| MATH 4332 Real Analysis (Section 10644) | |
| Time: | 10:00-11:30am TTH, 309 PGH |
| Instructor: | M. Friedberg |
| Prerequisites: | Math 4331 or consent of instructor. |
| Text(s): | Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, 3nd Edition. |
| Description: | Sequences and series of functions, Implicit and Inverse Function Theorems, Stone-Weierstrass Approximation Theorem, and Lebesgue Theory. |
| MATH 4333 Advanced Abstract Algebra (Section 10645) | |
| Time: | 11:30-1:00 TTH, 301 AH |
| Instructor: | J. Hausen |
| Prerequisites: | Prerequisites: MATH 3330 and consent of instructor |
| Text(s): | Gilbert & Gilbert, Elements of Modern Algebra, the fifth edition, published by Brooks/Cole. (2000) |
| Description: | Direct products, Sylow theory, ideals, extensions of rings, factorization of ring elements, modules, and Galois theory. |
| MATH 4335 PDE (Section 12911) | |
| Time: | 9:00-10:00 am, MWF, 348 PGH |
| Instructor: | D. Wagner |
| Prerequisites: | Math 3331 and 2433. Math 3334 recommended |
| Text(s): | Partial Differential Equations: An Introductions by Walter A. Strauss, John Wiley & Sons, 1992, ISBN 0471548685 |
| Description: | Existence and uniqueness for Cauchy and Dirichlet problems; classification of equations; potential-theoretic methods; other topics at the discretion of the instructor. |
| MATH 4340 Nonlinear Dynamics and Chaos (Section 12921) | |
| Time: | 4:00-5:30 pm, MW, 347 PGH |
| Instructor: | Timofeyev |
| Prerequisites: | Prerequisite: MATH 3331 or consent of instructor. |
| Text(s): | Nonlinear Dynamics and Chaos , by Steven Strogatz, Perseus Books. |
| Description: | Dynamical systems associated with one-dimensional maps of the interval and the circle; elementary bifurcation theory; modeling of real phenomena. |
| MATH 4365: Numerical Analysis (Section 10647) | |
| Time: | 4:00-5:30 pm, TTH, 348 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 direct methods for linear systems of algebraic equations, initial-value problems for ordinay differential equations, eigenvalue problems, and an introduction to partial differential equations. This is an introductory course and will be a mix of mathematics and computing. |
| MATH 4377: Advanced Linear Algebra (Section 10648) | |
| Time: | 10:00-11:30 am, TTH, 347 PGH |
| Instructor: | J. Johnson |
| Prerequisites: | Math 2431 and minimum 3 hours of 3000 level mathematics. |
| Text(s): | Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition, Prentice-Hall. |
| Description: | Topics to be covered in this course include linear equations, vector spaces, polynomials, linear transformations and matrices. |
| MATH 4378: Advanced Linear Algebra (II) (Section 10649) | |
| Time: | 2:30-4:00 pm, TTH, 309 PGH |
| Instructor: | J. Hausen |
| Prerequisites: | Graduate standing or consent of instructor. |
| Text(s): | Linear Algebra, K. Hoffman and R. Kunze, 2nd Edition, Prentice-Hall. |
| Description: | Matrices, eigen-values, and canonical forms. |
| MATH 4383: Number Theory (Section 12922) | |
| Time: | 10:00-11:00 am, MWF, 121 SR |
| Instructor: | J. Hardy |
| Prerequisites: | MATH 3330 Abstract Algebra |
| Text(s): | Elementary Number Theory, 6th Edition by David M. Burton (ISBN:0-07-305188-8) McGraw-Hill Higher Education |
| 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. |
| MATH 4397: Introduction to Options (Section 12898) | |
| Time: | TTH 11:30-1:00, 202 SEC |
| Instructor: | D. Bao |
| Prerequisites: | MATH 2433 and 3338. |
| Text(s): | Will be provided by the instructor. The course will use an unpublished manuscript written by Prof. Bruce Lowe of Texas A&M. |
| Description: | This course uses mathematics to study certain financial instruments known as options. Let's say I expect gasoline prices to reach $4 a gallon by summer 2006. So, I ask Shell Oil to sell me the privilege (option) of buying gasoline at $3.70 per gallon on May 31, 2006. If by that time gasoline costs over $3.70 a gallon on the open market, then I exercise my privilege and make some money. If by that time the price is under $3.70, I simply flush that privilege granted by Shell because it's a right, not a contract. Obviously, Shell has to charge me up front (that is, now) a fair premium before it would issue me the sought agreement. Figuring out such a premium requires a good amount of mathematics (calculus, probability, and partial differential equations).
