600-level courses; Spring 1998

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as regular courses, or they may be offered by faculty as independent study courses, or the offered course may be withdrawn (i.e., cancelled).

The following 600-level courses were offered in the spring 1998 semester:

• MATH 620: Semigroups II (McAlister)
• MATH 630: Geometric Analysis (Waterman)
• MATH 640: Theory of Newtonian and Nonnewtonain Incompressible Fluids(Necas)

Descriptions follow.

Math 620: Semigroups II

Instructor: Don McAlister

This course is a continuation of Math 620: Semigroups which was offered in Fall 1997. The main emphasis will be on the structure of semigroups in the spirit of the book ``Fundamentals of Semigroup Theory'' by J.M. Howie.

Howie's book will be used as the text for the first part of the course which will emphasize the structure of inverse semigroups and other classes of regular semigroups which have proved to be important in semigroup theory.

The second part of the course will concentrate on finite semigroups and their relationship with formal languages and automata. In particular, the Krohn-Rhodes Theorem and Eilenberg's variety theorem will be discussed. There will be no formal text for this part of the course. The material can be found in books by Grillet, Eilenberg, and Almeida.

Math 630: Geometric Analysis

Instructor: Peter Waterman

• Analytic continuation
• Elliptic functions
• Riemann Surfaces
• Hyperbolic geometry

The course will describe interrelations between the geometry of surfaces and natural domains for analytic functions.

Not only does the plane possess a Euclidean structure but so does a cylinder and a torus. One can view the periodic functions sin and cos as defined on a cylinder and elliptic functions (doubly periodic functions which arise in number theory and differential equations) as defined on a torus.

Spherical and Hyperbolic geometries will also be discussed.

Prerequisite: Math 532

Text: G.A. Jones \& D. Singerman: Complex Functions. Cambridge UP, 1987.

References:

• L.V. Ahlfors: Complex Analysis. McGraw-Hill, 1979.
• A.F. Beardon: The Geometry of Discrete Groups. Springer, 1983.
• A.F. Beardon: A Primer on Riemann Surfaces, Cambridge UP, 1984.
• J.B. Conway: Functions of One Complex Variable. Springer, 1995.

MATH 640: Theory of Newtonian and Nonnewtonian incompressible fluids

Instructor: Jindrich Necas

Co-requisite: MATH 542

Theory of monopolar incompressible fluids modeled by Navier-Stokes equations or more general equations, when viscosity depends on the velocity of shear, is fundamental from mathematical point of view as well as for applications.

In the course there will be discussed:

• Existence and uniqueness problems
• Regularity of solutions
• Selfsimilar solutions
• Hausdorf and fractal dimension of the universal attractor
• Basic conservation laws and entropy inequality

Text: None

References:

• R. Temam: Navier-Stokes Equations, Theory and Numerical Analysis, 3rd rev. ed. North-Holland, Amsterdam 1984
• R. Temam: Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1988
• Ch. R. Doering, J. D. Gibbon: Applied analysis of the Navier-Stokes equations, Cambridge University Press, 1995
• P. Constantin, C. Foias: Navier-Stokes Equations, University of Chicago Press, 1988
• J. Necas, J. M\'{a}lek, M. Rohyta, M. Ruzicka: Weak and Measure-Valued Solutions to Evolutionary Partial Differential Equations, Chapman & Hall, 1996