Note: Math 4365, 6367, 6371, 6397 do not require the 2004 full semester courses as the prerequisites.


 
 
MATH 4332: INTRODUCTION TO REAL ANALYSIS (Section 10397) 
Time: 1:00-2:00 pm, MWF, 315 PGH  
Instructor: S. Ji
Prerequisites: Math 4331 or consent of instructor.  
Text(s): Principles of Mathematical Analysis, Walter Rudin, McGraw-Hill, Latest Edition (required).
Description: Sequences and series of functions, Contraction Mapping Principle, Implicit and Inverse Function Theorems, Lebesgue Theory for the Real Line. Rigorous proofs are an essential part of this course.

 
 
MATH 4333: ABSTRACT ALGEBRA (Section 12471) 
Time: 1:00-2:30 pm, TTH, 350 PGH  
Instructor: J. Hausen
Prerequisites: Math 3330 or equivalent.  
Text(s): Modern Algebra (Fourth Edition), John R. Durbin, John Wiley & Sons, Inc., 2000 (ISBN 0-471-32147-8)
Description: The students are assumed to be familiar with basic number theory and fundamental concepts from group theory. The course will cover most of Chapters VI through X of the text: Introduction to Rings, The Field of quotients of a domain, Polynomials, Quotient Rings, Field Extensions. Homework assignments will be an integral part of the course.

 
 
MATH 4351: DIFFERENTIAL GEOMETRY II (Section 12473) 
Time: 10:00-11:00 MWF, 347 PGH  
Instructor: M. Ru
Prerequisites: Math 4350 Differential geometry I or consent of instructor.  
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 4365: NUMERICAL ANALYSIS II (Section 10398)
Time: 4:00-5:30 pm, MW, 309 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, Pascal, Matlab, and Maple.
Text(s): Numerical Analysis (Seventh edition), R.L. Burden and J.D. Faires. 7th edition, Thomson, 2001.
Description: The first semester is not a prerequisite. We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on the iterative methods for solving linear systems, approximation theory, numerical solutions of nonlinear equations, iterative methods for approximating eigenvalues, and elementary methods for ordinary differential equations with boundary conditions and partial differential equations. This is an introductory course and will be a mix of mathematics and computing.

 
 
MATH 4377: ADVANCED LINEAR ALGEBRA I (Section 10399)
Time: 11:30-1:00, TTH, 301 AH  
Instructor: K. Kaiser
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 4378: ADVANCED LINEAR ALGEBRA II (Section 10400)
Time: 2:30-4:00 pm, TTh, 309 PGH  
Instructor: J. Johnson
Prerequisites: Math 4377, or consent of instructor. 
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 4397: STOCHEASTIC PROCESSES (Section 12618)
Time: 4:00-5:30 pm, MW, 345 PGH  
Instructor: K. Josic
Prerequisites: Math 3341 or 3338: Probability Theory, or Equivalent, or consent of instructor.
Text(s): Howard M. Taylor and Samuel Karlin, An Introduction to Stochastic Modeling, 3rd edition, Academic Press

Optional textbook: Linda Allen, Stochastic Processes with applications in biology, Pearson, Prentice Hall.

Description: Many processes in nature are intrinsically stochastic, a property that frequently needs to be reflected when they are modeled. This course introduces students to a variety of probabilistic techniques for mathematical modeling. The course will start with a review of the basics of probability theory. The mathematical topics covered in the course will include generating functions, Poisson and Markov processes (discrete and continuous), branching processes, renewal processes and an introduction to stochastic calculus and diffusion.

The use of each of these mathematical techniques will be illustrated in a variety of examples including many from biology. The background for each problem will be described, mathematical models will be developed and studied, and the implications of the mathematical results will be interpreted.  

 
 
Math 5332: ORDINARY DIFFERENTAIAL EQUATIONS (OnLine course) (Section 10422)
Time: On Line
Instructor: G. Etgen
Prerequisites: Consent of instructor.  
Text(s): Linear Algebra and Differential Equations , by Golubitsky and Dellnitz, Brooks/Cole.
Description: ?

