Senior and Graduate Math Course Offerings Fall 2003
- The time for Math 6397: THEORETICAL COMPUTATIONAL NEUROSCIENCES has been changed from "4:00-5:30 pm, TTH, 350 PGH" to "4:00-5:30 pm, MW, 204 AH".
- The time for Math 7394: NUMERICAL METHODS FOR SOME PDE has been changed from "2:30-4:00 pm, TTH, 350 PGH" to "11:30-1:00 pm, TTH, 162 F".
- The course Math 4385 Mathematical Statistics and MATH 4397 Theory of Computation have been canceled.
| MATH 4315: GRAPH THEORY WITH APPLICATIONS (Section 09890) |
| Time: |
1:00-2:00 pm, MWF, 154F |
| 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 09891) |
| Time: |
1:00-2:00 pm, MWF, 345 PGH |
| Instructor: |
M. Friedberg |
| Prerequisites: |
Math 3334 or consent of instructor. |
| Text(s): |
Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, 3nd Edition. |
| Description: |
Basic topology at metric spaces, Sequences and series, Continuity in metric spaces, Riemann-Stieltjes integrals. |
| MATH 4355 : MATHEMATICAL SIGNAL REPRESENTATION (Section 11918) |
| Time: |
5:30-7:00 pm, MW, 204 AH
|
| Instructor: |
M. Papadakis |
| Prerequisites: |
Linear algebra course Math 2431 and Calculus II. |
| Text(s): |
A. Pinkus, S. Zafrany, Fourier Series and Integral transforms, Cambridge UP, 1997.
Supplementary Textbooks: N. Kalouptsidis, Signal Processing and systems, J.Wiley, 1997. |
| Description: |
The linear aalgebraof inner product spaces. Finite orthogonal and orthonormal systems, Gram-Schmidt orthogonalization, norms, convergence in the norm, infinite orthonormal bases.
Fourier Series of real valued functions, uniform and pointwise convergence of Fourier Series, Parseval's theorem and the L2-convergence of Fourier Series.
The Integral Fourier transform: Definition, properties, Riemann-Lebesgue lemma, Inverse Fourier transform, convolutions and time-invariant linear systems, Plancherel's theorem, Band-limited and time-limited signals (functions), filtering and its connection with Fourier inversion. An intuitive introduction to tempered distributions and the Dirac delta function. Shannon's sampling theorem. The Discrete Fourier transform (Fourier transform on finite additive cyclic groups), and the Fast Fourier transform and the use of FFT in the computation of Integral Fourier transforms.
The following will be covered if time permits.
2-D Fourier transform and an introduction to JPEG. The Short-time Fourier transform and other windowed Fourier transforms. The Laplace transform and. its inversion. |
| Remarks: |
Math 4355 will not have a follow up course per, but it furnishes the basics for other courses, even for PDE courses. I plan on offering a 6395 course in Spring 2004 "Mathematics of Computerized Tomography". |
| MATH 4364: NUMERICAL ANALYSIS (Section 09896) |
| 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 either FORTRAN or C. |
| 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 of ordinary differential equations, and direct 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. |
| MATH 4370 : MATHEMATICS OF FINANCIAL DERIVATIVES (Section 11919) |
| Time: |
10:00-11:30 am, TTH, 350 PGH
|
| Instructor: |
E. Kao |
| Prerequisites: |
Math 3338, 3339, or equivalent background in probability and statistics. |
| Text(s): |
Options, Futures, and Other Derivatives, by John C. Hull, Prentice-Hall, 2003 |
| Description: |
Course Description: This course is an introduction to financial derivatives. It includes a survey of options, futures, and other derivatives used in equity, fixed income, and energy markets. We will study Black-Scholes model and its extensions for valuation of these contingent claims. We will cover the quantitative methods used for valuation and hedging. They include stochastic calculus, dynamic optimization, Monte Carlo simulation, and numerical methods. |
| MATH 4377: ADVANCED LINEAR ALGEBRA (Section 09897) |
| Time: |
10:00-11:00 am, MWF, 347 PGH |
| Instructor: |
J. Hausen |
| Prerequisites: |
Math 2431 and at least 3 semester hours of junior level math courses. |
| Text(s): |
Linear Algebra, K. Hoffman and R. Kunze, 3rd(?) 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 6198 : TEACHING PRACTICUM (Section 09906) |
| Time: |
TO BE ARRANGED |
| Instructor: |
D. Blecher |
| Prerequisites: |
No |
| Text(s): |
How to teach mathematics? Krantz, 2nd edition, AMS, ISBN 0-821-81398-6. |
| Description: |
Introduction to teaching and assisting at the 色花堂. |
| Remarks: |
One credit hour course. Combining with the 2 credit hours course Math 6298 next. Credit will not be counted as in the graduate requirement. |
| MATH 6298 : INTRODUCTION TO COMPUTING RESOURCES (Section 09912) |
