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Mathematics & Statistics

(Leader: Prof. Dmitry Khavinson )

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**Speaker**

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Minimal Discrete Energy on the Sphere, Part IV

E. A. Rakhmanov

3:00pm-4:00pm

PHY 118

**Title**

**Speaker**

**Time**

**Place**

Minimal Discrete Energy on the Sphere, Part III

E. A. Rakhmanov

4:00pm-5:00pm

PHY 118

**Title**

**Speaker**

**Time**

**Place**

Minimal Discrete Energy on the Sphere, Part II

E. A. Rakhmanov

4:00pm-5:00pm

PHY 118

**Title**

**Speaker**

**Time**

**Place**

Minimal Discrete Energy on the Sphere

E. A. Rakhmanov

4:00pm-5:00pm

PHY 118

**Abstract**

We will make a review of some results on distributions of \(N\) points on the \(2\)D sphere minimizing discrete Riesz Energy. In particular, we mention

- Numerical experiment: How to compute extremal points.
- Comparatively small \(N\) — symmetries of configurations.
- Asymptotics of energy as \(N\) goes to infinity: Smale problem.
- Open Problems

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**Speaker**

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Integrability and Laplacian growth: another view on the Schwarz potential, Part III

Razvan Teodorescu

4:00pm-5:00pm

PHY 118

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**Speaker**

**Time**

**Place**

On a Uniqueness Property of Harmonic Functions

Dmitry Khavinson

4:00pm-5:00pm

PHY 118

**Abstract**

We shall discuss the problem of uniqueness for functions \(u\) harmonic in a domain \(G\) in \(R^n\) and vanishing on some parts of the intersection \(V\) (not necessarily connected) of \(G\) with a line \(m\). The question originated more than a decade ago with N. Nadirashvili (private communication). For example, let \(G\) be a spherical shell, i.e., the region between two concentric spheres, and \(m\) is a line through the origin. Does \(u\) vanish on both segments along which \(m\) intersects \(G\) if it does so on one of them? To illustrate the cunning depth of the question note that if you let \(G\) to be the annulus with a sector cut out, the function \(u=\arg z\) in the plane does vanish on the positive part of the real axis, but not on the whole intersection. What happens if \(G\) is a spherical shell but m does NOT pass through the center? What if we replace harmonic functions by polyharmonic functions, or, more generally, solutions of analytic elliptic equations, or even worse, by linear combinations of Riesz potentials that satisfy no PDE altogether? The answers are by no means obvious and, in many cases, may be judged as surprising.

**Title**

**Speaker**

**Time**

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Integrability and Laplacian growth: another view on the Schwarz potential, Part II

Razvan Teodorescu

4:00pm-5:00pm

PHY 118

**Title**

**Speaker**

**Time**

**Place**

Integrability and Laplacian growth: another view on the Schwarz potential

Razvan Teodorescu

4:00pm-5:00pm

PHY 118

**Abstract**

The Schwarz potential introduced by Khavinson and Shapiro allows to describe Laplacian growth in terms of a Cauchy problem for the evolution in physical time \(t\). Under this evolution, a maximal set of parameters (harmonic moments) are preserved, hinting to the integrability of the system. The most elegant proof of integrability is an unexpected application of the Schwarz potential. I will discuss these connections and a number of examples.

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**Speaker**

**Time**

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Laplacian Growth in \(R^n\) and the Schwarz Potential

Erik Lundberg

4:00pm-5:00pm

PHY 118

**Abstract**

We consider the problem of describing the moving interface between an inviscid fluid (water) pushing against a region of viscous fluid (oil) where there is a sinc. Under the simplifying assumption that surface tension is zero, the pressure in the oil domain coincides with Green's function with singularity at the position of the sinc. The velocity of the moving boundary is then (a constant times) negative the gradient of pressure. It has been pointed out that the time derivative of the Schwarz function of the moving boundary coincides with \(z\)-derivative of the pressure. We discuss a generalization to arbitrary dimensions using Khavinson and Shapiro's generalization of the Schwarz function. We will then speculate on the prospect of this point-of-view leading to exact solutions of \(n\)-dimensional Laplacian growth for \(n > 2\).

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**Speaker**

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Wave flows in the capillary hydrodynamics: bifurcations and attractors

Grigori Sisoev

School of Mathematics

University of Birmingham, UK

4:00pm-5:00pm

PHY 118

**Abstract**

Experiments for films flowing down a vertical plane demonstrate a wide spectrum of wave regimes which are strongly dependent on flow conditions and liquid properties. In a wide range of physical parameters, the film flow is modelled by evolutionary partial differential equations for film thickness and local flow rate. Dynamical system generated by the model for space-periodic solutions is investigated. Manifold of its limit cycles is numerically constructed. Attracting invariant solutions, limit cycles and invariant tori, developing from different initial data are discussed. The attracting solutions are in a good agreement with experimental data.