Senior and Graduate Course Offerings Spring 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 4315 GRAPH THEORY WITH APPLICATIONS (Section 11611) | |
| Time: | 4-5:30 TTH Rm. 347 PGH |
| Instructor: | C. Larson |
| Prerequisites: | Discrete Mathematics (Math 3336) |
| Text(s): | None. The course will be based on my notes. |
| Description: | The course is an introduction to graph theory and some of its applications including trees, Eulerian and Hamiltonian graphs, probabilistic methods and fullerenes - the new form of carbon. I will emphasize algorithmic methods and describe a number of open computational problems. |
| MATH 4332: INTRODUCTION TO REAL ANALYSIS (Section 08436) | |
| Time: | 4-5:30 MW Rm. 315 PGH |
| Instructor: | A. Torok |
| Prerequisites: | Math 4331 or equivalent. |
| Text(s): | Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, Latest Edition. |
| Description: | This course is a continuation of Math 4331. We will prove, among others, the basic theorems of calculus in several variables. |
| MATH 4333: ADVANCED ABSTRACT ALGEBRA (Section 11129) | |
| Time: | 10-11:30 TTH Rm. 211 PGH |
| Instructor: | J. Hausen |
| Prerequisites: | MATH 3330 or equivalent. |
| Text(s): | Rings, Fields and Groups, R. B. J. T. Allenby, Butterworth-Heineman Books (ISBN 03405 44406). |
| Description: | Topics from Ring Theory, Field Theory and Group Theory including polynomial rings, quotient rings, field extensions and finite fields. Structure theorems from group theory as times permits . |
| MATH 4336: PARTIAL DIFFERENTIAL EQUATIONS (Section 08611) | |
| Time: | 2-3 MWF Rm. 209 PGH |
| Instructor: | S. Canic |
| Prerequisites: | Math 4335. |
| Text(s): | Partial Differential Equations, Walter Strauss. |
| Description: | The course discusses existence, uniqueness and structure of solutions of basic linear partial differential equations. |
| MATH 4365: NUMERICAL ANALYSIS (Section 08439) | |
| Time: | 5:30-7 TTH Rm. 209 PGH |
| Instructor: | J. He |
| Prerequisites: | It is assumed that the students have completed the standard college calculus sequence and have a good knowledge of a high-level programming language. Familiarity with the fundamentals of matrix algebra and differential equations is also required. |
| Text(s): | Numerical Analysis, Richard L. Burden and J. Douglas Faires, Brooks/Cole. |
| Description: | Iterative Techniques in Matrix Algebra, Approximation Theory, Approximating Eigenvalues, Numerical Solutions of Nonlinear Systems of Equations, Boundary-Value Problems for Ordinary Differential Equations, Numerical Solutions to Partial Differential Equations. |
| MATH 4377: ADVANCED LINEAR ALGEBRA (Section 08440) | |
| Time: | 12-1 MWF Rm. 309 PGH |
| Instructor: | M. Friedberg |
| Prerequisites: | Math 2331 and a minimum of three semester hours of 3000 level mathematics or consent of instructor. |
| Text(s): | Linear Algebra, K. Hoffman, P. Kunze, 2nd Edition. |
| Description: | Introduction to vector spaces, linear transformations, matrices and their applications to linear systems. |
| MATH 4378: ADVANCED LINEAR ALGEBRA (Section 11592) | |
| Time: | 1-2:30 MW Rm. 128 SR1 |
| Instructor: | C. Peters |
| Prerequisites: | Math 4377. |
| Text(s): | The Theory of Matrices, Peter Lancaster & Miron Tismenetsky, Academic Press, 2nd Edition. (RECOMMENDED). |
| Description: | Normal, Hermitian and positive definite matrices; inner products, quadratic forms, spectral theory, functions of a matrix; vector and matrix norms, bounds for eigenvalues, Perron-Frobenious theorem and applications. |
| MATH 4378: ADVANCED LINEAR ALGEBRA (Section 08441) | |
| Time: | 4-5:30 TTH Rm. 211 AH |
| Instructor: | J. Johnson |
| Prerequisites: | Math 4377 or consent of instructor. |
| Text(s): | Linear Algebra, Hofmann-Kunze, Prentice-Hall, 2nd Edition. |
| Description: | Topics to be covered in this course include linear equations, vector spaces, polynomials, linear transformations, and matrices . |
| MATH 4398: APPLIED MATH SEMINAR (Section 11590) | |
| Time: | 4-5:30 MW Rm. 315 PGH |
| Instructor: | B. Keyfitz |
| Prerequisites: | This is the capstone course in the Applied Analysis Option. It would be desirable if students had completed some of the program, including at least one course from the list: Linear Algebra, Intermediate Analysis, Advanced Multivariable Calculus, Differential Equations, Probability and Statistics. Students uncertain of their preparation may consult the instructor. |
