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Senior and Graduate Math Course Offerings 2004

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Current Course Offerings

Fall 2004

Time: 1:00-2:00 pm, MWF, 154-F
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: 9:00-10:00 am, MWF, 347 PHG  
Instructor: S. Ji
Prerequisites: Math 3333.  
Text(s): Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, Latest Edition (required).
Description: Elements of topology; sequences and series; continuity, differentiability and integrations of functions of one and several variables; the inverse function theorem and other fundamental results. Rigorous proofs are an essential part of this course.

Time: 10:00-11:00 am, MWF, 345 PGH  
Instructor: M. Ru
Prerequisites: Math 2433 (Calculus of Functions of Several Variables) and Math 2431 (Linear Algebra).  
Text(s): Differential Geometry of Curves and Surfaces by Manfredo Do Carmo (publisher: Prentice Hall)
Description: This year-long course will introduce the theory of the geometry of courves and surfaces in three-dimensional space using calculus techniques, exhibiting the interplay between local and global quantities. 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. We hope that we can cover up to Chapter 3 in the first semester. In the second semester, we will cover Chapter 4 and Chapter 5.

MATH 4364: NUMERICAL ANALYSIS (Section 10142)
Time: 4:00-5:30 pm, MW, 345 PGH  
Instructor: T. Pan
Prerequisites: Math 2331 (Linear Algebra), Math 3331 (Differential Equations). Ability to do computer assignments in one of the following: FORTRAN, C, Matlab, Mathematica, and Maple. But the publisher provides programs in Matlab.
Text(s): Elementary Numerical Analysis (Third edition), K.E. Atkinson and Weimin Han, Wiley, 2003.
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 of ordinary differential equations, and numerical methods for solving linear systems of algebraic equations. This is an introductory course and will be a mix of mathematics and computing.

Remarks: This is a first semester of a two semester course.

Time: 10:00-11:30 TTH, 350 PGH  
Instructor: E. Kao
Prerequisites: Math 3338, 3339, or equivalent background in probability and statistics.
Text(s): Derivative Securities, the second edition, by Robert Jarrow and Stuart Turnbull, South-Western College Publishing.
Description: The course is an introduction to financial derivatives. We study the roles played by options, futures, forwards, and swaps in risk management. We introduce the notions of geometric Brownian motion, risk-neutral pricing, binomial models, and martingales. We will also also examine interest rate contracts, the HJM Model,and non-standard options.

Time: 2:30-4:00 pm, TTH, 347 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: Syllabus: Topics to be covered in this course include linear equations, vector spaces, polynomials, linear transformations and matrices.  

MATH 4383: Number Theory (Section 12123)
Time: 10:00-11:30 am, TTH, 127 SR  
Instructor: J. Hardy
Prerequisites: Math 3330. or equivalent.
Text(s): To be determined.
Description: This course covers most of the material on classical number theory that a mathematics major/minor ought to know. Topics will include divisibility and factorization, congruences, arithmetic functions, primitive roots, quadratic residues and the Law of Quadratic Reciprocity, Diophantine equations, and other topics as time permits.  

Time: 4:00-5:30 pm, TTH, 309 PGH  
Instructor: C. Peters
Prerequisites: Math 3339, or the equivalent.  
Text(s): Introduction to Linear Regression Analysis, 3rd Ed., by Montgomery, Vining, and Peck, Wiley 2001.  
Description: Multiple linear regression, and linear models, regression diagnostics, model selections, other topics as time permits. Projects using data analysis software.  

Math 5397: Analysis (OnLine course) (Section 12222)
Time: On Line
Instructor: G. Etgen
Prerequisites: Consent of instructor.  
Text(s): Calculus, Michael Spivak, Publisher: Pulish or Perish.
Description: A survey of the concepts of limit, continuity, differentiation and integration for functions of one variable and functions of several variables; selected applications are used to motivate and to illustrate the concepts.

Math 5397: PROBABILITY (OnLine course) (Section 12221)
Time: On Line
Instructor: C. Peters
Prerequisites: Math 5331 or consent of instructor.  
Text(s): Concepts in Probability and Stochastic Modeling , by James J. Higgins & Sallie Keller-McNulty, Duxbury 1995.
Description: Probability, random variables, distributions, Markov chains, counting processes, continuous time processes.

Math 5397: Abstract Algebra (OnLine course) (Section 12223)
Time: On Line
Instructor: K. Kaiser
Prerequisites: 3330 or consent of instructor.  
Text(s): Abstract Algebra: A First Course by Dan Saracino, Waveland Press, Incorporated, Hardcover, ISBN: 0-88133-665-3 / 0881336653  
Description: The basic elements of groups, rings and fields will be covered with special emphasis on divisibility theory for rings.  

