600-level courses; Fall 2000

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may be offered as regular courses, or they may be offered by faculty as independent study courses, or the course may be withdrawn (i.e., cancelled).

The following 600-level courses are proposed for the fall 2000 semester:

Courses from previous semesters: Spring 2000   |   Fall 1999   |   Spring 1999   |   Fall 1998   |   Spring 1998

MATH 620 (Topics in Algebra)
Algebraic Combinatorics

Instructor: Harvey Blau

CRQ: Math 520

The use of linear algebraic methods to study combinatorial problems has not only produced insightful solutions but has generated some algebraic systems which raise new questions and are of interest in their own right. This course will focus on the structures variously known as hypergroups, C-algebras and (generalized) table algebras. We will define them, motivate their existence through a look at permutation groups and at association schemes, study their basic properties, survey some known results, and see how these objects may be applied to combinatorial configurations such as distance- regular graphs. We will sketch some open research problems which seem fairly accessible to the interested student. The necessary background from the theory of algebras and of nonnegative matrices will be covered in the course.

Text: Notes by the instructor

References:

MATH 640 (Topics in Applied Mathematics)
Mathematical Introduction to Fluid Mechanics

Instructor: Fred Bloom

Prerequisites: MATH 542, or MATH 442 and MATH 536, or consent of department.

A presentation of some of the basic ideas and problems of incompressible and compressible fluid dynamics in a form suitable for mathematicians. The course will present the physical background and motivation for some constructions that have been used in recent mathematical and numerical work on the Navier-Stokes and Euler equations.

Topics to include:

Text: Alexandre Chorin and Jerrold Marsden, A Mathematical Introduction to Fluid Mechanics (3rd Edition), Springer-Verlag, 1992.

References:

Math 650 (Topics in Geometry and Topology)
Geometric Aspects of Sturm-Liouville Problems

INSTRUCTOR: H. Wu

PREREQUISITE: Math 536 or equivalent.

A Sturm-Liouville problem consists of an ordinary differential equation of the form $-(py')'+qy=\lambda wy$ on an interval $(a,b)$ and a boundary condition such as $y(a)=y(b)=0$. Here $p$, $q$ and $w$ are given functions and $\lambda$ is the so called spectral parameter. The goal of study is to find out for which values of $\lambda$ the equation has non-trivial solutions satisfying the boundary condition. These values of $\lambda$ are called the eigenvalues of the problem and the corresponding non-trivial solutions their eigenfunctions. Such problems have a wide range of applications in engineering and science, as well as in mathematics.

The course will first introduce the basics about Sturm-Liouville problems and then discuss their geometric aspects. We will carefully define the geometric concepts needed here. A large part of the topics are motivated by the numerical computation of eigenvalues.

TEXT: none

REFERENCE

  1. E. A. Coddington & N. Levinson: Theory of Ordinary Differential Equations. McGraw-Hill, 1955.
  2. A. Zettl: Sturm-Liouville problems. In ``Spectral Theory and Computational Methods of Sturm-Liouville Problems" edited by D. Hinton & P. Schaefer. Marcel Dekker, 1997.
  3. M. S. P. Eastham, Q. Kong, H. Wu & A. Zettl: Inequalities among eigenvalues of Sturm-Liouville problems. J. Inequalities & Appl. 3 (1999), 25--43.
  4. Q. Kong, H. Wu & A. Zettl: Geometric aspects of Sturm-Liouville problems, I. Structures on spaces of boundary conditions. Proc. Royal Soc. Edinburgh, to appear.
  5. K. Haertzen, Q. Kong, H. Wu & A. Zettl: Geometric aspects of Sturm-Liouville problems, II. Subspace of boundary conditions for left-definiteness. In preparation.
  6. Q. Kong, Q. Lin, H. Wu & A. Zettl: A new proof of the inequalities among Sturm-Liouville eigenvalues. PanAmerican Math. J., to appear.
  7. Q. Kong, H. Wu & A. Zettl: Dependence of the $n$-th Sturm-Liouville eigenvalue on the problem. J. Differential Equations 156 (1999), 328--354.
  8. Q. Kong, H. Wu & A. Zettl: Left-definite Sturm-Liouville problems. Preprint.

MATH 680 (Topics in Number Theory)
Introduction to Elliptic Curves

Instructor: Chris Hurlburt

Prerequisites: Math 520 or Math 580 or consent of the department

By definition an elliptic curve is a plane, non-singular, cubic curve. Perhaps the simplest way to describe an elliptic curve over the rationals would be as the solutions to the equation y2 = P(x) where P(x) is a monic polynomial of degree three with rational coefficients. After conics, these are the simplest curves encountered in Diophantine geometry and their theory is both rich and diverse.

This course will explore the arithmetic and geometric aspects of elliptic curves including a study of the Weierstrass equations, the group law, rational points, elliptic curves over finite fields and number fields, and a summary of rank and height. Whenever possible students will be expected to construct/compute specific examples as appropriate. Prerequisites for this course are a first course in graduate algebra or a course in graduate number theory.

Text: Joseph H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer-Verlag, 1986.
Reference: Anthony W. Knapp, Elliptic Curves, Mathematical Notes 40, Princeton University Press, 1992.