| Text(s): | A Concrete Approach to Mathematical Modeling, Michael Mesterton-Gibbons, Wiley-Interscience, 1995. (REQUIRED). |
| Description: | In the seminar we will look at how mathematics is used in modeling, and describe applications of undergraduate mathematics. The issues of establishing and developing models, acquiring data, and then analyzing and interpreting the results will be discussed using specific modeling problems. Students, in teams of two to three, will develop and analyze a mathematical model for real-world situation of their choice; give a seminar on the topic, and write a technical report. Each student will complete two projects during the semester. Students are encouraged to include a computational component in their projects, and computer accounts and software are available. |
| MATH 4397: NONLINEAR ANALYSIS and CHAOS, II (Section 11629) | |
| Time: | 1-2:30 TTH Rm. |
| Instructor: | M. Field |
| Prerequisites: | Math 3321 of 3331; Math 3341. |
| Text(s): | Chaos, An Introduction to Dynamical Systems, K.T. Alligood, T.D. Sauer & J. Yorke, Springer-Verlag, 1996 . |
| Description: | The course is intended to be an introduction to nonlinear analysis and chaotic dynamics. Much of the course will be devoted to studying differential equations and part one of the course (Math 4340), which was devoted to iterated maps, will not be a prerequisite. We will be interested in long term behavior of differential equations, existence of periodic solutions (limit cycles), chaotic dynamics, and sensitive dependence on initial conditions. We will also investigate how dynamics change when we vary the equation (the subject of bifurcation theory). I expect that about 30% of course time will be spend in the Mathematics Computer Lab. |
| MATH 6303: MODERN ALGEBRA (Section 08492) | |
| Time: | 1-2:30 TTH Rm. 348 PGH |
| Instructor: | J. Johnson |
| Prerequisites: | Math 6302 or consent of instructor. |
| Text(s): | Algebra, Thomas W. Hungerfold, Springer-Verlag; Graduate Texts in Mathematics #73. |
| Description: | Topics from the theory of groups, rings, and fields with special emphasis on modules and universal constructions. |
| MATH 6321: FUNCTIONS OF A REAL VARIABLE (Section 08513) | |
| Time: | 4-5:30 MW Rm. 309 PGH |
| Instructor: | G. Auchmuty |
| Prerequisites: | Math 6320 or consent of instructor. |
| Text(s): | Lebesgue Integration on Euclidean Space, F. Jones, Jones & Bartlett. |
| Description: | The course will treat the following topics. The theory of Lp spaces; convolutions of functions on Rn; Fourier series and transforms; weak derivatives of functions and Sobolev spaces; Hausdorff measure and Gauss-Green theorems. |
| MATH 6343: POINT SET TOPOLOGY (Section 08515) | |
| Time: | 2:30-4 TTH Rm. 134 SR1 |
| Instructor: | I. Melbourne |
| Prerequisites: | Math 6342 or consent of instructor. |
| Text(s): | Topology, James R. Munkres, 2nd Edition. |
| Description: | Continuation of Fall semester. Will cover Chapters 5 and 6, some topics from Chapter 7 and 8, and some additional topics. Tychonoff\022s theorem, Stone-Cech compactification. Nagata-Smirnov Metrization theorem, paracompactness, the long line, partitions of unity inverse limits and characterization of the Cantor set space-filling curves. Baire spaces, dimension theory. |
| MATH 6361: APPLICABLE ANALYSIS (Section 11128) | |
| Time: | 10-11:30 TTH Rm. 350 PGH |
| Instructor: | W. Fitzgibbon |
| Prerequisites: | Math 4331-32. |
| Text(s): | Linear Operator Theory in Science & Engineering,, A. W. Naylor and G. R. Sell, Springer Verlag. |
| Description: | Metric spaces and the contraction mapping theorem. Applications to the solvability of finite dimensional equations. Existence and uniqueness of solutions of ordinary differential equations and integral equations. Introduction to Hilbert spaces and the solvability of linear operator equations. |
| MATH 6371: NUMERICAL ANALYSIS (Section 08516) | |
| Time: | 2:30-4 TTH Rm. 350 PGH |
| Instructor: | E. Dean |
| Prerequisites: | Graduate standing or consent of instructor. This is the second semester of a two semester course. The first semester is not a prerequisite. |
| Text(s): | Introduction to Numerical Analysis, J. Stoer and R. Bulirsch, Springer-Verlag, 2nd Edition. |
| Description: | We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on iterative methods for numerical linear algebra, including large systems of linear equations and eigenvalue problems, numerical methods for ordinary differential equations. This course will prepare students for other graduate level course in numerical mathematics. |
