MATH 4315: Graph Theory (Section 10422)
Time: 1:00-2:00 PM, MWF, 154-F 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 Real Analysis (Section 10423)
Time: 10:00-11:30 AM, TTH, 348-PGH
Instructor: M. Friedberg
Prerequisites: Math 3334 or consent of instructor.
Text(s): Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, 3nd Edition.
Description: Metric spaces, topology of Rn, Riemann-Stieltjes integrals, Lebesgue measure on R, Implicit and Inverse Function Theorems.  

 
 
MATH 4364: Numerical Analysis (Section 10425)
Time: 4:00-5:30 PM, TTH, 309-PGH
Instructor: E. Dean
Prerequisites: Math 2431 (Linear Algebra), Math 3331 (Differential Equations), Cosc 1301 or 2101 or equivalent. This is the first semester of a two semester course.
Text(s): Numerical Analysis (8th edition), by 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, 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 10427)
Time: 2:30-4:00 PM, TTH, 347-PGH (Hausen) and 344-PGH (Walker)
Instructor: J. Hausen and P. Walker
Prerequisites: Graduate standing or consent of instructor.
Text(s): Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition, Prentice-Hall.
Description: This is the first half of a 2-semester sequence. Topics covered include vector spaces, linear tramsformations, polynomial rings and determinants. Homework will be an integral part of the course.  

 
 
MATH 4397: Topology (Section 12442)
Time: 1:00-2:30 PM, TTH, 315-PGH
Instructor: M. Friedberg
Prerequisites: Math 3333 or Math 3334 or consent of instructor.
Text(s): Introduction to Topology by Baker, Crump W, Krieger Publishing Company.
Description: Metric space, completeness, general topologies, metric topologies, continuity, compactness, and connectedness.

 
   
MATH 4355: Signal Representation (Section 12413)
Time: 4:00-5:30 PM, MW, 315-PGH
Instructor: M. Papadakis
Prerequisites: MATH 2431 and one of the following: MATH 3333, MATH 3334, MATH 3330, MATH 3363.

MATH 3321 can be used instead of MATH 2431. Students may attempt the course without having one of the above junior-level courses but they first have to obtain the consent of the instructor.

Text(s): A first course in wavelets with Fourier Analysis by A. Boggess and F. Narcowich, Prentice Hall, 2001, ISBN 0-13-022809-5
Description: The linear algebra of inner product spaces: Linear Subspaces, linear independence, linear bases. Finite orthogonal and orthonormal systems, Gram-Schmidt, Orthogonalization, norms, convergence in the norm, Infinite orthonormal bases (this part will be covered from the Sham's series book "Linear algebra"). Fourier Series of real valued functions, Uniform and pointwise convergence of Fourier Series, Parseval's theorem and L-2 convergence of Fourier Series. Discrete Fourier transform (Fourier transform on finite cyclic groups) and Fast Fourier transform. Discrete 2-D Fourier transform Integral Fourier transform: Definition, properties, Riemann Lebesgue lemma, Inverse Fourier transform, convolutions and time-invariant linear systems, Plancherel's theorem, Analytic functions and power series, Band-limited and time-limited signals (functions), time-invariant filtering. Shannon's sampling theorem (Analog to Digital and digital to Analog conversions). A brief overview of Computerized tomography (Radon transform) and the back-projection algorithm.  

 
 
MATH 4362: ODE (Section 12414) Canceled !!!
Time: 2:30-4:00 PM, TTH, 345-PGH
Instructor: P. Walker
Prerequisites: Math 3331 or consent of instructor.
Text(s): An Introduction to Ordinary Differential Equations, by Earl A. Coddengton, ISNB 0-486-65942-9.

The Qualitaties Theory of Ordinary Differential Equations-an introduction by Fred Brauer and John A Nohel, ISBN 0 -486-65846-5.

Description: Existence and uniqueness of solutions to single equations and septems, phase space analysis, Sturm-Liouville theory.  

