For inquiries concerning the seminar send email to Jacek Szmigielski, szmigiel@math.usask.ca.

The main focus of the seminar will be the book

**
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
**

by Percy Deift.

Courant Lecture Notes in Mathematics, Vol. 3, Courant Institute of Mathematical Sciences, New York City, 1999.

Contents:

- Riemann-Hilbert Problems
- Jacobi Operators
- Orthogonal Polynomials
- Continued Fractions
- Random Matrix Theory
- Equilibrium Measures
- Asymptotics for Orthogonal Polynomials
- Universality

The seminar takes place in McLean Hall 242.1 on Thursdays at 2:30 till 4:00 unless advertised differently.

**Next meeting**:
January 30, 2003 .

Previous meetings: January 23, 2003 , January 16, 2003 , September 18, September 25, October 2, October 9, October 16, October 23 , October 30, November 6, November 13 , November 20, November 27

Forthcoming attractions: February 6 , February 8 Past attractions: October 24, November 1, November 15 .

Ergodic thery of continued fractions (second part) .

Ergodic thery of continued fractions.

Arithmetic continued fractions.

A Riemann-Hilbert Problem for orthogonal polynomials.

Chapter 3. Formulas for orthogonal polynomials.

Finishing chapter 2.

Spectral theory of self-adjoint unbounded operators

On the road again. Limit point/limit circle theory.

Review of some aspects of the theory of self-adjoint extensions of unbounded operators.

Jacobi operators, case of unbounded operators, self-adjoint extensions etc

Jacobi operators, spectral map

Jacobi operators.

Riemann-Hilbert problems. Beals-Coifman work on the scattering for the first order systems. Part 2.

Riemann-Hilbert problems. Beals-Coifman work on the scattering for the first order systems. Part 1.

Professor **Stanley Gudder**,
Department of Mathematics
University of Denver

Fixed Points of Quantum Operations

McLean Hall , room 142.1

1:00 PM - 2:30 PM

Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation and quantum information theory. If an operator A is invariant under a quantum operation A fixed point. Physically, the fixed points are the operators that are not disturbed by the action of the quantum operation. Our main purpose is to answer the following question.

If A is a fixed point, is A compatible with the operation elements of the quantum operation?

We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

This is a report on some recent work that I have done in collaboration with Alvaro Arias and Aurelian Gheondea.

Professor **Peter Gibson **,
Department of Mathematics and Statistics
University of Calgary,

Topology and discrete geometry arising from a simple mechanical model

ARTS, room 206

4:00 PM - 5:00 PM

In this talk I will explain how inverse problems based on a very simple, discrete, mechanical model lead naturally to some interesting topology and discrete geometry. The mechanical model consists of the linearized dynamics of a chain of oscillators. In analyzing ill-posed inverse problems for such a model, the discrete geometry of a special class of convex polytopes, the so-called permutohedra, comes into play. More precisely, the combinatorics of permutohedra makes it possible to analyze the topological structure of solution sets to the inverse problems.

Professor **David C. Brydges**,
Canada Research Chair
Department of Mathematics
The University of British Columbia
Vancouver, BC Canada
and
Department of Mathematics
University of Virginia
Charlottesville, USA

Branched Polymers and Dimensional Reduction

ARTS, room 206

4:00 PM - 5:00 PM

The problem of determining the expected end-to-end distance of a long self-avoiding random walk in three dimensions is old and unsolved. Thinking of self-avoiding walk as a model for a long chain molecule makes it natural to generalise the question to other topologies, such as self-avoiding trees. Curiously this problem has a version which is exactly solvable. This exact solution will be the subject of the colloquium. It is based on a paper posted in preliminary form at http://xxx.lanl.gov/abs/math-ph/0107005. I shall make every effort to make the colloquium accessible to a general audience.

Professor **Lance Littlejohn**,
Department of Mathematics and Statistics
Utah State University,

The BKS(N,M) Problem in Orthogonal Polynomials

McLean Hall 242.1

2:30 PM - 4:00 PM

Professor **Lance Littlejohn**,
Department of Mathematics and Statistics
Utah State University,

The Glazman-Krein-Naimark Theorem with Applications

McLean Hall 242.1

2:00 PM - 3:00 PM