This is the second of three core courses in a newly designed (pending administrative approval) Finance Option of the B.Sc. degree in Mathematics. The pre-requisites are Calculus III and Probability; an acquaintance with partial differential equations is useful, but is not essential. The topics covered include: calls and puts, American and European vanilla options, expiry, strike price, in the money versus out of the money, arbitrage, call-put parity, long and short positions, portfolio hedging, straddles and strangles and spreads, drift and volatility, Ito calculus, the Greeks, geometric Brownian motion, Black-Scholes theory, binomial model, filtration, and self financing strategy. |
| MATH 4397: Survey on Undergraduate Mathematics (Section 12845) | |
| Time: | ? |
| Instructor: | C. Peters |
| Prerequisites: | Math 3331, 3333, 3330 and 3 hours of 4000 level mathematics or consent of instructor. |
| Text(s): | None. |
| Description: | Brief reviews of analysis, algebra, differential equations, linear algebra, and other topics in the undergraduate mathematics curriculum. This course is approved for three hours credit forward the NS&M Capstone requirement. |
| MATH 5332: Differential Equations (Section 10675) | |
| Time: | On line course |
| Instructor: | G. Etegn |
| Prerequisites: | Math 5331 or consent of instructor. |
| Text(s): | Linear Algebra and Differential Equations , by Golubitsky and Dellnitz, Brooks/Cole. |
| Description: | Linear and nonlinear systems of ordinary differential equations; existence, uniqueness and stability of solutions; initial value problems; higher dimensional systems; Laplace transforms. Theory and applications illustrated by computer assignments and projects. |
| MATH 5382: Probability (Section 12925) | |
| Time: | On line course |
| Instructor: | C. Peters |
| Prerequisites: | Math 2431 and Math 1432 or consent of instructor. |
| Text(s): | Concepts in Probability and Stochastic Modeling, by James J. Higgins & Sallie Keller-McNulty, Duxbury 1995. |
| Description: | Sample spaces, events and axioms of probability; basic discrete and continuous distributions and their relationships; Markov chains, Poisson processes and renewal processes; applications. Applies toward the Master of Arts in Mathematics degree; does not apply toward Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees. |
| MATH 5397: Mathematical Modeling (Section 10678) | |
| Time: | On line course |
| Instructor: | Guidoboni |
| Prerequisites: | Some kowledge of ODE or consent of instructor. |
| Text(s): | A First Course in Mathematical Modeling, by F. R. Giordano, M.D. Weir and W.P. Fox. |
| Description: | Mathematical modeling is the process of creating a mathematical representation of a phenomenon in order to gain a better understanding of that phenomenon. The main goal of this course is to learn how to make a creative use of some mathematical tools, such as differential equations, probability theory and numerical analysis, to build a mathematical description of some physical problems. |
| MATH 5397: Abstract Algebra (Section 12924) | |
| Time: | On line course |
| Instructor: | 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. This course is meant for students who wish to pursue a Master of Arts |
| MATH 6303: Modern Algebra (II) (Section 10719) | |
| 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: | Topics from the theory of groups, rings, fields, and modules with special emphasis on universal constructions. |
| MATH 6321: Real Variables (II) (Section 10742) | |
| Time: | 10:00-11:00 am, MWF, 347 PGH |
| Instructor: | V. Paulsen |
| Prerequisites: | Math 6320 or its equivalent. |
| Text(s): | Gerald B. Folland, Real Analysis: Modern Techniques and their Applications , John Wiley and Sons, ISBN 0471317160. |
| Description: | This course is a continuation of Math 6320. We will cover the material in, roughly, Chapters 4--8 of Folland's text. Topics to be covered include: Basics of functional analysis, the $L^p$ spaces, measures and positive linear functionals, an introduction to Fourier analysis. |
| MATH 6323: Complex Analysis (Section 12918) | |
| Time: | 12:00-1:00 pm, MWF, 345 PGH |
| Instructor: | S. Ji |
| Prerequisites: | Math 6322. |
| 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 6361: Applicable Analysis (II) (Section 10745) | |
| Time: | 11:30-1:00, TTH, 315 PGH |
| Instructor: | Glowinski |
| Prerequisites: | Math 6360 or equivalent. |
| 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:
|
| MATH 6367: Optimization (Section 10746) | |
| Time: | 4:00-5:30 pm, MW, 309 PGH |
| Instructor: | J. He |
| Prerequisites: | Graduate standing or consent of the instructor. |
| Text(s): | The required textbook: Dimitri P. Bertsekas, Dynamic Programming and Optimal Control, Vol. I, 3rd Edition, 2005, Athena Scientific, ISBNs: 1-886529-26-4. |
