadratic Reciprocity Theorem, Pythagorean Triples. 

Note: Successful completion of Math 4383 and Math 4333 (Advanced Abstract Algebra) now satisfies the requirement of a         4000-level sequence for a bachelor's degree in mathematics. 


 
 
MATH 6198: TEACHING PRACTICUM (Section 09459)
Time: TO BE ARRANGED
Instructor: D. Blecher 
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 09465)
Time: TO BE ARRANGED, Rm. 651 PGH
Instructor: A. Torok
Prerequisites: none
Text(s): The material used for this course will be either available on the web (see www.math.uh.edu/~torok) or handed out in class. 
Description: 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, Mathematica, Matlab), web-publishing (HTML), and Internet use (mail, electronic archives, etc.). 

The course will consist of weekly workshops accompanied by hands-on applications in the computer lab of the Math Dept. 


 
 
MATH 6302: MODERN ALGEBRA (Section 09466)
Time: 1-2:30pm TTH,  Rm. 350 PGH
Instructor: K. Kaiser
Prerequisites: Graduate standing .
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 6320: FUNCTIONS OF A REAL VARIABLE (Section 09479)
Time: 2:30-4pm TTH, Rm. 350 PGH
Instructor: A. Torok
Prerequisites: Math 4331;4332 or consent of instructor. 
Text(s): Possible texts to be used are: Real Analysis, H.L. Royden, (3rd Edition), Prentice Hall.; 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: 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 6324: DIFFERENTIAL EQUATIONS (Section 11152 )

Time:
10-11:30am TTH, Rm. 128-SR
Instructor: M. Golubitsky
Prerequisites: Linear Algebra and an undergraduate ODE course. 
Text(s): Introduction to Differential Equations, Dynamical Systems, and Linear Algebra, Hirsch and Smale, Pure and Applied Mathematics, Vol. 60, Academic Press.
Description:  This course will emphasize:
       phase portrait analysis for linear systems, 
       general existence theorems for nonlinear systems,
       linearization theorems including the stable and unstable manifold theorems,
       theory of discrete dynamical systems,
       standard well known examples of systems of ODEs
as time permits.

 
 
MATH 6360: APPLICABLE ANALYSIS (Section 09483)
Time: 4-5:30pm MW, Rm 348 PGH (NOTE TIME AND ROOM CHANGE!)
Instructor: G. Auchmuty
Prerequisites: MATH 4332 or permission of instructor. 
Text(s): Linear Operator Theory in Science and Engineering, Naylor and Sell, Springer-Verlag. 
Description: This course will concentrate on the solvability theory for various types of equations. 

Background material required is a knowledge of the basic of metric and normed spaces. Topics to be covered will include:
  (i) the contraction mapping theorem, 
  (ii) applications of the contraction mapping theorem to finite dimensional problems, including the inverse and implicit function      theorems, 
  (iii) existence theorems for ordinary differential equations, 
  (iv) solvability of Volterra integral equations, 
  (v) some results on Fredholm integral equations. 

The second half of the semester we'll do an introduction to the theory of Hilbert spaces, leading to the fundamental theorem of linear analysis, the Fredholm alternative and the Lax-Milgram theorem. 
 


 
 
MATH 6370: NUMERICAL ANALYSIS (Section 09485)
Time: 5:30-7pmTTH, Rm. 348 PGH
Instructor: E. Dean
Prerequisites: Graduate standing or consent of instructor. Students should have had a course in Linear Algebra and an introductory course in analysis
(such as MATH 3333). This is the first semester of a two-semester course. 
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. The emphasis this semester will be on floating point arithmetic, error analysis, interpolation, numerical quadrature, and systems of linear algebraic equations. If time permits, we will also discuss systems of nonlinear equations. 

 
 
MATH 6372: NUMERICAL ORDINARY DIFFERENTIAL EQUATIONS (Section 11153)
Time: 5:30-7pm MW, Rm. 201 AH
Instructor: J. He
Prerequisites: Numerical analysis and an undergraduate ODE course.
Text(s): Computer Methods for Ordianry Differential Equations and Differential-Algebraic Equations, by Uri M. Ascher and Linda R. Petzold, SIAM, 1998, ISBN 0-89871-412-5. 
Description:  Introduction to ordinary differential equations; theory of general linear methods; stability analysis and linear multistep methods; Runge-Kutta methods and extrapolation methods; stiffness and nonlinear methods; Hamiltonian systems and symplectic methods; numerical methods for singular perturbation problems; numerical methods for boundary value problems; numerical methods for differential-algebraic equations. 

This course is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition. It also addresses reasons why existing software succeeds or fails. 


 
 
MATH 6377: BASIC TOOLS FOR THE APPLIED MATHEMATICIAN (Section 11442)
Time:  5:30-7:00 pm TTH, Rm 130-SR
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 6382:  PROBABILITY MODELS AND MATHEMATICAL STATISTICS (Section 11154)
Time: 2:30-4PM TTH,  Rm. 315-PGH
Instructor: E. Kao  (NOTE CHANGE OF INSTRUCTOR!)
Prerequisites: MATH 3334 and 4378
Text(s): A First Course in Probability  (sixth edition), Sheldon Ross, Prentice Hall , ISBN 0-13-033851-6
Description:  This is an introduction to probability theory. It covers random variables, probability distributions, expectations, limit theorems, and Monte Carlo simulation. The background developed in this course is essential to subsequent studies of mathematical statitstics and stochastic processes. 

