600-level courses during Fall 2001

Each semester, faculty propose some 600-level courses to possibly be offered to graduate students, primarily doctoral students. Some of these courses follow up on 500-level courses, and some are primarily focused on current research. Depending on several variables, including student interest, these courses may be offered as regular courses, or they may be offered by faculty as independent study courses, or the course may be withdrawn (i.e., cancelled).

Math 640: Multipliers for Fourier series and transforms

Instructor: Jindrich Necas

Special Guest-Star: Daniel Grubb

Prerequisite: Math 530

This course is a natural extension of Fourier series to Lp functions, with elements of Lacunar theory and an easy version of Marcinhiewicz's multiplier theorem. Other topics will include the relationship with the Fourier transform, especially to singular kernels, and some modern tools as maximal operators.

Texts:

• Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, 1970.
• Elias M. Stein, Harmonic Analysis, Princeton, 1993.

Math 650: Surfaces of Constant Mean Curvature

Instructor: Hongyou Wu

Prerequisite: Math 532

At each point p on a surface in space, the surface has a most curved direction and a least curved direction, yielding a maximum curvature and a minimum curvature. The surface is said to have constant mean curvature if the average H of its maximum curvature and minimum curvature does not change when p varies on the surface. In differential geometry, these surfaces are basic surfaces. These surfaces are also natural: among the surfaces with a common boundary, the one with the least area (if it exists) must have H identically zero, and is usually called a minimal surface. The surface with the minimum area under a volume constraint must have a non-zero constant mean curvature and is now commonly called a CMC surface.

In this course we will first motivate and define the above concepts carefully. Then we will examine some well-known examples of such surfaces in detail. Afterward, we will discuss construction of these surfaces from holomorphic (analytic) functions.

Interested? Visit the galleries of minimal surfaces and CMC surfaces at GANG.

Text: none.

References:

1. A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, 1998.
2. M. do carmo, Differential Geometry of Curves of Surfaces, Prentice-Hall, 1976.
3. J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633-668.
4. J. Dorfmeister and H. Wu, Construction of constant mean curvature trinoids from holomorphic potentials, preprint.

Math 660: Advanced Topics in Numerical Linear Algebra

Instructor: Biswa Datta

Prerequisite: Math 434, but Math 564 or its equivalent is preferred.

The course will cover important topics in numerical linear algebra that span a variety of practical applications, including signal processing and statistical analysis. The application, computation, and computer implementation of each topic will be covered. Topics will include:

• Generalized and quadratic eigenvalue problems
• Structured (Toeplitz, vandermonde, etc.) linear systems and eigenvalue problems
• Large-scale eigenvalue problems (Krylov subspace methods)
• Constrained and total least-squares
• Updating matrix factorizations (QR and Cholesky)
• Singular value problems, including the structured singular value problem

Text: None.

References:

• Gene Golub and C. Van Loan, Matrix Computations, Johns Hopkins University Press, 1996.
• Biswa Datta, Numerical Linear Algebra and Applications, Brooks/Cole Publishing Company, 1995.
• Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, 1992.
• Apropriate reprints and preprints from current literature

Math 680: Heights and Diophantine Approximation

Instructor: Jeffrey Thunder

Prerequisite: Math 520

This course will start with the study of valuations on the field of rational numbers and how they can be extended to finite algebraic extensions of the rationals (number fields). Heights on number fields will be defined and used to quantify some classical results in Diophantine approximation.

The majority of the sourse will be spent on Diophantine equations.  This is the study of how well real numbers can be approximated by rational numbers (\pi is pretty close to 22/7, right?). We'll

Text: No textbook will be required. Reasonable detailed lecture notes will be distributed.

References:

• E. Atrin, Algebraic Numbers and Algebraic Functions
• W. M. Schmidt, Diophantine Approximation and Diophantine Equations (LNM # 1467)