## Colloquium program from January 2008 to August 2008

**[colloque] Tue 26.02.08, Phys 2.52 at 17h15** | more |

### Dr. Jérôme BERTRAND (Toulouse): Notion de courbure minorée sur un espace métrique

Je présenterai quelques aspects d'une théorie initiée -indépendamment- par
Lott-Villani et Sturm d'espaces métriques mesurés "de courbure de Ricci minorée".
Lorsque l'espace métrique est une variété riemannienne, cette propriété
coïncide avec la définition usuelle. Ce nouveau point de vue utilise fortement le
transport optimal de masse. Après des rappels sur le transport optimal, j'expliquerai comment on peut le relier au comportement de la courbure de Ricci d'une variété riemannienne.

[Invited by F. Fillastre and Prof. Kellerhals]

**[colloque] Tue 4.03.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Özlem IMAMOGLU (ETHZ): Zeroes of Weierstrass P-function and Hypergeometric series

Surprisingly the location of the zeroes of the Weierstrass P-function was not known until 1982, when Eichler and Zagier found for them a beautiful integral formula. In this talk I will report on joint work with W. Duke where we were able to "deuniformize" the Eichler Zagier formula and write the zeroes in terms of generalized hypergeometric series.

[Invited by Dr. Viada]

**[colloque] Tue 11.03.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Gérard ANTILLE (Genève): Réduction de dimensions pour des suites temporelles de matrices de données

Au cours de ce colloque une technique d'analyse de suites temporelles de matrices de données basées sur les méthodes de réduction de dimensions, propres à l'analyse des données, sera présentée. L'objectif principal de la méthode proposée est de décrire l'évolution, par rapport au temps, des unités statistiques dans un espace résumant l'ensemble des matrices et de détecter des comportements similaires des unités. L'analyse duale peut être tout aussi pertinente comme le montrera, par exemple, une application basée sur l'évolution des taux de différentes causes de décès dans les cantons suisses.

[Invited by Prof. Gabriel]

**[colloque] Tue 18.03.08, Phys 2.52 at 17h15** | more |

### Dr. Pierre-Emmanuel CAPRACE (IHES): A few specimens from a zoology of simple groups

Simple groups are to group theory what primes are to number theory: they constitute the elementary particles. Although one observes that generically, the automorphism group of a highly symmetric objects tends to be simple, actual simplicity proofs are often delicate to obtain. In fact, until the early 20th century the only known examples of infinite simple groups were closely related to the simple Lie groups. The aim of this talk is to recall that many basic and familiar objects in mathematics admit an almost simple automorphism group.

[Invited by Prof. Kellerhals]

**[colloque] Tue 1.04.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Evgeny SPODAREV (Ulm): Mean curvatures of the Wiener sausage

The parallel r-neighborhood of a path of the Brownian motion in d dimensions is often
called a Wiener sausage. In this talk, some geometric properties
of Wiener sausages are studied. For instance, a formula for the mean surface
area and its asymptotics for small r are given. Approximations of the Wiener
sausage by polyconvex sets and the convergence of the corresponding curvature
measures are addressed as well.
We conclude by considering the Boolean model of Wiener sausages and its
geometric characteristics.
This is joint work with R. Cerny, S. Funken, D. Meschenmoser, J. Rataj and V. Schmidt.

[Invited by Prof. Bernig]

**[colloque] Tue 8.04.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Yanyuan MA (Neuchâtel): Locally Efficient Estimators for Semiparametric Models With Measurement Error

We derive constructive locally efficient estimators in semiparametric measurement error models. The setting is one where the likelihood function depends on variables measured with and without error, where the variables measured without error can be modelled nonparametrically. The algorithm is based on backfitting. We show that if one adopts a parametric model for the latent variable measured with error and if this model is correct, then the estimator is semiparametric efficient; if the latent variable model is misspecified, our methods lead to a consistent and asymptotically normal estimator. Our method further produces an estimator of the nonparametric function that achieves the standard bias and variance property. We extend the methodology to allow for parameters in the measurement error model to be estimated by additional data in the form of replicates or instrumental variables. The methods are illustrated via a simulation study and a data example, where
the putative latent variable distribution is a shifted lognormal, but concerns about the effects of misspecification of this assumption and the linear assumption of another covariate demand a more model-robust approach.

A special case of wide interest is the partially linear measurement error model. If one assumes that the model error and the measurement error are both normally distributed, then our estimator has a closed form. When a normal model for the unobservable variable is also posited, our estimator becomes consistent and asymptotically normally distributed for the general partially linear measurement error model, even without any of the normality assumptions under which the estimator is originally derived. We show that the method in fact reduces to a same estimator in Liang et al. (1999), thus showing a previously unknown optimality property of their method.

This is joint work with Raymond J. Carroll and Anastasios A. Tsiatis.