 
 
Math 5383: NUMBER THEORY (OnLine course) (Section 10421)
Time: On Line
Instructor: M. Ru
Prerequisites: None.  
Text(s): Discovering Number Theory, by Jeffrey J. Holt and John W. Jones, W.H. Freeman and Company, New York, 2001.
Description: Number theory is a subject that has interested people for thousand of years. This course is a one-semester long graduate course on number theory. Topics to be covered include divisibility and factorization, linear Diophantine equations, congruences, applications of congruences, solving linear congruences, primes of special forms, the Chinese Remainder Theorem, multiplicative orders, the Euler function, primitive roots, quadratic congruences, representation problems and continued fractions. There are no specific prerequisites beyond basic algebra and some ability in reading and writing mathematical proofs. The method of presentation in this course is quite different. Rather than simply presenting the material, students first work to discover many of the important concepts and theorems themselves. After reading a brief introductory material on a particular subject, students work through electronic materials that contain additional background, exercises, and Research Questions, using either mathematica, maple, or HTML with Java applets. The research questions are typically more open ended and require students to respond with a conjecture and proof. We then present the theory of the material which the students have worked on, along with the proofs. The homework problems contain both computational problems and questions requiring proofs. It is hoped that students, through this course, not only learn the material, learn how to write the proofs, but also gain valuable insight into some of the realities of mathematical research by developing the subject matter on their own.

 
 
Math 5386: REGRESSION AND LINEAR MODELS (OnLine course) (Section 10423 )
Time: On Line
Instructor: C. Peters
Prerequisites: Math 5385 Statistics and Math 5331 Linear algebra, or consent of instructor.  
Text(s): Introduction to Linear Regression Analysis , 3rd Edition Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining, 2001, Wiley. ISBN: 0-471-31565-6  
Description: Simple and multiple linear regression, normal inferece, regression diagnostics, model selection, robust regression, and other topics as time permits. Computing projects using R, Mathlab, or another package will be assigned.  

 
 
Math 5397: MATHEMATICAL MODELLING (OnLine course) (Section 10424)
Time: On Line
Instructor: G. Guidoboni
Prerequisites: some kowledge of odes or consent of instructor  
Text(s): None. The course notes will be provided.
Description: Mathematical modeling is the process of creating a mathematical representation of some 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 6303: MODERN ALGEBRA II (Section 10465)
Time: 10:00-11:30 am, TTH, 309 PGH  
Instructor: J. Hausen
Prerequisites: MATH 6302 (Modern algebra) or equivalent.
Text(s): W. J. Wickless, A FIRST GRADUATE COURSE IN ABSTRACT ALGEBRA, Marcel Dekker, Inc., New York, 2004. ISBN: 0-8247-5627-4.
Description: This is the second part of a two-semester course on Abstract Algebra. The first part covered most of the first three chapters (Groups, Rings, and Modules) including some results on infinite dimensional Vector Spaces. Chapters five and six (Fields and Galois Theory, Topics in Noncommutative Rings) will be the subject of the spring semester. Additional topics as time permits. Homework will be an integral part of the course.  

 
 
MATH 6321: FUNCTIONS OF A REAL VARIABLE II (Section 10487)
Time: 12:00-1:00 pm, MWF, 345 PGH  
Instructor: M. Friedberg
Prerequisites: Math 6320 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.  

 
 
 MATH 6325: ORDINARY DIFFERENTIAL EQUATION II (Section 12480 )

Time: 		11:00-12:00 am, MWF, 345 PGH
Instructor: J. Morgan
Prerequisites: Math 6324 (ODE I)  
Text(s): None. The course notes will be provided.
Description: This is the second semester of a two semester sequence. The topics from the spring semester will include:

 
 
 MATH 6343: Topology II (Section 10490)

Time: 		9:00-10:00 am, MWF, 350 PGH
Instructor: D. Blecher
Prerequisites: Math 6342 or consent of instructor.  
Text(s): Topology, A First Course, J. R. Munkres, Second Edition, Prentice-Hall Publishers (required).
Description: This is the second 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 mostly concentrate on algebraic topology. Thus in the text (Munkres) we hope to cover the main results in Chapters 9-13. Some additional material on homology will also be covered. We also hope to assign several course projects you can choose between.

