## Colloquium program from January 2009 to August 2009

**[colloque] Tue 17.02.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Philippe CLEMENT (TU Delft): An introduction to gradient flows on metric spaces

The aim of this lecture is to present a theorem of Ambrosio-Gigli-Savar on gradient flows in metric spaces. Starting from convex gradient flows in Hilbert spaces we arrive at a generalization which includes many interesting gradient flows in metric spaces of probability measures.

[Invited by Profs. J-P. Berrut and J-P. Gabriel]

**[colloque] Tue 24.02.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Carlo GASBARRI (Università di Roma II "Tor Vergata"): The analogy between the number fields and the function fields and the abc conjecture

We will explain the analogy between the arithmetic of the varieties over number fields and the arithmetic of the varieties over function fields and we will explain how this gives some insight on the abc conjecture.

[Invited by Dr. Evelina Viada]

**[colloque] Tue 03.03.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Gerhard WANNER (Université de Genève): Kepler, Newton, et l'analyse numérique

[Pour dire "au revoir" à mes amis de Fribourg, où j'ai donné en 1970
un des premiers cours de ma carrière]

En 1609, il y a 400 ans précisemment, Kepler a publié les premiers deux des "Lois de Kepler". On peut considérer ceci comme le premier grand progrès, dépassant le niveau scientifique des Anciens Grecs, qui a mené à Newton et à la science moderne.

[Invited by Profs. J-P. Gabriel and J-P. Berrut]

**[colloque] Tue 10.03.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Martin FÜRER (Pennsylvania State University): Schnelle Multiplikation grosser natürlicher Zahlen

In 1962, Karatsuba has presented an integer multiplication algorithm which multiplies two binary or decimal numbers in less than quadratic time, showing that school multiplication is not optimal. Faster methods have been designed quickly. All are based on some version of the Chinese Remainder Theorem. Most methods reduce integer multiplication to fast polynomial multiplication, which itself is a scheme for the evaluation of polynomials, followed by multiplication of their values and interpolation. For more than 35 years, the fastest known multiplication method has been the Schönhage-Strassen algorithm. It is based on the fast Fourier transform (FFT) and runs in time O(n log n log log n). Only in 2007 has a faster algorithm come closer to the conjectured optimal running time of order n log n. The running time bounds hold for multitape Turing machines. The same bounds are valid for the size of boolean circuits.

[Invited by Prof. Norbert Hungerbühler]

**[colloque] Tue 17.03.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Dagmar IBER (ETH Zürich): Quantitative, predictive models for cellular signaling networks

An increasing number of cellular signaling networks are well characterized in terms of components and interactions. Yet how functionality arises remains unclear. In close collaboration with experimental groups we develop detailed, quantitative models to unravel the mechanistic details and to arrive at a thorough understanding of biological signaling. Importantly, all our models have predictive power, and predictions are tested in experiments. By example of two bacterial signaling networks that regulate cell fate decisions in B. subtilis, and a stress response in E.coli I will illustrate the approach. Finally I will comment on how the method can be used to address similar questions also in higher organisms where less detail is known.

[Invited by Prof. Christian Mazza]

**[colloque] Tue 31.03.2009, Phys 2.52 at 17h15** | more |

### Dr. Norbert PEYERIMHOFF (Durham University): Two radius theorems for harmonic manifolds

In 1929, the Rumanian mathematician Dimitrie Pompeiu asked the following question: given a continuous function f on R² and a compact set K. Assume that the integral of f over all images of K under rigid motions vanishes. Does this imply that the function f itself is zero? Surprisingly, the answer is negative if K is a disk. (It is an open problem whether the answer is yes for all other possible shapes.) However, it can be shown that the conclusion holds if the funtion f vanishes on all disks of two radii r1 and r2, as long as r1 and r2 avoid a certain countable set of radii. It is natural to ask similar questions in more general geometries. In collaboration with E. Samiou, we investigated similar questions in non-compact harmonic spaces. Our results hold in this setting, but in this talk I will only consider the particular case of Damek-Ricci spaces for illustration.

