Sept. 23 at 4PM in PS302

Mark Colarussso (ISU) THE GELFAND-ZEITLIN INTEGRABLE SYSTEM ON gl(n,C)

ABSTRACT: For an n×n complex matrix X, we consider the eigenvalues of all the i×i submatrices in the top left hand corner of X. These are known as Ritz values and play an important role in numerical linear algebra. In this talk, we will see how questions about Ritz values naturally lead to a study of the Gelfand-Zeitlin integrable system on the Lie algebra of n × n complex matrices gl(n, C). This system was introduced in recent work of Kostant and Wallach and is a complexified version of the Gelfand-Zeitlin system on the n × n Hermitian matrices introduced by Guillemin and Sternberg as a geometric analogue of the Gelfand-Zeitlin basis of highest weight modules. We will describe our results about the geometric properties of the Gelfand-Zeitlin system and explain how they can be used to answer questions of Parlett and Strang about Ritz values.

Oct. 28 at 4PM in PS306

Professor H. Turner Laquer(ISU) 1-MANIFOLDS

ABSTRACT: Mathematics has a unity---there are many  branches but it is all one subject.

Interesting research in mathematics often occurs on the interface between branches of mathematics. This current work had its origins in the areas of partial differential equations and differential geometry, but it has now become a project in topology and graph theory. A special case of the earlier work has been a key source of examples for the current research.

The talk will be accessible to anyone with basic knowledge of topology

and graph theory.

Nov. 11 at 4 PM, Professor T. H. Steele, Weber state University

Continuity and chaos in discrete dynamical systems

T. H. Steele

Weber State University

In the latter part of the nineteenth century, there was a belief in the deterministic, clockwork precision of the universe. From this belief arose an interest in establishing the stability of the planetary motions in our solar system. Oscar II, King of Sweden and Norway, initiated a mathematical competition in 1887 to celebrate his sixtieth birthday in 1889. One of the problems, posed by Karl Weierstrass, dealt with this stability:”Given a system of arbitrarily many mass points that attract each other according to Newton’s laws, assuming that no two points ever collide, give the coordinates of the individual points for all time….”

Henri Poincare attacked this problem for our solar system but found this too complicated to solve. He switched to a three body problem and instead studied the discrete time trajectories of the point masses. From his analysis Poincare submitted a solution to the three body problem. He was awarded the prize but in the publication review of the work, the referee Phragmen found an error. Poincare came to understand that a concept we now call sensitive dependence on initial conditions was involved, rendering impossible the long term prediction of the three body problem.

Since Poincare’s seminal work, chaotic and dynamical systems have been used to model many systems in the physical and life sciences. Typically, these problems involve a continuous function f which maps a compact set X into itself, and study the dynamics of the sequence {x, f(x), f(f(x)), f(f(f(x))) . . . }, where x is an element of X.  For each x in X, one also can define (x, f), the -limit set of f at x, as the cluster set of the sequence {x, f(x), f(f(x)), f(f(f(x))) . . . }. In the first part of this talk we focus our attention on how -limit sets (x, f) are affected by perturbations in the initial condition x, the generating function f , or both. We then turn our attention to a description of the typical behavior for a function f on one of its -limit sets (x, f).

While we cannot solve the problem posed by Weierstrass, our results will allow us to give a very specific description of our solar system’s very likely long term behavior.

Dec. 2 at 4PM , Professor Matthew Housley (BYU)

Title: A Skein Theoretic Construction of Certain Kazhdan-Lusztig

Left Cell Representations

Abstract: Kazhdan and Lusztig constructed a remarkable canonical basis

of the Hecke algebra $\mathscr{H}$ for the symmetric group which

leads naturally to left cells and left cell representations. These

based representations have important applications in the study of

simple Lie algebras of type A. In this talk, I will define an action

of$\mathscr{H}$  on crossingless matchings via skein relations and

a graphical calculus. The representations arising in this way are

in fact certain left cell representations, where the crossingless

matching basis is equivalent to the canonical left cell basis.