 
 
MATH 5331: Linear Algebra with Applications (Section 12557)
Time: On line course
Instructor: G. Etegn
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 5397: Differential Geometry (Section 13330)
Time: On line course
Instructor: M. Ru
Prerequisites: Math 2433(or equivalent) or consent of instructor.
Text(s): A set of notes on curves and surfaces will be written by Dr. Ru.
Description: The course will be an introduction to the study of Differential Geometry-one of the classical (and also one of the more appealing) subjects of modern mathematics. We will primarily concerned with curves in the plane and in 3-space, and with surfaces in 3-space. We will use multi-variable calculus, linear algebra, and ordinary differential equations to study the geometry of curves and surfaces in R3. Topics include: Curves in the plane and in 3-space, curvature, Frenet frame, surfaces in 3-space, the first and second fundamental form, curvature of surfaces, Gauss's theorem egrigium, minimal surfaces.  

 
 
MATH 5397: Graph Theory with Applications (Section10439 )
Time: On line course
Instructor: S. Fajtlowicz
Prerequisites: Graduate standing or consent of instructor.
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 5397: Discrete Math (Section 12560)
Time: On line course
Instructor: Kaiser
Prerequisites: Graduate standing or consent of instructor.
Text(s): Kenneth H. Rosen, Discrete Mathematics, McGraw-Hill, current edition.

Additional and more advanced material will be posted in form of notes on the WEB.

Software: Homework has to be submitted as a LaTex file. The students should buy Scientific Notebook from McKichan ( http://www.mackichan.com/) for doing this.

Description: The course will cover topics from logic and set theory. The material will be covered first in an informal manner to provide students with a working knowledge on truth tables, quantifier logic, and na�e set theory. We then will introduce the axioms of Zermelo Fraenkel and revisit elementary set theory from an axiomatic point of view. The Axiom of Choice and its equivalent versions will be discussed.  

 
 
MATH 6302: Modern Algebra (Section 10447)
Time: 4:00-5:30 PM, MW, 345-PGH
Instructor: K. Kaiser
Prerequisites: Graduate standing or consent of instructor.
Text(s): Thomas W. Hungerford, Algebra, Springer Verlag (required).

I will also circulate my own classroom 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 also discuss the most important algebraic constructions from a universal algebraic as well 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: Real Variables (Section 10482)
Time: 10:00-11:00 AM, MWF, 350-PGH
Instructor: V. Paulsen
Prerequisites: 4331-4332 or equivalent.
Text(s): Gerald B. Folland, Real Analysis: Modern Techniques and their Applications, John Wiley and Sons, ISBN 0471317160.
Description:
  • Measures.
  • Integration.
  • Signed Measures and Differentiation.
  • Point Set Topology.
  • Elements of Functional Analysis.
  • Lp Spaces.
  • Radon Measures.
  • Elements of Fourier Analysis.
  • Elements of Distribution Theory.
  • Topics in Probability Theory.
  • More Measures and Integrals.
 

 
 
MATH 6322: Complex Analysis (Section 13364)
Time: 12:00-1:00 PM, MWF, 345-PGH
Instructor: S. Ji
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: PDE (Section 12417)
Time: 11:00-12:00 AM, MWF, 315-PGH
Instructor: J. Morgan
Prerequisites: Math 4331 (Real Analysis) or equivalent.
Text(s): Partial Differential Equations , Lawrence C. Evans, American Mathematical Society, 1998.
Description: This is an introduction to the theory of partial differential equations, and will emphasize the tools of analysis used to study existence, uniqueness and qualitative behavior of solutions. In the first semester we will cover chapters 2-5: examples of protoype equations, the definition of characteristics and their importance in PDE; basic techniques of separation of variables, transforms and asymptotics; and the definition and properties of Sobolev spaces. The second semester will apply this theory to two important classes of equations: second-order elliptic equations and linear evolution equations; further topics will be chosen to suit the interests of the class and the instructor.  