| MATH 6381: METHODS OF FUNCTIONAL ANALYSIS IN STOCHASTIC PROCESSES PART II (Section 11133) | |
| Time: | 5:30-7 TTH Rm. 350 PGH |
| Instructor: | A. Bobrowski |
| Prerequisites: | Graduate standing or consent of instructor |
| Text(s): | (Course notes).Stochastic Calculus, R. Durrett, CRC Press, 1996 (Recommended). For the students who did not take the first part of the course, the classic monograph, An Introduction to Probability Theory and Its Applications, W. Feller, Vol. II, Wiley 1971, highly recommended. |
| Description: | The course is a continuation of 6380 and focuses on the role of functional analytic arguments in Probability and Stochastic Processes. Having covered the basic material in the Fall we will go on to special topics, which may include: 1) Hilbert space methods (continued), theory of stochastic processes, square integrable martingales, stochastic integrals with respect to martingales, generalized stochastic processes (white noise), Banach space methods. 2) weak topologies in Banach spaces, the Alaoglu theorem, the Prokhorov theorem on tightness of measures, construction of the Brownian motion as a measure on the space of continuous functions, distributions as operators, applications to limit theorems, Markov processes and the related semigroups of contraction operators, Feller semigroups, diffusion processes, diffusion approximations, the Feynmann-Kac formula, diffusion approximation to the Fisher-Wright-Moran model in population genetics, Markov chains and the Kato's theorem on Markov semigroups, Markov chain models of population genetics, stochastic analysis. The choice of the topics will depend on the interests of the audience. |
| MATH 6383: PROBABILITY AND STATISTICS (Section 11604) | |
| Time: | 1-2:30 TTH Rm. |
| Instructor: | C. Peters |
| Prerequisites: | Math 6382 or consent of instructor. |
| Text(s): | None. Approximation Theorems of Mathematical Statistics, Serfling, will be a primary reference (Wiley). |
| Description: | Parameter and estimators, location and scale parameter, empirical distribution and quantile functions, parametric families, exponential families, optimal estimators, maximum likelihood, asymptotic theory of maximum likelihood estimator and likelihood ratio tests. |
| MATH 6395: INTRODUCTION to HARMONIC ANALYSIS AND ITS APPLICATIONS (Section 111627) | |
| Time: | 2:30-4 TTH Rm. 309 PGH |
| Instructor: | M. Papadakis |
| Prerequisites: | Math 4377, 4331-4332. Advanced engineering students may attend the course with the consent of the instructor. |
| Text(s): | An Introduction to Harmonic Analysis, Dover, any edition. |
| Description: | Baire's category theorem in functional analysis, Open Mapping and Uniform Boundedness theorems. Fourier series on the unit circle: Convolution in L1 (T). Norm and pointwise convergence, Fejer, Dirichlet and Poisson summability kernels. Absolutely convergent Fourier series, Fourier series of square-summable functions and the concept of orthonormal bases of Hilbert spaces. Divergent Fourier series. Fourier transforms on the line: Convolution in L1 (R) and approximate identities. The (integral) Fourier transform in L1 (R), Fourier transform in L2 (R), applications to signal processing; inversion and differentiability properties of the integral Fourier transform. Fourier transform in Lp (R). Positive definite functions and Bochner's theorem; a brief overview of the theory of tempered distributions. Almost periodic functions on the line and Wiener's generalized harmonic analysis. |
| MATH 6394: NUMERICAL METHODS FOR PDE CONTROL (Section 11501) | |
| Time: | 11:30-1 TTH Rm. 347 PGH |
| Instructor: | R. Glowinski |
| Prerequisites: | Numerical linear algebra; Numerical ODE's and NUMERICAL PDE's. |
| Text(s): | None. |
| Description: | The main goals of these lectures is to discuss the computational aspect of solution methods for the control of systems modeled by partial differential equations. The topics to be discusses include the adjoint equation methods for the gradient and conjugate gradient solutions of control problems and the Ricatti equation approach for linear/quadratic control problems. A particular attention will be given to the control of time-discrete models. Applications include the control of systems modeled by advection-reaction-diffusion equations and by wavetype equations. |