[Invited by Prof. Ruth Kellerhals]

**[colloque] Tue 07.04.2009, Phys 2.52 at 17h15** | more |

### Dr. David CIMASONI (ETH Zürich): Counting dimer configurations on graphs

A dimer configuration (aka perfect matching) on a graph G is a family of edges of G, called dimers, such that each vertex of G is adjacent to exactly one dimer. In the early 60's, Kasteleyn proved the following surprising theorem: the number Z(G) of dimer configurations on a PLANAR graph G is equal to the Pfaffian of some signed-adjacency matrix, and can therefore be computed easily. He also stated (without proof) that for a graph G of genus g, Z(G) can be written as a linear combination of 2^{2g} Pfaffians. In this talk, I shall first review the classical work of Kasteleyn. I will then outline a contribution of N. Reshetikhin and myself to this question: a general Pfaffian formula for Z(G), whose proof uses nothing but a little bit of geometry.

[Invited by Dr. Paul Turner]

**[colloque] Tue 21.04.2009, Phys 2.52 at 17h15** | more |

### Dr. George SZPIRO (NZZ Korrespondent, Jerusalem): Wahrheit, die ganze Wahrheit, nichts als die Wahrheit?

Soll sich ein Journalist, der über Mathematik schreibt, von diesem Grundsatz leiten lassen? Die Frage scheint auf den ersten Blick überflüssig, fast provokativ. Natürlich ist die Wahrheit ein Gut, das allen Menschen - nicht nur Journalisten -, in allen Lebenslagen als Leitmotiv dienen soll. Und für politische Berichterstatter zum Beispiel ist es gar keine Frage, dass sie unbedingt der Wahrheit, der ganzen Wahrheit und nichts als der Wahrheit verpflichtet sind. Wenn ich meinen anderen Hut trage, den des Israel-Korrespondenten der NZZ, bin ich natürlich stets bemüht, mich peinlich genau an alle drei Teile dieses Grundsatzes zu halten. Aber ist das, was Journalisten im Allgemeinen eine Selbstverständlichkeit ist, auch für das Gebiet der Mathematik angebracht?

[Invited by Prof. N. Hungerbühler and Dr. Ch. Leuenberger]

**[colloque] Tue 28.04.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Nicolas BERGERON (Université P. et M. Curie, Paris): Quelques problèmes de topologie des 3-variétés suggérés par des questions arithmétiques

Je commencerai l'exposé par la construction d'une famille de 3-variétés hyperboliques dites "arithmétiques". Ces variétés fournissent une riche source d'exemples (et contre-exemples), j'en décrirai quelques uns. Reste que leur topologie est encore bien mystérieuse. La "philosophie de Langlands" prédit certaines propriétés remarquables. J'en expliquerai certaines et énoncerai quelques résultats inconditionnels allant dans ce sens; résultats récemment obtenus par divers auteurs.

[Invited by Prof. R. Kellerhals and Dr. M. Gendulphe]

**[colloque] Thu 30.04.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Dirk LAURIE (University of Stellenbosch, South Africa): Rational approximations to the complex error function

We start from Weideman's 1994 paper on evaluation of the complex error function (also known as the Faddeeva function) by a formula valid in all of the upper half-plane. That formula can be viewed as an (n-1,n) rational approximation in which the denominator has a single pole of appropriate multiplicity, and can be derived by transforming a certain auxiliary function to the upper half-plane to the unit disk by a Moebius transformation that maps the pole in question to infinity, followed by Taylor expansion around the origin. Instead of Taylor expansion, we use near-best rational approximation on the unit circle to obtain the same accuracy with (n) reduced by a factor of more than two.
The technique used to obtain the near-best rational approximant is the Caratheodory-Fejer method of Trefethen and Gutknecht. A key step in this method is to find the n-th largest eigenvalue of a Hankel matrix formed from some of the Taylor coefficients of the auxiliary function. The free parameter in the Moebius transformation (i.e. the point is mapped to zero) is so chosen to minimize the magnitude of that eigenvalue.

[Invited by Prof. J-P. Berrut]

**[colloque] Tue 05.05.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Martin GANDER (Université de Genève): B-Methods: Numerical Integrators for Time Dependent Non-Linear Partial Differential Equations

Time dependent nonlinear partial differential equations, like for example reaction diffusion equations, are usually solved by classical time marching schemes, like Runge-Kutta methods, or linear multi-step methods. Such equations can however have solutions which blow up in finite time, and in the blowup regime, the behavior of the solution is dominated by the non-linearity. I will show two different approaches how one can construct specialized numerical time integrators which take into account the physics of the underlying non-linear problem. I will show both theoretically and numerically that their performance can be orders of magnitude better than the performance of classical time integrators for such problems.