 
 
MATH 6342: Topology/Geometry (Section 13544)
Time: 1:00-2:30 PM, MW, 350-PGH
Instructor: A. Torok
Prerequisites: MATH 4331 (mainly Chapter 2 of "Principles of Mathematical Analysis" by W. Rudin) or consent of the instructor.
Text(s): J. R. Munkres, Topology, Publisher: Prentice Hall; 2nd edition (December 28, 1999), ISBN: 0131816292

Remark: Students can use instead the first edition of this book: J. R. Munkres, Topology; A First Course, Publisher: Prentice Hall, 1974, ISBN: 0139254951

Description: An axiomatic development of point set topology: connectivity, compactness, separability, metrizability, function spaces.

Note: this course is listed in the graduate catalog as "Point Set Topology", but the prerequisites have been modified.  

 
 
MATH 6360: Applicable Analysis (Section 10485)
Time: 11:30-1:00 PM, TTH, 345-PGH
Instructor: Guidoboni
Prerequisites: Math 4331-4332.
Text(s): Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences, Vol.40) by Arch W. Naylor and Goerge R. Sell.
Description: Metric Spaces: convergence, completeness and compactness. Contraction mapping theorem, Arzel�Ascoli theorem and applications to differential equations and integral equations. Introduction to Hilbert spaces and the solvability of linear operator equations.  

 
 
MATH 6366: Optimization (Section 10486)
Time: 4:00-5:30 PM, MW, 348-PGH
Instructor: J. He
Prerequisites: Graduate standing or consent of the instructor. Students are expected to have a good grounding in basic real analysis and linear algebra.
Text(s): The required textbook: Leonard D. Berkovitz, Convexity and Optimization in Rn, Wiley-Interscience, 2001, ISBN: 0471352810.

The optional reference book: Stephen Boyd and Lieven Vandenberghe, Convex Optimization , Cambridge University Press, 2004 (available on the web at http://www.stanford.edu/~boyd/cvxbook.html)

Description: The focus is on key topics in optimization that are connected through the themes of convexity, Lagrange multipliers, and duality. The aim is to provide a rigorous treatment of the analytical/geometrical foundations of finite dimensional constrained optimization. The course is divided into three parts that deal with convex analysis, optimality conditions and duality, and computational techniques.

Part I Convexity

  • Convex sets
  • Convex functions
  • Convex optimization problems

Part II Optimality

  • Optimality conditions
  • Duality theory
  • Minimax/saddle point theory

Part III Algorithms

  • Unconstrained optimization
  • Equality constrained optimization
  • Inequality constrained optimization
 

 
 
MATH 6370: NUMERICAL ANALYSIS (Section 10487)
Time: 1:00-2:30 PM, TTH, 345-PGH
Instructor: R. Hoppe
Prerequisites: Graduate standing or consent of instructor.
Text(s): J. Stoer and R. Bulirsch; Introduction to Numerical Analysis , 3rd Edition. Springer, New York, 2002.
Description: This is the first semester of a two-semester course. We will consider
  1. Direct solution of linear algebraic systems,
  2. Iterative solution of linear algebraic systems,
  3. Error analysis,
  4. Numerical solution of nonlinear equations and systems,
  5. Polynomial and spline interpolation,
  6. Numerical integration,
  7. Computation of eigenvalues and eigenvectors.
 

 
 
MATH 6374: Numerical PDE (Section 12418)
Time: 5:30-7:00 PM, MW, 309-PGH
Instructor: Pan
Prerequisites: (i) Math6370-71 or consent of instructor; (ii) Ability to do computer assignments.
Text(s): Stig Larsson and Vidar Thomee, Partial Differential Equations with Numerical Methods, Springer-Verlag, Berlin, 2003, ISBN 3-540-01772-0.
Description: The purpose of this course is to give an elementary, relative short, and hopefully readable account for the basic types of linear elliptic, parabolic, and hyperbolic equations and their properties, together with the most commonly used methods for their numerical solution. The approach is to integrate the mathematical analysis of differential equations with the corresponding numerical analysis. We will focus on finite difference and finite element methods for solving each type of equations. Discretization, error estimates, stability analysis (if needed), properties of the discrete solution, and implementation of methodologies will be discussed. Algorithms to solve large sparse algebraic systems, classical iterative methods and conjugate gradient methods, will be discussed. Programming projects will be given.  

