# Hausdorff Kolloquium WS 2013/14

**Date:** October 30, 2013 - January 29, 2014

**Venue:** Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

## Wednesday, October 30

15:15 |
Artur Avila (UPMC, Paris / IMPA, Rio de Janeiro): The billiard on the regular polygon |

16:45 |
Anna Wienhard (Heidelberg): Higher Teichmueller spaces - from SL(2,R) to other Lie groups |

## Wednesday, November 6

15:15 |
Roman Kotecky (Warwick / Charles Univ., Prague): Long range order for random colourings |

16:45 |
Sergei Kuksin (Paris Diderot): Perturbed Hamiltonian PDE with resonances |

## Wednesday, January 29

15:15 |
Karsten Grove (Notre Dame, Indiana): Convexity and Symmetry in Positive Curvature and Beyond |

16:45 |
Ulrich Pinkall (TU Berlin): Decomposing smoke into smoke rings |

## Abstracts:

#### Sergei Kuksin: Perturbed Hamiltonian PDE with resonances

I will discuss long-time behaviour of small oscillations in a non-linear Shroedinger equation, perturbed by a random force and linear dissipation. The equation is scaled in such a way that its solutions are small, but their limiting dynamics is non-trivial. The limiting behaviour turns out to be described by another damped/driven Hamiltonian PDE. The new Hamiltonian is constructed out of the resonant terms of the original Hamiltonian. It is homogeneous and has infinitely many quadratic integrals; two of them are coercive. I will discuss relation of this result with the problem of weak turbulence.

#### Roman Kotecky: Long range order for random colourings

A uniform distribution of proper random colourings on some planar lattices features a long range order. This is a pure case of ``order by disorder” — the phase transition of purely entropic origin. The motivation and the main ideas of the proof will be explained.

#### Ulrich Pinkall: Decomposing smoke into smoke rings

We present an algorithm to approximate a given velocity field in space ("smoke") by the field generated by a finite number of closed vortex filaments (“smoke rings“). This discretization of smoke into smoke rings is useful both for visualization and for fluid simulation.

A two-dimensional version of this algorithm (which also has applications in Computer Graphics) concerns choosing an optimal flat metric with cone singularities on a given Riemann surface.