## Colloquium program from August 2007 to January 2008

**[colloque] Tue 25.09.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Jörg KRAMER (HU Berlin): Recent developments in Arakelov geometry

The determination of the number of linearly independent meromorphic functions with prescribed zeroes and poles on a compact Riemann surface is a classical problem. It is solved by means of the Riemann-Roch Theorem. The generalization of this theorem to the higher dimensional setting is due to Hirzebruch. Asymptotically, the number of the linearly independent meromorphic functions in question is governed by the degree of the underlying holomorphic line bundle on the complex manifold under consideration. In particular, these numbers play a significant role in the theory of Siegel modular forms/automorphic forms. From an arithmetic point of view the number of Siegel modular forms with "small", integral Fourier coefficients is of interest. This problem is basically solved by an application of the Arithmetic Riemann-Roch Theorem which incorporates the main results of Arakelov Geometry developped by H. Gillet and C. Soulé. However, a direct application of Arakelov Geometry is not possible in this context since the hermitian metrics under consideration are logarithmically singular. In our talk we will give a survey on the topics mentioned above as well as the latest contributions to incorporate singular metrics into Arakelov Geometry.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 2.10.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Jost ESCHENBURG (Augsburg): Submanifolds and the normal Gauss map

The geometry of submanifolds in n-dimensional euclidean space (higher dimensional "surfaces") has two aspects: Inner and outer geometry. While the inner geometry measures the arc length and the angles of curves on the "surface", the outer geometry shows how the tangent and normal spaces move from point to point. This is measured by the normal Gauss map which assigns to each point of the "surface" its normal space as an element of a Grassmannian. We ask how much of the geometry of the "surface" is encoded by the Gauss map. In particular we will display a situation where only the range of the Gauss map determines the "surface" completely. This is related to a rich class of very beautiful submanifolds (including spheres, Grassmannians, orthogonal and unitary groups and many others), the so called extrinsic symmetric spaces: submanifolds which are preserved by the reflection at each normal space.

[Invited by Prof. Anand Dessai]

**[colloque] Tue 09.10.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Jean-Pierre ECKMANN (Université de Genève): A topological glass

I will describe a very simple dynamical model which has a behavior known from the theory of glasses. The state space of the model is given by all triangulations of a sphere with n nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles (Pachner-move) with a temperature dependent probability. This model has aging properties of glasses and also the ultrametric landscape known from other glass models.

[Invited by Prof. Mazza]

**[colloque] Tue 16.10.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Laurent BARTHOLDI (EPF Lausanne): Growth and amenability of groups and algebras

Amenability of groups (a notion introduced by von Neumann in his study of the Banach-Tarski paradox) is a far-reaching generalization of "finiteness".
The (word) growth of a group, on the other hand, is an important geometric invariant. It can be exponential or subexponential; a classical result is that groups of subexponential growth are amenable.
I will define a natural notion of amenability for associative algebras (first considered by Gromov), and explain its relation to amenability of groups.
Let G be a finitely generated group. Call the algebraic growth of G the function computing the rank of successive quotients along the lower central series of G. This function is a lower bound for the usual (word) growth function of G.
I will discuss the following result: if G is amenable, then its algebraic growth is subexponential. This answers a question by Vershik, and is a partial converse to the classical statement that groups of subexponential (word) growth are amenable.
As a corollary, all the groups constructed by Golod and Shafarevich groups are non-amenable. This gives the first examples of non-amenable, residually-finite torsion groups.

[Invited by Dr. Laura Ciobanu]

**[colloque] Tue 23.10.2007, Phys 2.52 at 17h15** | more |

### Dr. Sebastian BAADER (ETH Zürich): Die asymptotischen Bahnen von Vektorfeldern

Gewisse dynamische Systeme lassen sich durch Vektorfelder auf R^3 beschreiben, man denke z.B. nur an eine Luftzirkulation. Die Bahnen eines solchen Vektorfelds sind auch von einem knotentheoretischen Standpunkt interessant. So misst die Helizität eines Vektorfelds, wie schnell sich zwei Bahnen im Durchschnitt umeinander winden. Wir werden das asymptotische Verhalten einiger Knoteninvarianten für nicht-periodische Bahnen untersuchen.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 30.10.2007, Phys 2.51 at 17h15** | more |

### Dr. Gauthier BERCK (Fribourg): Areas and volumes in Finsler spaces

If in Euclidean or Riemannian geometry the notions of areas and volumes are well defined, there exist several natural but different areas and volumes for normed or Finsler spaces. We will present the Busemann (or Hausdorff) and Holmes-Thompson (or symplectic) definitions and explain some of their properties related to convexity and the calculus of variation.

