## Colloquium program from August 2008 to January 2009

**[colloque] Tue 16.8.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Michael S. FLOATER (University of Oslo): Barycentric coordinates and transfinite interpolation

Recent generalizations of barycentric coordinates to polygons and polyhedra, such as Wachspress and mean value coordinates, have been used to construct smooth mappings that are easier to compute than harmonic amd conformal mappings, and have been applied to curve and surface modelling. We will summarize some of these developments and then discuss how these coordinates naturally lead to smooth transfinite interpolants over curved domains, and how one can also match derivative data on the domain boundary.

[Invited by Prof. Jean-Paul Berrut]

**[colloque] Tue 30.9.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Katrin TENT (Universität Münster): Algebraic groups and Tits-Buildings

I will explain how buildings arise as combinatorial structures from algebraic groups. Using group actions on buildings I will give group theoretic characterizations of algebraic groups.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 14.10.08, Phys 2.52 at 17h15** | more |

### Dr. Tatiana SMIRNOVA-NAGNIBEDA (Université de Genève): Volume entropy in Outer space

In a seminal paper published in 1986, Culler and Vogtmann initiated a study of automorphisms of free groups by methods analogous to powerful geometric techniques invented by Thurston to study mapping classes of surfaces. In particular, they introduced (for each n>1), a contractible space, now called "Outer space", on which the group Out(F(n)) of outer automorphisms of the free group F(n) acts properly. Outer space can be thought of as a combinatorial counterpart of Teichmüller space. In the talk I'll discuss how analysis on metric trees helps to investigate this analogy and its limitations.

[Invited by Dr. Laura Ciobanu]

**[colloque] Tue 21.10.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Felix SCHLENK (Université de Neuchâtel): Hamiltonian dynamics via symplectic geometry

Hamiltonian systems describe dynamical phenomena without friction. Such systems are often complicated, since small oscillations never decay. I will try to explain how modern methods from symplectic geometry and elliptic PDEs nevertheless allow one to better understand Hamiltonian systems.

[Invited by Prof. Andreas Bernig]

**[colloque] Tue 28.10.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Frank KUTZSCHEBAUCH (Universität Bern): Holomorphic factorization of maps into the special linear group

We report on recent progress in a joint work with B. Ivarsson
about solving the Vaserstein problem posed by Gromov in the 1980s:
Does every holomorphic map Cn ! SLm(C) decompose into a finite product
of holomorphic maps sending Cn into unipotent subgroups in SLm(C)?
The introduction of the problem will be on the level of a first year linear
algebra course. To describe the methods used we try to give an introduction
to the Oka-Grauert-Gromov-h-principle in Complex Analysis.

[Invited by Prof. Anand Dessai]

**[colloque] Tue 4.11.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Simon BRENDLE (Stanford / ETHZ): Ricci flow and the classification of 1/4-pinched manifolds

I will describe the recent classification (joint with Rick Schoen) of manifolds with pointwise 1/4-pinched curvature. This proof employs the Ricci flow, but the curvature condition we consider first arose from consideration of the index form for minimal surfaces. I will also discuss the borderline case of manifolds with weakly 1/4-pinched curvature. This relies on a new strong maximum principle for the Ricci flow.

[Invited by Dr. Patrick Ghanaat]

**[colloque] Tue 11.11.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Thomas WIHLER (Universität Bern): Efficient Numerical Methods for Partial Differential Equations

In recent years, discontinuous Galerkin (DG) finite element methods for the numerical solution of partial differential equations have gained considerable interest. They provide an ideal framework for so-called hp-type discretizations, i.e., for approaches that feature both local mesh refinements as well as local adjustments of the approximation orders. Under certain conditions such schemes are even able to achieve exponential rates of convergence and thereby to compute highly accurate solutions in a very efficient way. The talk is divided into three parts:

I) Introduction to finite element methods and hp-discretizations for linear elliptic PDEs.

II) Discontinuous Galerkin methods as an attractive framework for hp-approaches.

III) Review of a recently developed technique to derive error indicators for DG methods. Error indicators are locally computable quantities which can be used to improve the numerical solutions automatically thereby leading to adaptive methods. A number of applications and numerical experiments will be given.

[Invited by Prof. Norbert Hungerbühler]

**[colloque] Tue 25.11.08, Phys 2.52 at 17h15** | more |

### Dr. Martin KOLÁR (Masaryk University Brno): The local equivalence problem in CR geometry

The talk will discuss various aspects of the local equivalence problem for real hypersurfaces in the complex n-dimensional space. We will consider an algebraic approach, originated by J. Moser, which leads to a normal form construction, and provides complete understanding of symmetries of such manifolds. Recent results for Levi degenerate manifolds (both of finite and infinite type) and their applications will be also discussed.

[Invited by Dr. Francine Meylan]

**[colloque] Tue 2.12.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Pascal LAMBRECHTS (Louvain-la-Neuve): Counting closed geodesics on compact manifolds

Fix a compact riemannian manifold M (without boundary). A closed geodesic on M is a geodesic that is periodic as a function. Many natural questions arise around the existence of such closed geodesics. Actually it is known in almost all cases that there are infinitely many closed geodesics. Gromov has studied this problem more quantitatively and asked about the asymptotic behaviour of the function counting the number of closed geodesics of length smaller than t, when t goes to infinity. He conjectures that for "most" manifolds this function grows at least exponentially with t. In this talk I will explain these problems and some of their solutions. At the end, I will show that Gromov's conjecture is true for non trivial connected sums.

[Invited by Prof. Anand Dessai]

**[colloque] Tue 9.12.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Elisha FALBEL (Université Paris VI): Triangulations of three manifolds and the dilog

In this talk I will explain how to define certain geometric structures associated to a triangulation of a manifold. I will discuss the particular cases of real hyperbolic geometry and complex hyperbolic geometry and certain relations between them. The dilog function defines the volume of simplices in real hyperbolic geometry and I will show how to define volumes in a more general context incluing complex hyperbolic geometry.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 16.12.08, Phys 2.52 at 17h15** | more |

### Dr. Uta FREIBERG (Universität Jena): Einstein Relation on Fractal Objects

Many physical phenomena proceed in or on irregular objects which are often modeled by fractal sets. Using the model case of the Sierpinski gasket, the notions of Hausdorff, spectral and walk dimension are introduced in a survey style. These "characteristic" numbers of the fractal are essential for the Einstein relation, expressing the interaction of geometric, analytic and stochastic aspects of a set.

[Invited by Prof. Andreas Bernig]