## Colloquium program from August 2006 to March 2007

**[colloque] Tue 31.10.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Martin BRIDSON (London): Non-positive curvature and complexity for finitely-presented groups

This is a version of the Invited Lecture that I gave at the ICM in Madrid this summer. A universe of finitely presented groups is sketched and explained, leading to a discussion of the fundamental role that manifestations of non-positive curvature play in group theory. The geometry of the word problem and associated filling invariants are discussed. The subgroup structure of direct products of hyperbolic groups is analysed and a process for encoding diverse phenomena into finitely presented subdirect products is explained. Such an encoding is used to solve problems of Grothendieck concerning profinite completions and representations of groups. In each context, explicit groups are crafted to solve problems of a geometric nature.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 7.11.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Joachim ROSENTHAL (Zürich): Three challenges of Claude Shannon

In 1948/1949 Claude Shannon wrote two papers [1,2] which became the foundation of modern information theory. The papers showed that information can be compressed up to the `entropy', that data can be transmitted error free at a rate below the capacity and that there exist provable secure cryptographic systems. These were all fundamental theoretical result. The challenge remained to build practical systems which came close to the theoretical optimal systems predicted by Shannon. In this overview talk we will explain how the first two challenges concerning coding theory have resulted in practical solutions which are very close to optimal. Then we explain why the gap between the practical implementation of cryptographic protocols with the theoretical result of Shannon is largest. The talk will be tutorial in nature and should be accessible to advanced undergraduate students.

References:

[1] C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379-423 and 623-656.

[2] C. E. Shannon, Communication theory of secrecy systems, Bell System Tech. J. 28 (1949), 656-715.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 14.11.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Christiane TRETTER (Bern): Spectral theory of block operator matrices and applications in mathematical physics

Block operator matrices are matrices the entries of which are linear operators. They arise frequently, for example, when coupled systems of partial differential equations have to be considered. In the talk will study the spectral properties of block operator matrices. The topics will include the location of the spectrum, description of the essential spectrum, existence of invariant subspaces, solutions of Riccati equations, and block diagonalization. The results presented will be illustrated by applications to various problems from mathematical physics.

[Invited by Prof. Norbert Hungerbuhler]

**[colloque] Tue 21.11.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Lutz DÜMBGEN (Bern): Nonparametric estimation involving log-concave densities

An interesting alternative to fitting parametric models to empirical data is to fit a distribution with log-concave density. In this talk I shall discuss and illustrate this aproach (a) in the simple setting of independent identically distributed observations, and (b) in the setting of censored observations.

[Invited by Prof. Christian Mazza]

**[colloque] Tue 28.11.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Elena KLIMENKO (Gettysburgh / visiting MPI Bonn): Recent progress in two-generator Kleinian groups

I will talk about PSL(2,C), which can be identified with the full group of orientation preserving isometries of hyperbolic 3-space. The discrete subgroups of this group are called Kleinian groups, and the corresponding quotient spaces are Kleinian orbifolds. Kleinian orbifolds are similar to 3-manifolds (with or without boundary), but the orbifolds can have a singular set that looks like a graph with vertices of degree 3. We discuss recent progress in the study of two-generator Kleinian groups and then we concentrate on the two-generator groups/orbifolds with real traces of the generators and their commutator. In particular, we give a complete classification of such discrete groups in terms of orbifolds, group presentations, and parameters. This is a joint work with Natalia Kopteva.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 5.12.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Ludwig BRÖCKER: Axiomatische semialgebraische Geometrie

Typische Objekte der reellen algebraischen Geometrie sind semialgebraische Mengen. Wieviele polynomiale Ungleichungen sind erforderlich, um eine gegebene semialgebraische Menge zu beschreiben? Es zeigt sich, dass sich Fragen dieser Art unter Verwendung von nur wenigen Grundprinzipien der reellen Algebra behandeln lassen. Wir bilden, hiervon ausgehend, eine axiomatische Theorie, demonstrieren diese an einfachen Beispielen, erläutern die Hauptsätze und kehren zurück zu erstaunlichen geometrischen Anwendungen.

[Invited by PD. Dr. Andreas Bernig]

**[colloque] Tue 12.12.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Bernhard BECKERMANN (Lille): Image numérique, GMRES et polynômes de Faber

Soit $F$_{n} le polynôme de Faber de degré $n$ associé à
l'image numérique d'un opérateur linéaire continu $A$ sur un
espace de Hilbert. Nous montrons dans un premier temps que
$|F$_{n}(A)| ≤ 2. Nous en déduisons ensuite, en terme d'image
numérique, de nouvelles estimations d'erreur pour la méthode
GMRES, méthode itérative adaptée à la résolution des
systèmes linéaires non-hermitiens.