[Invited by Prof. Andreas Bernig]

**[colloque] Tue 6.11.2007, Phys 2.52 ay 17h15, Plancherel Lecture (BeFri Program in Maths)** | more |

### Prof. Dr. Nicholas HIGHAM (Manchester): Functions of a Matrix: Past and Present

The year 2008 marks the 150th anniversary of "A Memoir on the Theory of Matrices"
by Arthur Cayley [1]--the first paper on matrix algebra. In that paper Cayley
investigated the square root of a matrix, and it was not long before definitions of
f (A) for general f were proposed by Sylvester and others. From their origin in pure
mathematics, matrix functions have broadened into a subject of study in applied
mathematics, with widespread applications in science and engineering. Drawing on
my recently completed book [3], I will outline the early history of matrix functions,
give an introduction to the underlying theory, mention some applications, and then
describe some recent work on the matrix square root.

References:

[1] Arthur Cayley. A memoir on the theory of matrices. Philos. Trans. Roy. Soc. London, 148:17-37, 1858.

[2] Nicholas J. Higham. Cayley, Sylvester, and early matrix theory. MIMS EPrint 2007.119, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, October 2007. 7 pp. To appear in Linear Algebra Appl.

[3] Nicholas J. Higham. Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. Book to appear.

[Invited by Prof. J-P. Berrut]

**[colloque] Tue 13.11.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Sinnou DAVID (Paris): Diophantine equations, transcendance methods, special values

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 Dr. Viada]

**[colloque] Tue 20.11.2007, Pérolles 2, room G120, 16h15-19h** | more |

### Peter Thullen Centennial Colloquium

The two lectures highlight the important contributions
of Peter Thullen to complex analysis and to the
development of social security systems in Europe and
in Latin America.

More information here.

[Organized by Profs. Hungerbühler, Holmann, Kaup and Rummler]

**[colloque] Tue 27.11.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Alain-Sol SZNITMAN (ETH Zürich): Random motions in random media

The mathematical analysis of models of particles moving randomly in disordered environments has been very active over the last thirty years. It has uncovered a number of surprising effects and mathematical challenges. We will discuss some of these effects and challenges in this lecture, which is aimed at a general mathematical audience.

[Invited by Prof. Mazza]

**[colloque] Tue 4.12.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Daniel HUG (Essen-Duisburg): Zufällige Polytope und Isoperimetrie

Abstract in PDF here.

[Invited by Prof. Bernig]

**[colloque] Tue 11.12.2007, Phys 2.52 at 17h15** | more |

### Dr. Oliver RÖHRLE: A Three-Dimensional Finite Element Framework for Investigating Functional Electrical Stimulation Protocols

Functional Electrical Stimulation (FES) is a technique that uses applied electric fields to elicit a functional response from nerve and muscle tissue. Mathematical models are an important tool to understand the effects of such electric fields on the neuromuscular system i.e., on muscle fibre recruitment or fatigue. In this talk, I will present an integrated three-dimensional biomechanical Finite Element (FE) model that is based on the anatomy of the right lower limb of the Visible Male data and includes mathematical models of the nerve, of a single muscle fibre, and the whole muscle.
The mathematical model of the nerve is based on the CRRSS model for warm blooded nerves. The electrical activity within a single skeletal muscle fibre is modelled using a biophysically-based cell model, which is itself an amalgamation of several existing cell models. To incorporate the cellular properties of skeletal muscle fibres within the whole muscle, homogenized values of key physiological parameters, e.g. the pre- and post-power stroke concentration of crossbridge attachments, are computed at the Gauss points of the FE integration scheme. These values are then included within a modified transversely isotropic constitutive relationship to obtain the contractile response of the whole muscle by solving the governing equations of finite elasticity theory using the FE method and tri-cubic Hermite basis functions.

[Invited by Prof. Gabriel]

**[colloque] Tue 18.12.2007, Phys 2.52 at 17h15** | more |

### Prof. Dr. Wilderich TUSCHMANN (Kiel): Almost nonnegative curvature

Mannigfaltigkeiten mit fast nichtnegativer Krümmung verallgemeinern fast
flache wie nichtnegativ gekrümmte Mannigfaltigkeiten und spielen in der
Strukturtheorie von Mannigfaltigkeiten mit unteren Krümmungsschranken
eine zentrale Rolle. Im Vortrag werde ich einen Überblick über einige aktuelle Entwicklungen, Resultate und Fragen der globalen Geometrie und
Topologie nichtnegativ und fast nichtnegativ gekrümmter Räume geben.

[Invited by Prof. Dessai]