The final grade is aproximately based on a total score of 400 or 500 points consisting of homework (100 points), a semester test (100 points), maybe a course project (100 points?), and a final exam (200 points). The instructor may change this at his discretion.


 
 
 MATH 6346: Topological Groups (Section 12472)

Time: 		10:00-11:00 am, MWF, 348 PGH
Instructor: V. Paulsen
Prerequisites: Some knowledge of group theory and linear algebra.  
Text(s): Symmetry Groups and Their Applications, Willard Miller, Academic Press, Series in Pure and Applied Math, vol 50. ISBN 0-12-497460-0.
Description: This course will be loosely based on the above text. Lecture notes will be distributed. The course will begin with a brief introduction to some important groups. We will then introduce the concept of a group acting on a set and of group algebras. We will study the basics of group representation theory, including Schur's theorem, the theory of irreducible representations, and the characters of a group. In the second part of the course, we will introduce topological groups, Haar measures and generalize many of the results of the first semester to continuous representations.

 
 
MATH 6361: APPLICABLE ANALYSIS II (Section 10491)
Time: 11:30-1:00 TTH, 315 PGH
Instructor: R. 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:
  • 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 6367: OPTIMIZATION II (Section 10492)

Time:		 5:30-7:00 pm, MW, 350 PGH
Instructor: E. Dean
Prerequisites: Math 4331 and 4377 or consent of instructor.
Text(s): Numerical Optimization,. Jorge Nocedal, and Stephen J. Wright, Wiley, Springer, 2000.
Description: This course will be a mix of mathematics and practicalities in numerical optimization. We will look at the following topics: linear programming, (small and large scale) nonlinear programmming, and (depending on the students' interests) a short introduction to dynamic programming. This is the second semester of a two semester course but it will be self-contained and so the first semester is not a prerequisite.

 
 
MATH 6371: NUMERICAL ANALYSIS II (Section 10493)
Time: 4:00-5:30 pm, MW, 350 PGH
Instructor: J. He
Prerequisites: Graduate standing or consent of the instructor. Students should have had a course on linear algebra and an introductory course on analysis and ODEs. This is the second semester of a two semester course. The first semester is not a prerequisite, but some familiarity with numerical solution of linear system is assumed.
Text(s): Numerical Mathematics, Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Springer Verlag, 2000, ISBN: 0387989595.
Description: This is the second semester of a two semester course. The focus in this semester is on approximation theory and numerical solution of ODEs. The applications of approximation theory to interpolation, least-squares approximation, numerical differentiation and Gaussian integration will be addressed. The concepts of consistency, convergence, stability for the numerical solution of ODEs will be discussed. This is the first semester of a two-semester course.

 
 
 
 
MATH 6378: BASIC SCIENTIFIC COMPUTING (Section 10495)
Time:  4:00-5:30 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 MODELS AND MATHEMATICAL STATISTICS II (Section 10496)
Time: 8:30-10:00 am, TTH, 348 PGH.
Instructor: M. Nicol
Prerequisites: Math 6382 or consent of instructor.
Text(s): Recommended Texts:

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 Sheldon Ross (A First Course in Probability, Prentice Hall) which is the set text for Math 6382. Topics covered include random samples, principles of data reduction, point and interval estimation, hypothesis testing, regression models and asymptotic evaluations.

 
 
MATH 6397: CONTINUOUS-TIME MODELS IN FINANCE (Section 12623)
Time: 4:00-5:30 pm, TTH, 350 PGH
Instructor: E. Kao
Prerequisites: MATH 6397 Discrete-Time Models in Finance MATH 6282 Probability and Statistics
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 enetitled "Discrete-Time Models in Finance." The course studies the roles played by continuous-time stochastic processes in pricing derivative securities. Topics incluse stochastic calculus, martingales, the Black-Scholes model and its variants, pricing market securities, interest rate models, arbitrage, and hedging.

