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Mathematics & Statistics

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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.

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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.

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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.

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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.

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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.

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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.