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Mathematics & Statistics

Colloquia — Fall 2007

Friday, December 14, 2007

Title
Speaker

Time
Place
Sponsor

Sub-Riemannian geometry in examples
Irina Markina
University of Bergen
3:00pm-4:00pm
PHY 108
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 \(\mathbb{S}^3\) as a sub-Riemannian manifold. We shall see how the Lagrangian and Hamiltonian formalisms work. The relation between the sub-Riemannian geometry of \(\mathbb{S}^3\) sphere and the Hopf fibration will be presented. We also give an example where the Riemannian metric is replaced by the Lorentzian one.

Title
Speaker

Time
Place
Sponsor

Pattern Recognition: Energy of the Laplace Evolution
Alexander Vasiliev
University of Bergen
11:00am-12:00pm
ENB 313
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
Speaker

Time
Place
Sponsor

The Dual of a Subnormal Operator
John Conway
George Washington University
3:00pm-4:00pm
PHY 130
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
Speaker

Time
Place
Sponsor

Schwarzian Derivatives of Analytic and Harmonic Functions
Peter Duren
University of Michigan
3:00pm-4:00pm
PHY 130
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
Speaker

Time
Place
Sponsor

How to Measure the Complexity of Singularities
Nero Budur
Notre Dame
4:00pm-5:00pm
PHY 141
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 \(y^2=x^3\) and \(y^2=x^2(x+1)\) 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
Speaker

Time
Place
Sponsor

Invariant Subspaces of the Hardy and Bergman Spaces
Brent Carswell
Allegheny College
3:00pm-4:00pm
PHY 141
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.