[Invited by Profs. J-P. Berrut, Ch. Mazza]

**[colloque] Tue 12.05.2009, Phys 2.52 at 17h15** | more |

### Women in Science Colloquium: Prof. Dr. Frances KIRWAN (University of Oxford): Symplectic implosion and non-reductive group actions

The symplectic reduction of a Hamiltonian action of a Lie group on a symplectic manifold plays the role of a quotient construction in symplectic geometry. It has been understood for several decades that symplectic reduction is closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Symplectic implosion (due to Guillemin, Jeffrey and Sjamaar) is much more recent, and is related to a generalised version of GIT which provides quotients for non-reductive group actions in algebraic geometry. The aim of this talk is to give a brief survey of symplectic reduction and symplectic implosion and their relationship with GIT, and (if time permits) describe an application of non-reductive GIT to Demailly's theory of jet differentials.

[Invited by Prof. R. Kellerhals and Dr. L. Ciobanu]

**[colloque] Tue 19.05.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Martin WALLMEIER (Universität Fribourg): Valuation of Multi-Asset Barrier Reverse Convertibles

The market for structured financial products in Switzerland
ranks among the largest in the world. A unique characteristic of the Swiss
market is that its most successful products are reverse convertibles on multiple
assets with conditional capital protection (multiple barrier reverse convertibles,
MBRC). The valuation of MBRCs is not straightforward, and
pricing tools are not yet publicly available. Thus, transparency with respect
to fair values might be poor, and it is not obvious that competition
between issuers is strong enough to ensure "fair" pricing. We provide an
empirical study on market pricing of MBRCs based on a database of 468
certificates outstanding in April 2007. Using a numerical, tree-based valuation
method, we obtain an average overpricing of at least 3.4%. This
premium on the entire product corresponds to a price discount of 29%
on the embedded short put. The overpricing is positively related to the
coupon level, indicating that investors tend to overweight the sure coupon
and underestimate the risk involved. This behavioral bias appears to be
important in explaining the success of the product. We check robustness
of our results with respect to the stock price process by valuing the options
based on a multivariate Variance Gamma process.

[Invited by Prof. Christian Mazza]

**[colloque] Tue 28.05.2009, Phys 2.52 at 17h15** | more |

### Prof. Dr. Michael JOACHIM (Universität Münster): On the Gromov-Lawson-Rosenberg Conjecture

The Gromov-Lawson-Rosenberg Conjecture states that a closed
connected spin manifold of dimension n greater or equal to 5 admits a
Riemannian metric with positive scalar curvature if and only if a certain
index theoretically constructed obstruction class vanishes. It turned out
that the answer to the conjecture very much depends on the fundamental
group of the manifold, so that one can ask wether the conjecture holds
for manifolds as above with an a priori given fundamental group. For
infinite fundamental groups Schick found various counterexamples to the
conjecture, however no counterexample with a finite fundamental group is
known. In our talk we will address some of the major results in the area
and present a new approach for calculating certain bordism groups which
shall be useful to prove the Gromov-Lawson-Rosenberg Conjecture for finite
groups.

[Invited by Prof. Anand Dessai]

## Other events from January 2009 to August 2009

**[workshop] Jan 21-23, 2009, Phys 2.52** | more |

### Workshop: Integral geometry and Finsler geometry

The aim of the workshop is to strengthen the interactions between two different and very active areas of geometry: Integral geometry and Finsler geometry. Both areas are deeply related and use similar tools, like symplectic geometry, Gelfand and Crofton transforms, geometric measure theory, geometric inequalities and convex geometry. This workshop is therefore the perfect occasion to create fruitful collaborations between leading mathematicians in these fields.

**[workshop] Jul 6-10, 2009, Math 0.102 at 10h-15h, Chem 433 at 15h-18h** | more |

### Minicourse: Introduction to continuum mechanics

The aim of this SystemsX seminar is:

a) to introduce basic continuum mechanics models

b) to show how to solve numerically some typical problems by finite
elements

c) discuss how it might be used in cell-growth modelling

More details here.