| MATH 6397: MATHEMATICAL LOGIC WITH APPLICATIONS (Section 12723) | |
| Time: | TBA |
| Instructor: | K. KAISER |
| Prerequisites: | Graduate standing in Mathematics or Computer Science. |
| Text(s): | Logic for Applications, Anil Nerode and Richard A. Shore, Text and Monographs in Computer Science, Springer-Verlag. |
| Description: | The course is meant as a two semester course. First Semester: Propositional Logic, Predicate logic, Relational systems and Ultraproducts with applications to algebra and the calculus. Second Semester: Horn Clause Logic (Prolog), Modal Logic, Special topics, e.g., the logic of knowledge and belief. |
| MATH 6398: GRAPH THEORY VIA CONJECTURE-MAKING PROGRAM (Section 08526) | |
| Time: | 2:30-4 TTH |
| Instructor: | S. Fajtlowicz |
| Prerequisites: | Graduate standing of consent of instructor. |
| Text(s): | None |
| Description: | Graffiti is a computer program for making discoveries and hypotheses some of which will be discussed in this course. These findings of the program inspired many papers by prominent mathematicians and a few by equally accomplished computer scientists. A more recent version of the program (Minuteman) also makes discoveries in chemistry. The course will be an experiment in learning graph theory, Texas style, by discovering the subject via conjectures of an offshoot of Graffiti the Ranger. Unlike with the original method developed by UT Professor R. L. Moore, students will have the opportunity to select problems with they like most, since the program is capable of generating plenty of them on a short notice. Later participants will have an opportunity to use the Ranger to make discoveries related to their own interests necessarily restricted to just computer science and mathematics. |
| MATH 6398: AUTOMATA AND COMPUTABILITY (Section 08524) | |
| Time: | 2:30-4 TTH Rm. 269 PGH |
| Instructor: | K. Kaiser |
| Prerequisites: | Math 3336 (Discrete Math) and Cosc 2320 (Data Structures). I have strictly enforced Math 3336 or equivalent. |
| Text(s): | Introduction to the Theory of Computation, Michael Sipser, PWS Publishing Company (REQUIRED). Introduction to Automata Theory, Languages, and Computation Hopcroft, Ullman (highly recommended. |
| Description: | The course is about automata (finite state machines, push-down automata, Turing machines) and the various languages (i.e., regular languages, context free langues, recursive sets) they recognize. Decidability, computability, and Church's thesis are also discusses. Remark: The course as well as the text books are at the senior level. |
| MATH 6398: ADVANCED TOPICS OF HARMONIC ANALYSIS (Section 08525) | |
| Time: | TBA |
| Instructor: | S. Papadakis |
| Prerequisites: | Math 6320, 6321. Math 7320 recommended. This course supplements the previous one. It has to be attended in parallel with the previous course otherwise a proper course in Harmonic analysis is a prerequisite. It is designed for advanced graduate students with backgrounds in real analysis and measure theory. |
| Text(s): | An Introduction to Harmonic Analysis, Y. Katznelson, Dover, any edition. |
| Description: | Fourier series on the unit circle: order of the magnitude of Fourier coefficients, Fourier coefficients of linear functionals. The Hardy Littlewood maximal function; Hardy spaces. An overview of interpolation of norms and linear operators; the Hausdorff-Young theorem, Lacunary series. Fourier transforms on the line: Tempered distributions and pseudomeasures; Paley Wilener theorems. Fourier analysis on locally compact abelian groups, Haar measure characters and the dual group, Fourier transforms on L1 (G). An overview of Fourier transforms in multidimension. Applications of the theory of commutative Banach algebras in abstract Harmonic analysis: Wiener's Tauberian theorems and the Fourier transforms as restrictions of the Gelfand transform. |
| MATH 7321: FUNCTIONAL ANALYSIS (Section 11127) | |
| Time: | 4-5:30 TTH Rm. 345 PGH |
| Instructor: | D. Blecher |
| Prerequisites: | Math 7320. |
| Text(s): | None. |
