600-level courses; Spring 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may be offered as regular courses, or they may be offered by faculty as independent study courses, or the course may be withdrawn (i.e., cancelled).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 Spring 2000 semester:

• MATH 620: The Combinatorics and Representation Theory of the Symmetric Group (Harald Ellers)
• MATH 640: Mathematical Introduction to Fluid Mechanics (Fred Bloom)
• MATH 660: Iterative Methods for Large-Scale Matrix Computations and Their Applications (Biswa Datta)

MATH 620
The Combinatorics and Representation Theory of the Symmetric Group

Instructor: Harald Ellers

Prerequisites: MATH 520.

Suppose a group $G$ acts on a finite dimensional vector space $V$ over a field $F$. If the action is linear (i.e. $g(v+w) = gv + gw$ and $g(av) = a gv$ for all $g \in G$, $v$, $w \in V$, and $a \in F$), the action is called a representation of $G$ over $F$. Many familiar groups such as dihedral groups, Galois groups of field extensions, and general linear groups, come already equipped with vector spaces on which they act naturally. For another example, consider the group of rotations of a cube. This group is isomorphic to $S₄$. The isomorphism provides a $3$-dimensional real representation of $S₄$.

The first goal of this course is to classify and construct all representations of the symmetric group $S_n$ over fields of characteristic $0$. In other words, we will construct all possible examples like the $3$-dimensional representation of $S₄$ above. The problems involved in doing this are largely combinatorial. A link between the natural action of $S_n$ on the set $\{1,2, \ldots,n\}$ and the action of $S_n$ on various vector spaces is provided by the combinatorics of certain diagrams called Young Tableaux.

After achieving this goal, we will turn to representations of $S_n$ over finite fields, and to the interaction between charcteristic $0$ and characteristic $p$. This is an area which still has many open problems.

As part of the course, students will use the computer algebra system GAP to explore examples.

This course provides many concrete examples of the structures studied in Math 521. Students who have taken or are simultaneously taking 521 should find that the courses support each other, but 521 is not a prerequisite or corequisite.

Required textbook: Group Characters, Symmetric Functions, and the Hecke Algebra, David M. Goldschmidt, AMS, Providence, 1991.

This book is available from the AMS virtual bookstore at www.ams.org for \$ 15.00 (\$12.00 for AMS members.) It covers the characteristic $0$ part of the course.

References:

• J.Beachy, Introductory Lectures on Rings and Modules, Cambridge University Press, Cambridge, 1999. (The last chapter is an introduction to general representation theory.)
• W.Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982.
• W.Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.
• G.James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer-Verlag, Berlin, 1978.
• G.James and A.Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, 1981.

MATH 640
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:

• Equations of Motion: Euler's Equations, Rotation and Vorticity, The Navier-Stokes Equations.
• Potential Flow and Viscous Flow: Potential Flows, Boundary Layers, Vortex Sheets, Stability an Bifurcation.
• Gas Flow: Characteristics, Shocks, The Riemann Problem, Combustion Waves

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

References:

• G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
• R. E. Meyer, Introduction to Mathematical Fluid Dynamics, Wiley-Interscience, 1971.

MATH 660
Iterative Methods for Large-Scale Matrix Computations and Their Applications

Instructor: Biswa N. Datta

Prerequisites: MATH 564, or consent of department.

The course will study the modern Krylov subspace iterative methods such as the Lanczos, Arnoldi, Block Lanczos and Block Arnoldi, Generalized Minimal Residual (GMRES), Quasi-Minimal Residual (QMR), Truncated RQ-Iteration (TRQ), preconditioned conjugate gradient (PCG), etc., for large-scale linear systems and eigenvalue problems, and their applications to applied problems such as the Lyapunov and Sylvester equations, model reduction and balanced realization, partial eigenvalue assignment problem, controllability and observability problems, and others arising in control theory and signal processing.

Text and Reference Books:

• Y. Saad, Numerical Methods for Large Eigenvalue Problems, Wiley, 1992.
• Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, 1996.
• A. Greenbaum, Iterative Methods, SIAM.
• Reprints and Preprints from current literature.