Math 5397: Graph Theory with Application (OnLine course) (Section 12324)
Time: On Line
Instructor: S. Fajtlowicz
Prerequisites: Graduate standing or consent of the instructor. Graduate standing in engineering departments is also enough.  
Text(s): No textbook  
Description:  Participants of this course will study Texas style the basics of graph theory by exclusively working on conjectures of the computer program Graffiti. A version of Graffiti will be available for individual use, so that students can learn or expand their knowledge of several subjects of their own choice, including: trees, planar graphs, independence and matching theory, network flows, chemical graphs, Ramsey Theory and eigenvalues of graphs. More information about the program is available on the web pages of the instructor, and Craig Larson.

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 than to be led to the rediscovery of known results, the participants will work exclusively on conjectures of selected versions of Graffiti, without getting any hints whether these conjectures are true or false. Another difference is that unlike in traditional Texas style courses the participants will be allowed, to read textbooks and even solutions of previous conjectures of Graffiti, because the problems they will encounter are unlikely to be found in textbooks anyway. This will create a more realistic setting for acquisition of research experience. 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. 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 6302: MODERN ALGEBRA (Section 10160)
Time: 10:00-11:30 am, TTH, 345 PGH  
Instructor: J. Hausen
Prerequisites: MATH 3330 (Abstract Algebra) or equivalent.
Text(s): W. J. Wickless, A FIRST GRADUATE COURSE IN ABSTRACT ALGEBRA, Marcel Dekker, Inc., New  
Description: This is a two-semester course on Abstract Algebra. It is anticipated that most of the first four chapters (Groups, Rings, Modules, (infinite dimensional) Vector Spaces) will be covered in the fall and chapters five and six (Fields and Galois Theory, Topics in Noncommutative Rings) in the spring. Additional topics as time permits. Homework will be an integral part of the course.  

MATH 6304: THEORY OF MATRICES (Section 12125)
Time: 10:00-11:00 am, MWF, 314 PGH  
Instructor: V. Paulsen
Prerequisites: Math 4377 and 4331 or Math 6377.
Text(s): Matrix Analysis, Horn and Johnson, Cambridge University Press NOTE: This book is available in paperback.  
Description: We will present topics in linear algebra and matrix theory that have proven to be important in analysis and applied mathematics. We assume that the student is familiar with standard concepts and results from linear algebra and basic analysis. We will study canonical factorizations of matrices, including the QR, triangular and Cholesky factorizations. We will develop ways to acheive the Jordan canonical form. We will study eigenvalue perturbation and estimation results and we will study special families of matrices such as positive definite, Hermitian, Hankel, Toeplitz. Matrix analysis is in a sense an approach to linear algebra that is willing to use concepts from analysis, such as limits, continuity and power series to get results in linear algebra.  

Time: 11:00-12:00 am, MWF, 315 PGH  
Instructor: M. Friedberg
Prerequisites: Math 4331; 4332 or consent of instructor  
Text(s): Real Analysis, 3nd Ed., H.L. Royden, Prentice Hall.  
Description: Lebesgue Measure and Integration, functions of bounded variabtion, obsolute continuity, the classical Lp spaces, general measure theory.  


12:00-1:00 pm, MWF, 345 PGH
Instructor: J. Morgan
Prerequisites: Math 4331 and a first course in linear algebra.  
Text(s): Differential Equations, Dynamical Systems and Linear Algebra , 2nd Edition, by Dr. Morris W. Hirsch , University of Wisconsin, Madison, USA Dr. Stephen Smale , University of California, Berkeley, USA, and Dr. Robert Devaney , Boston University, Massachussetts, USA. ISBN 0123497035 . Hardback . 400 Pages Academic Press . Published December 2003.
Description: This is the first semester of a two semester sequence. The topics from the fall semester will include:
  • A review of linear algebra.
  • Autonomous first order linear systems, steady states and stability.
  • An introduction to function spaces and the contraction mapping theorem.
  • Well posedness for general first order nonlinear systems.
  • Continuous dependence on initial data and parameters.
  • Stability theory, and linearized stability.
  • The implicit function theorem.
  • The stable manifold theorem.
  • Elementary bifurcation theory.

 MATH 6342: Topology (Section ?)

9:00-10:00 am, MWF, 350 PGH
Instructor: D. Blecher
Prerequisites: Math 4331 and Math 4337 or consent of instructor.  
Text(s): Topology, A First Course, J. R. Munkres, Second Edition, Prentice-Hall Publishers (not absolutely required).
Description: This is the first semester of a two-semester introductory graduate course in topology (the second semester is largely devoted to algebraic topology). This is a central and fundamental course and one which graduate students usually enjoy very much! This semester we discuss a little set theory, the basic definitions of topology and basis, separation properties, compactness, connectedness, nets, continuity, local compactness, Urysohn's lemma, Tietze, the characterization of separable metric spaces, and basic constructions such as subspaces, quotients, and products.