| Time: |
TO BE ARRANGED, Rm. 648 PGH |
| Instructor: |
A. Torok |
| Prerequisites: |
Graduate standing or consent of instructor |
| Text(s): |
No |
| Description: |
The purpose of this course is to familiarize students with the computer tools that are relevant for mathematical research in today's environment. It is intended primarily for graduate students and math majors, but it is useful for anybody interested in these topics. The topics we plan to discuss include the Unix and Linux operating systems, a multi-functional text editor (emacs), software for mathematical publications (TeX and its dialects), languages for formal and numerical computations (Maple, Mathemtica, Matlab), web-publishing (HTML) and Internet use (mail, electronic archives etc.). We will also mention a few principles of writing and presenting a mathematical paper. The course will consists of weekly workshops accompanied by hands-on applications in the computer lab of the Math Department, followed by individual projects. These projects (e.g., typesetting a short mathematical paper, designing a web-page, writing programs in various languages) will give the students the opportunity to practice the notions they are being taught. The material used for this course will be either available on the web or handed out in class. |
| Remarks: |
Two credit hours course. Combining with the 1 credit hour course Math 6198 above. Credit will not be counted as in the graduate requirement |
| MATH 6302: MODERN ALGEBRA (Section 09913) |
| Time: |
11:30 am-1:00 pm, TTH, 134 SR |
| Instructor: |
J. Johnson |
| Prerequisites: |
Math 4333 or Math 4378, or consent of instructor. |
| Text(s): |
Algebra, by Thomas W. Hungerford, Springer-Verlag; Graduate Texts in Mathematics #73. Lastest edition and printing. |
| Description: |
This couese in modern algebra will include topics from the theory of groups, rings, fields with special emphasis on modules and universal constructions. |
| Remarks: |
This is a first semester of a two semester course. |
| MATH 6320: FUNCTIONS OF A REAL VARIABLE (Section 09941) |
| Time: |
4:00-5:30 pm, MW, 350 PGH |
| Instructor: |
V. Paulsen |
| Prerequisites: |
4331-4332 or equivalent. |
| Text(s): |
Foundations of modern analysis, by Avner Friedman, Dover. |
| Description: |
This course begins with an introduction to the concept of measure, followed by a detailed examination of Lebesgue measure and integration. |
| MATH 6322: Theory of Functions of a Complex Variable (Section 11824) |
| Time: |
9:00-10:00 am, MWF, 347 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. |
| Remarks: |
This is a first semester of a two semester course. |
| MATH 6326: PARTIAL DIFFERENTIAL EQUATIONS (Section 11825) |
Time: |
5:30-7:00 pm, MW, 309 PGH |
| Instructor: |
B. Keyfitz |
| 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 (Section 11826) |
| Time: |
1:00-2:30 pm, TTH, 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 (required). |
| Description: |
This is the first semester of a two-semester introductory graduate course in 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 axioms, compactness, connectedness, nets, continuity, local compactness, Urysohn's lemma, Tietze, and basic constructions such as subspaces, quotients, products, the fundamental group. |
| MATH 6360: APPLICABLE ANALYSIS (Section 11827) |
| Time: |
10:00-11:30 am, TTH, 348 PGH |
| Instructor: |
Jeffrey Morgan |
| Prerequisites: |
Math 4331-32. |
| Text(s): |
Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences, Vol 40) by Arch W. Naylor, George R. Sell. |
| 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 6366 : OPTIMIZATION AND VARIATIONAL METHODS (Section 11828) |
| Time: |
5:30-7:00 pm, TTH, 348 PGH |
| Instructor: |
E. Dean |
| Prerequisites: |
Math 4331 and 4377 or consent of instructor. |
| Text(s): |
Numerical Optimization, by J. Nocedal and S.J. Wright. |
| Description: |
This is the first semester of a two semester course. The topics for this first semester will include the theory of finite dimensional linear and nonlinear optimization and numerical methods. This course will be a mix of mathematics and practicalities. The second semester, depending on student interest and background, will include topics in dynamic optimization and/or more general optimization theory. |
| MATH 6370: NUMERICAL ANALYSIS (Section 09944) |
| 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 (such as MATH 3333). This is the first semester of a two-semester course. |
| Text(s): |
Numerical Linear Algebra, Lloyd N . Trefethen and David Bau, SIAM, 1997, ISBN: 0898713617 (required).