 
 
MATH 6394: HOMOLOGICAL ALGEBRA (Section 12425)
Time: 12-1:30pm MW, Rm. 132-SR
Instructor: A. Helemskii (visiting from Moscow State University)
Prerequisites: TBA
Text(s): TBA
Description:  This will be a first course in homological algebra. Homological algebra is the study of commuting diagrams, exact sequences and complexes of maps between algebraic objects. This material is basic to many parts of mathematics, but is especially used in the study of algebra and algebraic topology. 

The course will begin with the first basic propositions about commuting diagrams such as the five lemma and snake lemma and proceed to develop the homological functors, Ext and Tor. 

Professor Helemski is one of the leading experts in the application of these ideas to the study of topological algebras. 


 
 
MATH 6395: INTRODUCTION TO COMPLEX ANALYSIS AND GEOMETRY (Section 09486)
Time: 11:30-1pm TTH, Rm.  314-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 I (Section 11339)
Time: 4-5:30pm TTH, Rm.  350-PGH
Instructor: M. Papadakis
Prerequisites: The course will be offered for students at the graduate level attending one of the following graduate programs: ECE, COSC, MATH, 
Applied Math, GEOL, PHY; the following combination may also work: ECE 6364 and 6342, or GEOL 7341 and 7336,  or MATH $332 and 4377, or ELEE 6330 and 6334, or PHY 6303 and MATH 4377, or consent of instructor. 
Text(s): A First Course  on Wavelets, by E. lHernandez and G. Weiss, CRC, 1996, if a text is used. 
Description: Fundamentals of Hilbert space theory (subspaces, bounded operators, projections, orthonormal bases); orthogonal direct sums of subspaces; multiresolution analysis, scaling functions and multiresolution (MRA) wavelets; Examples of MRAs, Shannon's sampling theorem; the construction of compactly supported wavelets; fast wavelet transforms and filter banks, 2-D separable multiresolution analysis and 2-D Fast Wavelet transform; Franklin and Spline wavelets on the real line; frames and frame wavelets. 

This is aproof oriented course. Knowledge of measure theory is not necessary. 


 
 
MATH 6397: NUMERICAL METHODS FOR PDE CONTROL (OPTIMAL CONTROL OF ADVECTION-REACTION-DIFFUSION SYSTEMS) (Section 11157)
Time: 2:30-4pmTTH, Rm. 314-PGH 
Instructor: R. Glowinski
Prerequisites: Numerical Linear Algebra, Numerical Methods for ODEs and PDEs. 
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: The main goal of this course is to introduce the student to adjoint equation based methods for the solution of problems originating from the optimal control of systems modeled by advection-reaction-diffusion equations. The following topics will be discussed:
    1. Formulation of the control problem.
    2. Derivation of the objective function by adjoint equation based methods. 
    3. Time-discrete control problems and derivation of the related adjoint equations. 
    4. Finite element approximations.
    5. Conjugate gradient methods for the solution of the discrete two-point boundary value problems.
    6. Brief discussion of recursive methods a la Kalman. 

 
 
MATH 6397: LINEAR ALGEBRA WITH APPLICATIONS(Section 12820)
Time: Web CT 
Instructor: G.J. Etgen and M. Golubisky
Prerequisites: Calculus I  
Text(s): Linear Algebra and Differential Equations Using MATLAB by Golubitsky and Dellnitz. Brooks-Cole Publ., Pacific Grove, 1999. Student Edition of Matlab also required.  
Description: Systems of linear equations, matrices, vector spaces, linear independence and linear dependence, determinants, eigenvalues; applications of the linear algebra concepts will be illustrated by a variety of projects.  This course will apply toward the Master of Arts in Mathematics degree; it will not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.  

 
 
MATH 7320: FUNCTIONAL ANALYSIS (Section 11155)
Time: 1-2pm MWF, Rm. 345-PGH
Instructor: Michael Friedberg
Prerequisites: Graduate standing.
Text(s): TBA
Description: TBA

 
 
MATH 7394: REACTION DIFFUSION SYSTEMS (Section 11340)
Time: 10-11:30am TTH,  348 PGH
Instructor: W. Fitzgibbon
Prerequisites: One year of real analysis or one year of applicable analysis, some exposure to partial differential equations either at the graduate or the undergraduate level. 
Text(s): none
Description: We will present the basic theory of reduction diffusion equasions and systems of reaction diffusion equations including well posedness and quantifiable behavior. The construction and use of a priory estimates will be carefully examined. We shall also investigate the existence and behavior of distinguished solutions. 

 
 
MATH 7394: MIXED AND HYBRID FINITE ELEMENT METHODS (Section 12576)
Time: 1-2:30pm MW 314, Rm 314-PGH 
Instructor: Y. Kuznetsov
Prerequisites: Graduate Standing
Text(s): Mixed and Hybrid Finite Element Methods, by F. Brezzi and M. Fortin (recommended).
Description: The main goal of this course is to give an extended introduction to the mixed and hybrid formulations of the elliptic differential problems as well as to the finite element methods for their numerical solution. The first part of the course is based on the book "Mixed and Hybrid Finite Element Methods" by Brezzi and Fortin. The second part of the course is devoted to some recent results and new applications of mixed and hybrid finite element methods for diffusion problems in strongly heterogeneous media. In particular, we shall discuss new efficient iterative solvers for large scale algebraic saddle-point systems. 

*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.