 
 
MATH 6377: Basic Tools in Applied Mathematics (Section 10488)
Time: 4:00-5:30 PM, TTH, 204-AH
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 and Statistics (Section 10489)
Time: 11:30-1:00, TTH, 301-AH
Instructor: Matthew Nicol
Prerequisites: Math 3333 or consent of instructor.
Text(s): Mathematical Statistics with Applications, 6th Edition Wackerly, Mendenhall III and Scheaffer, Duxbury Press
Description: Emphasis will be placed on a thorough understanding of the basic concepts as well as developing problem solving skills. Topics covered include: combinatorial analysis, independence and the Markov property, the major discrete and continuous distributions, joint distributions and conditional probability, modes of convergence. These notions will be examined through examples and applications.  

 
 
MATH 6395: C* Algebra (Section 12445)
Time: 9:00-10:00 AM, MWF, 350-PGH
Instructor: D. Blecher
Prerequisites: Basic notions from functional analysis or consent of instructor.
Text(s): Not required.

Instructor will supply notes and a list of good books. A cheap resource which will periodically overlap with the course is J. B. Conway's "A course in operator theory", Springer-Verlag.

Description: This is a one semester course containing some beautiful, and more advanced, topics in functional analysis and operator theory. These topics are very algebraic in nature, indeed often the most inspiring feature is the way in which the algebra, analysis, and topology, all fit and work together. We begin by quickly discussing Banach algebras, spectral theory, and operator theory (complete notes will be provided so that we can get through this fast). Then we develop the theory of C*-algebras, von Neumann algebras, and related spaces such as C*-modules. Finally we discuss some connections and applications to other subjects. It would be best for the student to have taken the basics of functional analysis, although it would be possible to read up these basics as we proceed.  

 
 
MATH 6384: Discrete-Time Models (Section 13582)
Time: 2:30-4:00 PM, TTH, 301-AH
Instructor: Sayit
Prerequisites: Math 6362, or equivalent background in probability.
Text(s): Introduction to Mathematical Finance: Discrete-Time Models, by Stanley R. Pliska, Blackwell, 1997.
Description: This course is for students who seek a rigorous introduction to the modern financial theory of security markets. The course starts with single-periods and then moves to multiperiod models within the framework of a discrete-time paradigm. We study the valuation of financial, interest-rate, and energy derivatives and optimal consumption and investment problems. The notions of risk neutral valuation and martingale will play a central role in our study of valuation of derivative securities. The discrete time stochastic processes relating to the subject will also be examined. The course serves as a prelude to a subsequent course entitled Continuous-Time Models in Finance.  

 
 
MATH 6396: Riemannian Geometry (Section 12468)
Time: 10:00-11:00 AM, MWF, 345-PGH
Instructor: M. Ru
Prerequisites: Graduate standing.
Text(s): Lectures on Differential Geometry by S.S. Chern, W.H. Chen and K.S. Lam, World Scientific, ISBN 981-02-3494-5.
Description: We hope to cover Chapter one to Chapter Five in the textbook. Topics include: Differentiable Manifolds, tensors, exterior algebra, vector bundles, exterior differentiation, integrals of differential forms, Stokes' formula, connections, the fundamental theorem of Riemannian Geometry, geodesic normal coordinates, sectional curvature, the Gauss-Bonnet theorem.  

 
 
MATH 6397: Dynamical Systems (Section 12443)
Time: 1:00-2:30 PM, TTH, 348-PGH
Instructor: K. Josic
Prerequisites: An undergraduate course in analysis and differential equations, or consent of instructor.
Text(s): J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dyanmical Systems and Bifurcations of Vector Fields, Springer Verlag.
Description: This course is a graduate level introduction to the mathematical theory of nonlinear dynamical systems. It is designed for students who want to begin research in dynamical systems, as well as for those who wish to apply dynamical systems techniques in their research. The topics listed below will be illustrated using nonlinear examples from the literature in several areas: Review of ODE basics (existence, uniqueness, smooth dependence on initial conditions and parameters, Gronwall inequalities, fixed points, linearization of vector fields, Jordan normal form of matrices), limit cycles, Poincaré-Bendixson Theorem, Floquet theory and Poincaré maps, bifurcation theory, the center manifold theorem, the Lorenz Equations, the method of averaging, homoclinic orbits and chaos (including the Melnikov method and Smale's horseshoe construction), Šilnikov's Theorem, Wazewski's principle, normal forms, Lyapunov exponents, and elementary perturbation theory.  