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 6360: APPLICABLE ANALYSIS (Section 10195)
Time: 11:30-1:00 TTH, 345 PGH
Instructor: R. Glowinski
Prerequisites: Math 4331 and 4332. Real Analysis, Mathematics for Engineers 
Text(s): Suggested Textbook: K.E. Atkinson and W.Han, Theoretical Numerical Analysis, Springer-Verlag, 2001 (this book contains a large section on Applicable Functional Analysis).
Description: The main objective of this course is to provide the students with mathematical tools, which have proved useful when addressing the solution of applied problems from Science and Engineering. Among the topics to be addressed let us mention:
  • 1. Functional Spaces with a particular emphasis on Hilbert spaces and the projector theorem. Weak convergence.
  • 2. Minimization of functional in Hilbert spaces.
  • 3. Iterative solution of linear and nonlinear problems in Hilbert spaces.
  • 4. The Lax-Milgram theorem and Galerkin methods in Hilbert spaces.
  • 5. Some notions on the Theory of Distributions.
  • 6. Application to the solution of variational problems from Mechanics and Physics.
  • 7. Time dependent problems and operator-splitting.
  • 8. Constructive methods for linear and nonlinear eigenvalue problems.
  • 9. Boundary value problems and their approximation.

 MATH 6366: OPTIMIZATION (Section 10196)

5:30-7:00 pm, MW, 309 PGH
Instructor: G. Auchmuty
Prerequisites: M4332 and M4377 or consent of instructor.
Text(s): Convexity and Optimization in R^n . Leonard D. Berkowitz, Wiley, 2002.
Description: This course will cover the major issues in the theory of unconstrained finite dimensional optimization, and of nonlinear and convex programming. We will develop the theory of quadratic programming and the analysis of finite-dimensional convex sets and functions. Steepest descent and conjugate gradient algorithms will be described and analyzed. Lagrangian methods, duality theory and saddle point methods will also be treated.

MATH 6370: NUMERICAL ANALYSIS (Section 10197)
Time: 4:00-5:30 pm, MW, 348 PGH
Instructor: J. He
Prerequisites: Graduate standing or consent of instructor. Students should have had a course in Linear Algebra and an introductory course in analysis. Familiarity with Matlab is also required.
Text(s): Numerical Linear Algebra, Lloyd N . Trefethen and David Bau, SIAM, 1997, ISBN: 0898713617
Description: This is the first semester of a two-semester course. The focus in this semester will be on numerical linear algebra. A short introduction to iterative solution of nonlinear systems and numerical optimization will also be given.  

Time: 5:30-7:00 TTH, 348 PGH
Instructor: E. Dean
Prerequisites: Graduate standing or consent of the instructor.
Text(s): Iterative Methods for Sparse Linear Systems , by Y. Saad, (2nd edition).
Description: This semester we will develop and analyze iterative methods for the solution of large systems of linear equations. Some of the topics to be covered include: basic iterative methods, conjugate gradient, Krylov subspace methods for nonsymmetric problems, and multigrid methods. We will also look at eigenvalue problems including the QR algorithm, divide-and-conquer technique, Lanczos and Arnoldi procedures.  

Time:  4:00-5:30 pm, TTH, 345 PGH
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.

Time: 2:30-4:00 pm, TTH, 350 PGH
Instructor: M. Nicol
Prerequisites: Math 3334, Math 3338 and Math 4377, or consent of instructor.
Text(s): A First Course in Probability, Sixth Edition by Sheldon Ross, 2002, Prentice Hall.
Description: This course is intended to help students build a solid foundation in probability. Emphasis will be placed on a thorough understanding of the basic concepts as well as developing problem solving skills. Topics covered include: axioms of probability; conditional probability and independence; discrete and continuous random variables; main discrete and continuous probability distributions (Bernoulli, Binomial, Poisson, Exponential etc); jointly distributed random variables; conditional expectation; moment generating function; classical limit theorems (strong and weak law of large numbers, central limit theorem etc); techniques of simulation, including Monte Carlo simulation.

Time: 2:30-4:00 pm, TTH, 301 AH
Instructor: E. Kao
Prerequisites: Math 6382, or equivalent background in probability.
Text(s): Introduction to Mathematical Finance: Discrete Time Models , by Stanley R. Pliska, Blackwell, 1997.
Description: The course an introduction to discrete-time models in finance. We start with single-period securities markets and discuss arbitrage, risk-neutral probabilities, complete and incomplete markets. We survey consumption investment problems, mean- variance portfolio analysis, and equilibrium models. These ideas are then explored in multiperiod settings. Valuation of options, futures, and other derivatives on equities, currencies, commodities, and fixed-income securities will be covered under discrete-time paradigms.