| Description: | This is an introduction to the modern control theory of dynamic systems, focusing on typical and characteristic results. Linear and nonlinear continuous-time and discrete-time systems are dealt with for finite state space sets, in either a deterministic or a stochastic framework. Continuous-time stochastic control problems, encountered in modern control theory, and discrete-time Markovian decision problems, typical in operations research, are both treated. Simulation-based approximation techniques for dynamic programming are discussed. |
| MATH 6371: NUMERICAL ANALYSIS (Section 10747) | |
| Time: | 1:00-2:30 pm, TTH, 350 PGH |
| Instructor: | R. Hoppe |
| Prerequisites: | Graduate standing in mathematics or consent of instructor. |
| Text(s): | J. Stoer and R. Bulirsch; Introduction to Numerical Analysis , 3rd Edition. Springer, New York, 2002. |
| Description: | We will deal with the construction, analysis and implementation of numerical methods for the solution of initial- and boundary-value problems for systems of ordinary differential equations. In particular, we will consider:
Nonstiff initial value problems
Stiff and differential-algebraic equations
Boundary value problems
Further Literature
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| MATH 6376: Numerical Linear Algebra (Section 12920 ) | |
| Time: | 10:00-11:30 am, TTH, 345 PGH |
| Instructor: | E. Dean |
| Prerequisites: | Math6370, 6371 or consent of instructor. |
| Text(s): | No textbook. |
| 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, and Krylov subspace methods for nonsymmetric problems. We will also look at eigenvalue problems including the QR algorithm, divide-and-conquer technique, Lanczos and Arnoldi procedures. If there is time we will also look at least squares problems. |
| MATH 6378: Basic Scientific Computing (Section 10748) | |
| Time: | 4:30-5:00 pm, TTH, 309 PGH |
| Instructor: | R. Sanders |
| Prerequisites: | Elementary Numerical Analysis. Knowledge of C and/or Fortran. Graduate standing or consent of instructor. |
| Text(s): | High Performance Computing, O'Reilly, Kevin Dowd & Cbarles Severance, the 2nd edition. |
| Description: | Fundamental techniques in high performance scientific computation. Hardware architecture and floating point performance. Pointers and dynamic memory allocation. Data structures and storage techniques related to numerical algorithms. Parallel programming techniques. Code design. Applications to numerical algorithms for the solution of systems of equations, differential equations and optimization. Data visualization. This course also provides an introduction to computer programming issues and techniques related to large scale numerical computation. |
| MATH 6383: Probability and Statistics (II) (Section 10749 ) | |
| Time: | 11:30-1:00, TTH, 345 PGH |
| Instructor: | Matthew Nicol |
| Prerequisites: | Math 6382 or consent of instructor. |
| Text(s): | Mathematical Statistics with Applications, Sixth Edition, by Wackerly, Mendenhall III and Scheaffer, Duxbury Press. (required textbook)
Statistical Inference, 2nd Edition by George Casella and Roger Berger, 2002, Duxbury Press. |
| Description: | This course is an introduction to mathematical statistics. It assumes a knowledge of probability at the level of Wackerly, Mendenhall and Sheaffer (which is one of the set texts for Math 6382). Topics covered include random samples, principles of data reduction, point and interval estimation, hypothesis testing, regression models and asymptotic evaluations. |
| MATH 6384: Continuous-Time Models (Section 12910) | |
| Time: | 2:30-4:00 pm, TTH, 345 PGH |
| Instructor: | Sayit |
| Prerequisites: | Math 6383 or consent of the instructor. |
| Text(s): | Arbitrage Theory in Continuous Time , 2nd edition, by Tomas Bjork, Oxford University Press, 2004, ISBN 0-19-927126-7. |
| Description: | This is a continuation of the course entitled "Discrete-Time Models in Finance." The course studies the roles played by continuous-time stochastic processes in pricing derivative securities. Topics include stochastic calculus, martingales, the Black-Scholes model and its variants, pricing market securities, interest rate models, arbitrage, and hedging. |
| MATH 6395: Operators on Hilbert Spaces (Section 12927 ) | |
| Time: | 1:00-2:00 pm, MWF, 315 PGH |
| Instructor: | V. Paulsen |
| Prerequisites: | Some familiarity with Hilbert spaces and complex analysis. |
| Text(s): | Course notes will be distributed. |
| Description: | We will focus primarily on the theory of reproducing kernel Hilbert spaces and operators on these spaces. These spaces play a central role in many parts of analysis, especially, approximation theory and complex interpolation theory. |
| MATH 6396: Reimannian Geometry (II) (Section 12900) | |
| Time: | 10:00-11:00 am, MWF, 348 PGH |
| Instructor: | M. Ru |
| Prerequisites: | Riemannian Geometry I |