 
 
MATH 6397: Statistical Computing (Section 13298)
Time: 1:00-2:30 PM, TTH, 202-SEC
Instructor: C. Peters
Prerequisites: Math 4377 and Math 4331 or consent of instructor.
Text(s): Introduction to Statistics through Resampling Methods and R/Splus, by Philip Good, Wiley, 2005.
Description: An introduction to methods and software for basic computing tasks in statistics, descriptive, graphical and exploratory techniques, sampling and simulation, modeling and fitting, and inference procedures. Most of the computing will be done with the packages R and Splus, although some use will be made of spreadsheet programs and MATLAB.  

 
 
MATH 7304: Theory of Group (Section 13436)
Time: 10:00-11:30 AM, TTH, 345-PGH
Instructor: J. Hausen
Prerequisites: Graduate standing or consent of instructor.
Text(s): An introduction to the Theory of Groups, by Joseph J. Rotman, 4th ed., corrected second printing! Graduate Texts in Mathematics, Springer, New York, 1999.
Description: This is a one-semester course in Group Theory. Topics to be covered include basic group theory, groups acting on sets, the Sylow theorems, normal series, solvable and nilpotent groups, free groups and presentations.  

 
 
MATH 7394: Financial and Energy Data (Section 12446)
Time: 10:00-11:30 AM, TTH, 350-PGH
Instructor: E. Kao
Prerequisites: MATH 6397 Continuous-Time Models in Finance, and MATH 6383 Probability and Statistics. Math 6397, Section 11831, Discrete-Time Models in Finance.
Text(s): Analysis of Financial Time Series, by Tsay, Ruey S., Wiley, 2002.
Description: This course deals with statistical analysis of financial and energy data. We discuss parameter estimation of Brownian motion and their variants (e.g., jump diffusion, mean-reversion processes etc.) We study time series models, ARCH/GARCH models for modeling volatilities and their roles in computing Value-at_Risk. The course uses S-Plus as the computing platform.  

 
 
MATH 7396: Algebraic Multigrid Methods (Section 12419) ( Canceled )
Time: 1:00-2:30 PM, MW, 348-PGH
Instructor: Y. Kuznetsov
Prerequisites: Graduate Courses on Numerical Analysis and Numerical PDEs
Text(s): None
Description: Multigrid (MG) methods are among the most efficient iterative solvers for systems of linear algebraic equations which arise from mesh discretizations of partial differential equations. In this course we discuss a special group of MG methods which are based on multilevel coarsening of the underlying vectors and matrices.We consider several approaches to the constraction of multiplicative and additive algebraic MG preconditioners as well as their applications to the numerical solution of the diffusion type equations.  

 
   

 
 
MATH 7397: Stochastic Calculus & Martingales (Section 13808)
Time: 5:30-7:00 pm, TTH, 348 PGH
Instructor: Sayit
Prerequisites: MATH6397 Continuous-Time Models in Finance, solid background on measure theory and some familarity with Hilbert space theory.
Text(s): Introduction to Stochastic Integration, by Chung and Williams, Birkhauser, 1990.

Reference books:

  • Revus and Yor, Continuous Martingales and Brownian Motion, Springer Verlag, 1991.
  • Karatzas and Shreve, Brownian Motion and Stochastic Calculus, Springer, 2nd ed. 1991.
  • Jacod and Protter, Probability Essentials, Springer Verlag.
Description: This course is an introduction to stochastic integration theory. The main purpose of this course is to develop stochastic integration for right continuous local L2-martingales. The topics to be covered in this course are: predictable and optional σ - algebras, filtrations, stopping times, local Martingales, quadratic variation processes, Ito's formulas, Tanaka's formula, Local times.

We will spend first several weeks to review some elements from measure theoretic probability theory.  

*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.