Colloquia — Fall 2007
Friday, December 14, 2007
| Title |
Sub-Riemannian geometry in examples |
| Speaker |
Irina Markina
University of Bergen |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Dmitry Khavinson |
| Abstract |
The basic definition of sub-Riemannian geometry will be given and some
examples will be considered. The main examples will be the Heisenberg group
and its generalizations, the unit sphere S3 as a sub-Riemannian manifold.
We shall see how the Lagrangian and Hamiltonian formalisms work. The relation
between the sub-Riemannian geometry of S3 sphere and the Hopf fibration will
be presented. We also give an example where the Riemannian metric is replaced
by the Lorentzian one. |
Friday, December 14, 2007
| Title |
Pattern Recognition: Energy of the Laplace Evolution |
| Speaker |
Alexander Vasiliev
University of Bergen |
| Time |
11:00 a.m.-12:00 p.m. |
| Place |
ENB 313 |
| Sponsor |
Dmitry Khavinson |
| Abstract |
In order to establish the patterns for the inter-phase line of the
(brain) tumor growth, the latter could be modeled by the mathematical model
known as Laplacian growth. Laplacian growth possesses many interesting
features, in particular, integrable evolution as it has been established
recently. We discuss connections between the Laplacian growth and general
models of quantum mechanics (QFT). In particular, we are interested in
energy characteristics of this evolution. |
Friday, November 30, 2007
| Title |
The Dual of a Subnormal Operator |
| Speaker |
John Conway
George Washington University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Sherwin Kouchekian |
| Abstract |
Using a result of James Thomson it is shown that a problem involving the
dual of a pure subnormal operator essentially becomes a function theory
problem. The talk will start by a discussion of normal operators and proceed
to a discussion of the problem. There will be a heavy emphasis on examples
rather than proofs. A graduate student who knows the Spectral Theorem should
be able to follow the talk. |
Friday, November 16, 2007
| Title |
Schwarzian Derivatives of Analytic and Harmonic Functions |
| Speaker |
Peter Duren
University of Michigan |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Dmitry Khavinson |
| Abstract |
After a brief account of the Schwarzian derivative of an analytic function and some of its classical applications, the talk will focus on criteria for univalence and estimates of valence. Generalizations to harmonic mappings will then be described, using a definition of Schwarzian recently proposed and developed in joint work with Martin Chuaqui and Brad Osgood. Here it is often natural to identify a harmonic mapping with its canonical lift to a minimal surface. |
Monday, October 22, 2007
| Title |
How to Measure the Complexity of Singularities |
| Speaker |
Nero Budur
Notre Dame |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 141 |
| Sponsor |
Masahiko Saito |
| Abstract |
This talk regards the geometry of spaces of solutions of polynomial
equations. Singularities are the places where these objects are not smooth.
We will explore some ways of measuring how far singularities are from being
smooth. For example, the solution (0,0) is a singular point for both
y2 = x3 and y2
= x2(x+1) since locally their space of
solutions does not look like a line. A certain numerical measure of its
complexity, the log canonical threshhold, gives 5/6 for the first equation
and 1 for the second equation, showing that the first curve is “more
singular” than the second. |
Friday, October 12, 2007
| Title |
Invariant Subspaces of the Hardy and Bergman Spaces |
| Speaker |
Brent Carswell
Allegheny College |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 141 |
| Sponsor |
Dr. Catherine Bénéteau |
Abstract: A classical theorem of Buerling from 1949 asserts that, for
the Hardy space, every closed subspace invariant under multiplication by the
identity function is singly generated by an inner function. When considered from
an operator theory point-of-view, this result characterizes the closed subspaces
of the space of absolutely square summable sequences of complex numbers which are
invariant under the forward shift operator. In the past two decades, several
people have obtained results inspired by Beurling, and noticable among these
accomplishments is the breakthrough of Aleman, Richter, and Sundberg who obtained
what can be viewed as a Bergman space version of Beurling's theorem. In this talk,
some results which were motivated by the aforementioned work will be presented.