| 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. This is a second part of our year-long course. We will cover chapter six and chapter seven in the textbook. After that, we will discuss some selected topics. |
| MATH 6397: Dynamical Systems (Section 12897) | |
| Time: | 1:00-2:30 pm, TTH, 315 PGH |
| Instructor: | K. Josic |
| Prerequisites: | Consent of the instructor. |
| Text(s): | Brin & Stuck, Introduction to Dynamical Systems, published by Cambridge University Press, ISBN number is 0-521-80841-3. |
| Description: | ? |
| MATH 6397: Stochastic Process (Section 12899) | |
| Time: | 2:30-4:00 pm, TTH, 301 AH |
| Instructor: | M. Nicol |
| Prerequisites: | Consent of instructor. |
| Text(s): | A First Course in Stochastic Processes, Karlin and Taylor, Second Edition, Academic Press. |
| Description: |
|
| Math7350 Topology/Geometry II (Section 12923) | |
| Time: | 1:00-2:30 am, MW, 348 PGH |
| Instructor: | Min Ru |
| Prerequisites: | Math6342 or consent of the instructor |
| Text(s): | I will circulate my own classroom notes |
| Description: | This course intends to cover the geometry part of the syllabus in Topology/Goemetry preliminary examination. It includes: manifolds, the inverse and implicit function theorems, submanifolds, partitions of unity; tangent bundles, vector fields and Lie derivatives, vector bundles; differential forms, tensors and tensor fields on manifolds; exterior algebra, orientation, integration on manifolds, Stokes' theorem; a brief introduction to Riemannian geometry. |
| MATH 7394: Multi-Variable Harmonic Analysis (Section 13095 ) | |
| Time: | 4:00-5:30 pm, MW, 343 PGH |
| Instructor: | M. Papadakis |
| Prerequisites: | Math 6320-21, Math 4355 or a similar mathematics course. |
| Text(s): | Classical and Modern Fourier Analysis, Loukas Grafakos, Prentice Hall. |
| Description: | Lp and week Lp spaces, convolution and approximate identities, interpolation theorems between Lp spaces, maximal functions, the Schwartz class, tempered distributions, convolution operators and multipliers, oscillatory integrals, Fourier coefficients pointwise convergence of Fourier series, Bochner-Riesz summability, the conjugate function and the norm convergence of Fourier series, Hilbert transform, Riesz transforms, Calderon-Zygmund decomposition and singular integrals, Littlewood-Paley theory.
The students and the instructor share presentations. |
| MATH 7396: Mixed Finite Element Methods for Diffusion Equations (Section 13097) | |
| Time: | 1:00-2:30 pm, MW, 350 PGH |
| Instructor: | Y. Kuznetsov |
| Prerequisites: | Graduate Courses on PDEs and Finite Element Methods |
| Text(s): | None |
| Description: | The mixed finite element method is one of the most efficient discretization techniques for elliptic partial differential equations with numerous applications in science and engineering. In this course,we consider mixed finite element methods for the discretization of the diffusion equations on general polygonal and polyhedral meshes.We provide the basic discretization algorithms,stability analysis,and error estimates. We investigate the matrix properties and propose efficient preconditioned iterative solvers for the underlying algebraic systems.We discuss applications of the mixed finite element method for numerical simulation in geosciences and evironmental sciences. |
| MATH 7397: Valuation of Credit Derivatives (Section 13096) | |
| Time: | 10:00-11:30 am, TTH, 350 PGH |
| Instructor: | E. Kao |
| Prerequisites: | Prerequisite: MATH 6382-6383 Mathematical Statistics MATH 6397, Discrete-Time Models in Finance and MATH 6397, Continuous-Time Models in Finance Expreience in Matlab |
| Text(s): | Credit Derivatives Pricing Models P. J. Schonbucher, 2003, ISBN 0-470-84291-1 |
| Description: | In this course, we cover important modelling approaches for valuation of credit derivatives. They include hedge-based pricing to stochastic intensity models, credit rating models and firms' value based models. Topics also include firm value and share price based models, and models for default correlation. |
| MATH 7397: Stochastic Calculus & Martingales II (Section 13152) | |
| Time: | 5:30-7:00 pm, TTH, 315 PGH |
| Instructor: | Sayit |
| Prerequisites: | Math7693 or consent of the instructor. |
| Text(s): | Stochastic Integration and Differential Equations, by Philip Protter, 2nd edition, corrected, Version 2.1 2005. |
| Description: | This course will be based on the first four chapters of Protter's book.We begin with a discussion of Lévy processes. Then we define stochastic integration for semimartingales and show Kunita-Watanabe inequality and Itô's formula. We discuss Lévy's charaterization theorem, Change of time theorem, Doob-Meyer decomposition theorem, Girsanov-Meyer theorem, and martingale representation theorem. |
*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.