| Description: | This is suppose to be the second semester of a two part course, although a student with a knowledge of the basic theory of Banach and Hilbert spaces is welcome to join the class. (The best prerequisite is probably an enjoyment of, and talent for, abstract concepts, and a desire to absorb mathematics). After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter. If you wish to do some preliminary reading you could try Nick Young's book An Introduction to Hilbert Space, which also has a good historical development. We will probably move the class-time to a time slot that is convenient for everybody. Outline of the syllabus. I. Operator theory on Hilbert space {Bounded operators: selfadjoint, positive, normal, compact, Fredholm, trace class.] II. Banach Algebras and spectral theory. [Algebras of continuous functions. Banach and C*-algebras. The Gelfand transform. Stone-Weierstrass. The spectral theorem.] III. Unbounded operator theory. Some basics. IV. Topics of interest to the class. Distributions and Sobolev spaces have been mentioned as a possible topic. |
| MATH 7325: BIFURCATION THEORY (Section 11130) | |
| Time: | 10-11:30 TTH Rm. 218 AH |
| Instructor: | M. Golubitsky |
| Prerequisites: | Some familiarity with ODEs, linear algebra, and (if possible) 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 departments, as well as in mathematics. |
| Text(s): | M. Golubitsky and D.G. Schaeffer and I.N. Stewart, Singularities and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, 1988. |
| Description: | Steady-state and time periodic-solutions to differential equations in the presence of symmetry. Pattern formation in several contexts: the disk (cellular structures and spirals) and the plane (spatially doubly periodic). Structurally stable dynamics in symmetric differential equations (such as heteroclinic cycles) and the dynamics of coupled cell systems will be discussed. Applications to speciation, animal gaits, reaction-diffusion equations, and convection, among others, will be presented. |
| MATH 7351: GEOMETRY OF MANIFOLDS (Section 11132) | |
| Time: | 3:30-10 TTH Rm. 268 PGH |
| Instructor: | S. Ji |
| Prerequisites: | Math 7350 or equivalent. |
| Text(s): | Riemannian Geometry, M.P. Do Carmo, Birkhauser, Boston, Basel, Berlin, 1992. |
| Description: | Geodesics, curvatures, complete manifolds, and other topics. |
| MATH 7394: MACRO-HYBRID FINITE ELEMENT METHODS (Section 11643) | |
| Time: | 1-2:30 MW Rm. 315 PGH |
| Instructor: | Y. Kuznetsov |
| Prerequisites: | Graduate standing. |
| Text(s): | Check with instructor. |
| Description: | Macro-hybrid finite element methods is a new advanced area of computational mathematics. The method has been inspired by the hybrid formulations of elliptic problems and domain decomposition methods with nonoverlapping subdomains. Recent variants of the method are based on coupling of domain decomposition and fictitious domain methods with distributed Lagrange multiplier technique. The purpose of this course is to give an extended introduction to the macro-hybrid formulations of elliptic problems, finite element methods with non-matching grids on the interfaces between subdomains (the mortar element method), the boundary and distributed Lagrange multiplier techniques, efficient preconditioning and solution methods for the arising large scale algebraic systems with the saddle-point matrices as well as to applications of macro-hybrid finite element methods to advanced problems in physics, mechanics and engineering. |
| MATH 7396: NUMERICAL SOLUTION OF TIME-DEPENDENT PROBLEMS (Section 11603) (Section 11270) | |
| Time: | 2:30-4 MW Rm. 345 PGH |
| Instructor: | S. Nepomnyashchikh |
| Prerequisites: | Graduate standing. |
| Text(s): | Difference methods for initial-value Problems, Richtmyer R., Morton K., The finite element method, Zienkiewicz, O. |
| Description: | This is a one semester course on numerical methods for time-depending problems. In a great number of practical problems the conditions are unsteady (i.e., time dependent), and effects of the time dimension have to be considered. Typically we are given the state of a system at some initial time, and required to determine the state of the system at subsequent times. Problems of this type include parabolic and hyperbolic equations, unsteady Stokes and Navier-Stokes problems. Finite difference and finite element methods are used for approximation of these problems. Approximation, stability, and convergence of discrete schemes are studied. The aim of this course is to provide both theoretical and practical aspects of modern numerical methods for the solution of time-dependent equations. It should be useful for students interested in using numerical methods as well as those working in the area of numerical methods. |
*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.