Introduction to Numerical Analysis , J. Stoer and R. Bulirsch, Springer-Verlag, 3rd Edition, New York, 2002, ISBN 038795452X ((recommended). |
| 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. |
| Remark: |
This is a first semester of a two semester course. |
| MATH 6377: BASIC TOOLS IN APPLIED MATHEMATICS (Section 09946) |
| 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. |
| MATH 6382: PROBABILITY MODELS AND MATHEMATICAL STATISTICS (Section 09947) |
| Time: |
2:30-4:00 pm, TTH, 347 PGH |
| Instructor: |
C. Peters |
| Prerequisites: |
Math 3334 and Math 4377, or consent of instructor. |
| Text(s): |
An Introduction to Stochastic Processes, by Edward Kao, Duxbury, 1997. |
| Description: |
Probability theory, distributions, Markov processes, Counting processes, 2nd order stationary processes, Brownian motion, and applications. |
| MATH 6397 : DISCRETE TIME MODELS IN FINANCE (Section 11829) |
| Time: |
4:00-5:30 pm, TTH, 309 PGH |
| 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: |
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 6397: THEORETICAL COMPUTATIONAL NEUROSCIENCES (Section 11831) |
| Time: |
4:00-5:30 pm, MW, 204 AH. |
| Instructor: |
Kresimir Josic |
| Prerequisites: |
Math 3331 ODE |
| Text(s): |
Alwyn Scott, Neuroscience: A Mathematical Primer.
Hugh Wilson, Spikes, Decisions and Actions. |
| Description: |
This course introduces students to standard mathematical models of individual neurons (Hodgkin-Huxley, `integrate and fire', etc.) and the synaptic events by which neurons communicate. We will also spend some time discussing simple models of signal propagation along neurons. We will spend most of the first semester of the course learning about models of single neurons, and models of communication between neurons, both chemical and electrical. If there is sufficient interest, there will be a second semester of the course which will deal with the modeling of neuronal networks.
The course is designed for advanced undergraduate and graduate students in mathematics, physics, engineering and the biological sciences and will be centered on differential equation models of neurons. The software package XPP will be used to simulate the behavior of small networks of neurons. There are no biological pre-requisites. The course material will prepare interested students for continuing research projects in the area of theoretical and computational neurobiology. |
| Remarks: |
for more information on Mathematical Biology in this ddepartment to see the instructor's webpage of this course. |
| MATH 6397 : KNOWLEDGE BASED ALGORITHMS (Section 11832) |
| Time: |
4:00-5:30 pm, MW, 315 PGH |
| Instructor: |
S. Fajtlowicz |
| Prerequisites: |
Graduate standing or consent of instructor. |
| Text(s): |
The course will be based on the instructor's notes. |
| Description: |
The purpose of this course is to discuss computer programs capable of making mathematical conjectures and scientific hypotheses. These ideas are based mostly on experiences with computer program Graffiti whose conjectures inspired number of papers, some by the most prominent mathematicians and more recently a few papers in chemistry.