[Invited by Prof. Jean-Paul Berrut]

**[colloque] Tue 19.12.2006, Phys 2.52 at 17h15** | more |

### Prof. Dr. Stefan BAUER (Bielefeld): Almost complex 4-Manifolds with vanishing first Chern class

It is not difficult to decide whether or not a given closed differentiable 4-dimensional manifold can be equipped with an almost complex structure with vanishing first Chern class. The answer depends solely on its cohomology structure. Indeed, almost complex 4-manifolds with vanishing first Chern class are abundant: Fix any first Betti number. Then there are infinitely many such manifolds, distinguished by, say, their signature.

It is an unsolved problem, which of these manifolds do support symplectic structures. The special case of Kähler structures, however, is completely known by Kodaira's classification. Such Kähler manifolds are minimal and of Kodaira dimension zero. The list is explicit and rather short, K3-surfaces being the only ones with non-vanishing signature.

The lecture will be about how Seiberg-Witten theory, combined with equivariant topology, can be used to drastically reduce the candidates for symplectic structures.

[Invited by Prof. Anand Dessai]

**[colloque] Tue 9.01.2007, Phys 2.52 at 17h15** | more |

### Dr. Pierre PATIE (Bern): Fonctions harmoniques associées à des semigroupes autosimilaires et à des généralisations du semigroupe d'Ornstein-Uhlenbeck

Classically, solving a diophantine equation amounts to finding integral solutions of a (set of) polynomial equation(s). Starting with concrete examples, we shall explain how imposing other conditions can lead to deep applications in number theory. We shall conclude with an introduction to the Pink-Zilber conjectures.

[Invited by Prof. Christian Mazza]

**[colloque] Tue 16.01.2007, Phys 2.52 at 17h15** | more |

### Dr. Peter LISCHER (Bern): Quantifying uncertainty of Measurement in environmental studies and food control

Abstract in PDF here

[Invited by Prof. Jean-Pierre Gabriel]

**[colloque] Tue 23.01.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Uwe SEMMELMANN (Köln): Weitzenböck formulas on manifolds with special holonomy

Weitzenboeck formulas are an important tool for linking differential geometry and topology. They may be used for proving the vanishing of Betti numbers under suitable curvature assumptions or for proving the non existence of metrics of positive scalar curvature on manifolds satisfying certain topological conditions. Moreover they are often applied in the proof of eigenvalue estimates for Laplace and Dirac operators.

In my talk I will consider Weitzenboeck formulas on Riemannian manifolds with a fixed compact structure group. I want to show how to derive all such formulas in a certain recursive procedure. It turns out that finding all possible Weitzenboeck formulas may be reformulated into a problem of linear algebra depending on the representation theory of the structure group. In the end the structure of the universal enveloping algebra of the Lie algebra of the structure group determines the existence of Weitzenboeck formulas.

[Invited by Prof. Anand Dessai]

**[colloque] Tue 30.01.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Francesco AMOROSO (Caen): Small points on subvarieties of algebraic tori: results and methods

Abstract in PDF here

[Invited by Prof. Kellerhals and Dr. Viada]

**[colloque] Tue 6.02.2007, Phys 2.52 at 15h15** | more |

### Prof. Dr. Alexei LOZINSKI (Houston): Numerical techniques for polymer kinetic theory models

Polymer kinetic theory is a valuable tool that can provide models for some classes of non-Newtonian viscoelastic fluids which are encountered in industry, food processing and biology. However, the kinetic theory models are usually much more complicated than traditional constitutive models and present therefore new challenges for theoretical analysis and numerical simulations. We shall talk about two approaches - stochastic simulations and deterministic methods based on the Fokker-Planck equation - on the example of models which do not possess the closed-form constitutive equations, like the model of finitely extensible dumbbells. Finally, some ongoing work on efficient numerical Fokker-Planck-based techniques for the models with high dimensional configuration space will be addressed.

[Invited by Dr. Ales Janka]

**[colloque] Tue 19.02.07, Phys 2.52 at 17h15** | more |

### Dr. Paul TURNER (Edinburgh): Khovanov homology for knots and links

Around 1998 M. Khovanov invented a homology theory for links which has nice functorial properties with respect to link cobordisms. His theory is bi-graded and recovers the Jones polynomial on taking the graded Euler characteristic. Over the last few years various generalisations have appeared and there are interesting connections with symplectic geometry, representations of simple Lie algebras, mathematical physics and knot Floer homology. In this talk I will give an overview of Khovanov homology discussing basic definitions, elementary properties and some applications if time permits.

[Invited by Prof. Kellerhals]