MATH 6397: DYNAMICS (Section 12120 )
Time: 10:00-11:30 am, TTH, 348 PGH
Instructor: M. Golubitsky
Prerequisites: ODEs (MATH 6324-6325) including some bifurcation theory, or permission of the instructor
Text(s): No required text.

Reference texts:

  • The Symmetry Perspective (Golubitsky and Stewart, Birkhauser)
  • Singularities and Groups in Bifurcation Theory Vol. II (Golubitsky, Stewart, and Schaeffer, Springer)
Description: This course will be a combination lecture and seminar course with students expected to read and present research level papers on theory and applications in pattern formation and/or coupled cells sytems.

Time: 4:00-5:30 MW, 315 PGH
Instructor: S. Canic
Prerequisites: Multivariable Calculus, Real and Complex Analysis.
Text(s): None required.

(Texbooks that will be used: W. Strauss's: "Partial Differential Equations" , R. Glowinski: "Numerical Methods for Fluids (Part 3)", Chorin and Marsden: "Mathematical Introduction to Fluid Mechanics", Y.C. Fung: "Circulation", Y.C. Fung: "Biomechanics: Mechanical properties of living tissues." R. LeVeques: "Conservation Laws", Research Papers)

Description: Topics Covered:
  • Review of basic linear PDEs.
  • Analysis of quasilinear PDEs with concentration to hyperbolic conservation laws.
  • Introduction to fundamentals of fluid mechanics (basic equations of motion: continuity, momentum, energy, vorticity).
  • Incompressible/compressible flow examples (derivation of the incompressible, viscous Navier-Stokes equations).
  • A brief introduction to Sobolev spaces. Fluid-structure interaction arising in blood flow modeling (effective models).
  • Energy estimates.
  • Special topics related to the study of blood flow through compliant blood vessels.

MATH 7320: FUNCTIONAL ANALYSIS (Section 12118 )
Time: 1:00-2:30 pm, MW, 350 PGH
Instructor: A. Torok
Prerequisites: Math 6320-6321, or consent of the instructor
Text(s): A course in functional analysis by John B. Conway. 2nd ed, New York: Springer-Verlag, c1990. SERIES of Graduate texts in mathematics ; 96. ISBN 0387972455 (alk. paper)

Also notes will be handed out in class.

Description: We will discuss Hilbert spaces, Banach spaces and topological vector spaces, bounded linear operators, the basic principles (Hahn-Banach, Uniform boundedness, Open mapping theorem) and their consequences. We will continue with the spectral theory of compact and Fredholm operators. We intend to also discuss applications of Functional Analysis to PDE's, or other topics of interest for the students.

Math 7394: ALGEBRAIC ITERATIVE METHODS (Section 12117)
Time: 1:00-2:30 pm, MW, 348 PGH
Instructor: y. Kuznetsov
Prerequisites: Graduate course on linear algebra and matrix analysis
Text(s): None
Description: This is an advanced course on the general theory and applications of the basic iterative methods for the numerical solution of large scale systems of linear algebraic equations. After a brief introduction to matrix analysis and the theory of linear algebraic equations we consider the general convergence theory for the stationary iterative methods including those which are applied to systems with singular matrices. The basic part of the course is devoted to the iterative methods based on minimization of quadratic functionals and orthogonaliza- tion ideas:generalized minimal residual (GMRES),preconditioned gradients (PCG) and preconditioned Lanczos (PL)methods. The major sources of large scale systems are mesh discretizations of partial differential equations.We shall use some of them to illustrate applications of iterative methods and preconditioning techniques.

Time: 1:00-2:30 pm, TTH, 345 PGH
Instructor: R. Hoppe
Prerequisites: Calculus, Linear Algebra, Numerical Analysis.
Text(s): P. Deuflhard; Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms . Springer, Berlin-Heidelberg-New York, 2004 (ISBN 3-540-21099-7)
Description: Large-scale nonlinear algebraic systems arise, for instance, from the discretization of differential and integral equations, in the framework of inverse problems as nonlinear least-squares problems, or as optimality conditions for nonlinear optimization problems.

We will consider local and global Newton and Gauss-Newton methods and variants thereof. Emphasis will be put on a thorough affine invariant convergence analysis as well as on appropriate damping strategies and monotonicity tests for convergence monitoring. Compared to traditional approaches, the distinguishing affine invariance concept leads to shorter and more transparent proofs and permits the construction of adaptive algorithms. We will also address parameter dependent nonlinear problems and focus on path-following continuation methods for their numerical solution.


*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.

Archive of Previous Course Offerings

2004 Spring
2003 Fall   2003 Spring
2002 Fall   2002 Spring
2001 Fall   2001 Spring
2000 Fall

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