# Research

## Analysis

### Friday, April 20, 2012

Title: An introduction to the rolling problem, Part II
Speaker: Mauricio Godoy Molina, École Polytechnique
FRANCE
Time: 4:00pm–5:00pm
Place: PHY 130

### Friday, April 13, 2012

Title: An introduction to the rolling problem
Speaker: Mauricio Godoy Molina, École Polytechnique
FRANCE
Time: 4:00pm–5:00pm
Place: PHY 130

#### Abstract

As far back as the 1890s, mathematicians have tried to understand the system of two bodies with smooth boundaries rolling one against each other. The first case studied was the so-called “plate-ball problem”, which is a classical question in control theory and asks whether one can (and how to) roll a sphere on the plane from any given configuration to any other configuration, under two natural restrictions: the ball cannot slip (slide) and cannot twist around its vertical axis. Starting with this example we will study the dynamics and the geometry of the system of two surfaces rolling as mentioned above. Though most of the results I will present have been known for a while, we will also discuss some new answers to old problems, which were obtained in a joint work with Erlend Grong (Geometric Conditions for the Existence of an Intrinisic Rolling).

### Friday, April 6, 2012

Title: Functions of Smirnov class in multiply connected domains with real boundary values
Speaker: Lisa De Castro
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

Let $$G$$ be an $$n$$-connected domain. I will show that if $$G$$ has certain boundary characteristics then there do exist analytic functions of Smirnov class $$E^p$$ in $$G$$ with real boundary values a.e., where $$p$$ depends on $$G$$. I will also discuss a physical application related to functions that have real boundary values a.e.

### Friday, March 23, 2012

Title: Orthogonal polynomials and determinantal processes in the complex plane, Part II
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, March 2, 2012

Title: Orthogonal polynomials and determinantal processes in the complex plane
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

We will continue the discussions about orthogonal polynomials and spectral theory, in particular their connection to problems of mathematical physics.

### Friday, February 24, 2012

Title: Orthogonal polynomials in Analysis and Math. Physics, Part II
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, February 17, 2012

Title: Orthogonal polynomials in Analysis and Math. Physics
Speaker: Evguenii Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

I will try to make a review of various types of problems and selected results related to OP on the real line.

### Friday, February 3, 2012

Title: A survey of some open problems in Dirichlet space, Part II
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: PHY 120

### Friday, January 27, 2012

Title: A survey of some open problems in Dirichlet space
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: PHY 120

#### Abstract

In this talk, I will discuss some open problems in the Dirichlet space, which are the analytic functions in the unit disk whose derivatives are area integrable. In particular, I will examine the issue of cyclicity and what is known about the connection between cyclicity, boundary zero behavior, and logarithmic capacity, and will give some explicit examples.

### Friday, January 20, 2012

Title: The Isoperimetric Inequality and Free Boundary Problems, Part II
Speaker: Dmitry Khavinson
Time: 4:00pm–5:00pm
Place: PHY 130

### Friday, January 13, 2012

Title: The Isoperimetric Inequality and Free Boundary Problems
Speaker: Dmitry Khavinson
Time: 4:00pm–5:00pm
Place: PHY 130

#### Abstract

The isoperimetric theorem states that, among all simple closed curves of a given length, the curve that surrounds the region of largest area is a circle. In this talk, I will examine some history and some early geometric and analytic proofs of this result. I will then turn to approximation theory and to the concept of analytic content. This approach will reveal ties to overdetermined boundary value problems and hydrodynamics. I will discuss the free boundary problem related to the isoperimetric inequality, solid contour quadrature identities, and analytic content. In particular, I will focus on the connection between this free boundary problem and the problem of determining the shape of an electrified droplet of perfectly conducting fluid in an electric field, or equivalently, that of an air bubble in fluid flow. In some special cases, this free boundary problem has been solved. Recent progress has been achieved in 2011 in a joint work with A. Abanov, C. Bénéteau and R. Teodorescu. There are still many easily formulated open problems that remain unsolved.