A version of Graffiti will be used by participants to learn or to expand their knowledge of one of several possible subjects of their own choice Texas style - the method developed by UT Professor L. A. Moore. These subjects will include expanders, number theory, hypergraphs, eigenvalues, and fullerenes. One significant difference will be that rather than to be led to discovery of known results, the students will work exclusively on conjectures of Red Burton - an educational version of Graffiti, without getting any hints whether these conjectures are true or false. This will create more realistic scenario for research experience. Active participants will have opportunity to discover new original results. |
| Remark |
Click to see more information about this course. |
| Math 6397: PROBABILITY (OnLine course) (Section 12069) |
| Time: |
OnLine |
| Instructor: |
C. Peters |
| Prerequisites: |
Math 2431 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 6397: ANALYSIS (OnLine course) (Section 12070) |
| Time: |
OnLine |
| 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 7324: BIFURCATION THEORYMATH (Section 11834) |
| Time: |
1:00-2:30 pm, TTH, 345 PGH |
| Instructor: |
M. Golubitsky |
| Prerequisites: |
Math 3334, 4362, 4378 or consent of instructor. |
| Text(s): |
Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, 1984. |
| Description: |
Steady-state and Hopf bifurcation, Liapunov-Schmidt reduction, singularity theory including unfolding theory, and classification of bifurcations by codimension.
Both theory and applications will be considered. |
| Remarks: |
This is a first semester of a two semester course. |
| MATH 7350: GEOMETRY OF MANIFOLDS (Section 11835) |
| Time: |
12:00-1:00 pm, MWF, 345 PGH |
| Instructor: |
M. Field |
| Prerequisites: |
Calculus and basic topology. Click for more details. |
| Text(s): |
Topology from the Differentiable Viewpoint by John Milnor. Paperback, 1997. Available from at the princely price of $14.95 (plus shipping). |
| Description: |
The first semester of this course will be an introduction to differential calculus on spaces other than Euclidean space. Even for calculus of functions of one real variable, critical point theory ('maxima and minima of functions) is a amazingly powerful tool. When we look at functions defined on more interesting spaces, such as the two dimensional sphere or projective space, we find that the topology or geometry of the space is intertwined with critical point theory. Much of the first semester will be spent in developing the tools and techniques for critical point theory on manifolds. As well as developing the basic theory, we will devote a fair amount of time to careful examination of some inspiring examples of manifolds. |
| Remarks: |
This is a first semester of a two semester course. For more information about this course, please click .
|
| MATH 7394: NUMERICAL METHODS FOR SOME NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS FROM MATHEMATICAL PHYSICS, CONTINUUM MECHANICS, AND DIFFERENTIAL GEOMETRY (Section 11836) |
| Time: |
11:30-1:00 pm, TTH, 162 F |
| Instructor: |
R. Glowinski |
| Prerequisites: |
Check with instructor. |
| Text(s): |
No. |
| Description: |
The main goal of this course is to introduce the students to computational methods for the solution of nonlinear partial differential equations playing in important role in physics, mmechanics and differential geometry. Indeed, th key idea will be find a unifying paradigm in these various models so that a relatively sasmallumber of techniques will be necessary to solve problem which, at first glance, hvhaveittle in relation with each other. The equations to be considered will include the Eikonal equation, the Monge-Ampere equation, the total curvature equation, the Ginzburg-Landau equation, and also the Navier-Stokes equations, since surprisingly, at first glance, all of the above can be reduced to mathematical models very close to those modeling incompressible viscous flow. |
| MATH 7396: COMPUTATIONAL ELECTROMAGNETICS (Section 11837) |
| Time: |
1:00-2:30 pm, MW, 314 PGH |
| Instructor: |
R. Hoppe |
| Prerequisites: |
Prerequisites: Calculus, Linear Algebra, and Numerical Analysis. |
| Text(s): |
A. Bossavit, Computational Electromagnetism. Academic Press, Boston, 1998. J. Lin, The Finite Element Method in Electro- magnetics, Wiley, New York, 1993 |
| Description: |
We give an introduction to the basic problems in electromagnetic field theory associated with Maxwell's equations in the low frequency case (eddy current equations) and in the high frequency regime (time-harmonic approach; waveguides, scattering). Then, we focus on numerical solution techniques with emphasis on the Finite Element Method (FEM) including edge elements, the Boundary Element Method (BEM), and the hybrid FEM/BEM Method for coupled interior- exterior domain problems. We will also address the Finite Integration Technique (FIT) and multipole 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.