 
 
MATH 6397: DYNAMICS II (Section 12622)
Time: 10:00-11:30 am, TTH, 350 PGH
Instructor: M. Golubitsky
Prerequisites: ODEs Math 6324 and undergraduate courses in group theory and senior-level linear algebra. Some familiarity with group representation theory would be helpful.

Dynamics I (Fall '04) is not a prerequisite for this course

Text(s): None required

Background mathematics can be found in

  • "The Symmetry Perspective", M. Golubitsky and I. Stewart, Birkhauser, Basel, 2002.
  • the first two chapters of "Singularities and Groups in Bifurcation Theory: Vol. II", M. Golubitsky, I.N. Stewart and D.G. Schaeffer, Applied Mathematical Sciences 69. Springer-Verlag, New York, 1988.
Description: This course will focus on the patterns of synchrony in coupled systems of ODEs and on the sometimes surprising dynamics that such systems can produce. Most of the reading material will be from recent papers.

 
 
MATH 6397: MATHEMATICAL HEMODYNAMICS II (Section 12621 )
Time: 4:00-5:30 pm, MW, 347 PGH
Instructor: S. Canic
Prerequisites: Consent of instructor
Text(s): None.

A list of reference books that will be used in the course is on the Syllabus. They include: Strauss: PDEs, Chorin and Marsden: Fluid Mechanics McOwen: PDEs LeVeque: Conservation Laws Fung: Curculation McDonald: Hemodynamics.

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 7321: FUNCTIONAL ANALYSIS II (Section 12470)
Time: 1:00-2:30 MW, 350 PGH
Instructor: A. Torok
Prerequisites: Mth 7320 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 continue from Conway's book: weak topologies, compact and Fredholm operators. Afterward, we will consider applications to differential equations (distribution theory, solution to elliptic PDE's) and stochastic differential equations (infinitesimal generator of a semigroup). Depending on the interest of those enrolled, other applications can also be discussed.

 
 
Math 7374: Finite Element Methods (Section 12482)
Time: 1:00-2:30 pm, TTH, 315 PGH
Instructor: R. Hoppe
Prerequisites: Calculus, Linear Algebra, and Numerical Analysis.
Text(s): The following reference books are not required.
  • D. Braess; Finite Elements Theory, Fast Solvers and Applications in Solid Mechanics. 2nd Edition. Cambridge Univ. Press, Cambridge, 2001.
  • S.C. Brenner and L. Ridgway Scott; The Mathematical Theory of Finite Element Methods, 2nd Edition. Springer, New York, 2002.
Description: Finite Element Methods are widely used discretization techniques for the numerical solution of PDEs based on appropriate variational formulations. We begin with basic principles for the construction of Conforming Finite Elements and Finite Element Spaces with respect to triangulations of the computational domain. Then, we study in detail a priori estimates for the global discretization error in various norms of the underlying function space. Nonconforming and Mixed Finite Element Methods will be addressed as well. A further important issue is adaptive grid refinement on the basis of efficient and reliable a posteriori error estimators for the global discretization error.

 
 
Math 7379: MONTE CARLO METHOD (Section 12620)
Time: 10:00-11:30 TTH, 347 PGH
Instructor: E. Kao
Prerequisites: MATH 6382-6383 Mathematical Statistics MATH 6397, Discrete-Time Models in Finance and Continuous-Time Models in Finance Expreience in Matlab
Text(s): Monte Carlo Methods in Financial Engineering by Paul Glasserman, Springer, New York, 2003, ISBN 0-387-00451-3.
Description: In this course, we first cover the fundamentals of Monte-Carlo methods. We then review pricing of contingency claims. The course is about the use of MonteCarlo methods for valuation of a variety of contingency claims. We will study techniques for improving accuracy and efficincy. Students in the course are expected to write computer programs in Matlab for implementation.

 
 
Math 7396: DECOMPOSITION DOMIAN METHOD ( Section 12619 )
Time: ? PGH
Instructor: Y. Kuznetsov
Prerequisites: ?
Text(s): No required textbook.
Description:
  

*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.