[Invited by Prof. Mazza]

**[colloque] Tue 15.04.08, Phys 2.52 at 17h15** | more |

### Dr. Tatiana SMIRNOVA-NAGNIBEDA (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. Ciobanu]

**[colloque] Tue 22.04.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. René SPERB (ETHZ): Knotenlinien von Eigenfunktionen und Fellzeichnung von Tieren

Abstrakt deutsch:

Es wird ein Modell von J.D. Murray vorgestellt, das einen Zusammenhang herstellt zwischen der Fellzeichnung von gewissen Tieren (Zebra, Leopard etc.), und den Knotenlinien der Eigenfunktionen der freien Membran (d.h. des Laplace Operators mit Neumann Randbedingungen). Zugrunde liegt ein Diffussions-Reaktions Prozess der Substanzen,die für die Melaninproduktion verantwortlich sind. Daraus ergeben sich interessante Konsequenzen für die möglichen Muster.

Abstract english: Nodal lines of eigenfunctions and animal coat patterns

A model of J.D. Murray is presented which gives a connection between the nodal lines of the free membrane(i.e. Laplacian with Neumann boundary conditions) and animal coat patterns(e.g. Zebra,Leopard etc.)It is based on a reaction-diffusion process of the substances responsible for the melanin production.There are interesting consequences for the possible patterns.

[Invited by Prof. Hungerbühler]

**[colloque] Tue 29.04.08, Phys 2.52 at 17h15** | more |

### Dr. Clément DOMBRY (Poitier): Convergence des schémas de récompense aléatoire

De nouveaux processus stables, autosimilaires et à accroissements stationnaires ont récemment été étudiés par S.Cohen et G.Samorodnitsky ([1]). Ces processus sont construit comme intégrale stochastique du temps local d'un mouvement Brownien fractionnaire contre une mesure stable. Nous étudions la convergence des schémas de récompenses aléatoires vers ces processus. Ces schémas discrets sont basés sur la convergence des marches aléatoires étudiée par H.Kesten et F.Spitzer ([2]) et repris par W.Wang ([3]).

[Invited by Prof. Mazza]

**[colloque] Tue 06.05.08, Phys 2.52 at 17h15** | more |

### Dominik COLANGELO (Lugano): Modelling and Forecasting Implied Volatility Surfaces

Despite the discrepancy between the Black and Scholes (BS) theory and reality, the concept of implied volatility surfaces (IVS) is still very popular. The mapping from observed market prices to implied volatilities (IV) is used as a way to make option prices more comparable. IVs of options are calculated and stored in financial databases. Market makers, traders, and risk managers rely on IV to calibrate their pricing models.

In efficient markets, new information has an instantaneous influence on option prices, which reflect expectations about future prices of the underlying stock. Hence IV can be seen as predictor for future volatility. This concept was formalized by Dupire (1994) and Derman et al. (1996). Conditional on the market level of the underlying stock at some future date, T, which is set as equal to K, and conditional on the available information up to time t, the market's expectation of the instantaneous volatility under the risk-neutral measure is defined as the local volatility. The Dupire formula represents local volatility as a function of observed call prices and their derivatives. Under certain conditions, IV can be thought of as an average of local volatility through the lifetime of the option.

Classical approaches assume that instantaneous volatility is a deterministic function in spot price and time. This implies that local and instantaneous volatility coincide. Dumas, Fleming, and Whaley (1998) find that estimated parameters are highly unstable over time, allowing only for short time predictions. As Poon and Granger (2003) point out, classical time series-based methods do not perform well in predicting instantaneous volatility. In Chapter 6 they also report that forecasting the instantaneous volatility based on option implied standard deviation has superior performance across different assets and over quite long forecast horizons (up to three years). This approach needs a strategy to decide which point on the IVS or which weighting scheme should be used to obtain a forecast of the instantaneous volatility, used as input in a option pricing framework.
In contrast to previous approaches, we model the whole implied volatility surface in order to price options simultaneously via BS formula as mapping from option prices to IV. Goncalves and Guidolin (2005) combine a cross-sectional approach similar to that of Dumas, Fleming, and Whaley (1998) with vector autoregressive models, but with exactly this idea in mind. The findings are mixed, and provide only a good in-sample fit, but questionable out-of-sample performance. Even the more complex and computational demanding dynamic semiparametric factor model (DSFM) of Fengler, HŠrdle and Mammen (2007) has quite limited predicting power. In a comparison of the one-day prediction error, the DSFM performs 10 percent better than a simple sticky-moneyness model, where IV is taken to be constant over time at a fixed moneyness.

Following this direction, we propose a new semi-parametric model based on an additive expansion of simple fitted regression trees, which can be easily estimated using boosting techniques. Any starting model can be enhanced with the help of our framework by including exogenous factors. The relevant ones are chosen automatically by the regression trees used in our model to minimize the residuals of observed and estimated IV. Our model improves the out-of-sample prediction of the IVS and can handle very high values of IV, of both in- and out-of-the-money options.

[Invited by Prof. Mazza]

**[colloque] Tue 13.05.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Nigel RAY (Manchester): Topological aspects of weighted projective spaces

In this talk, I shall begin by discussing complex projective space as a complex manifold that is equipped with a well-behaved action of the torus. Its generalisation to weighted projective space is a source of many examples for algebraic and symplectic geometers, and even theoretical physicists, but is less well-known to topologists. So I shall introduce some of the basic topological properties of weighted projective spaces, as background to recent joint work with Tony Bahri and Matthias Franz. If time permits, I hope to outline the relevance of one or more key concepts such as the Borel construction, homotopy colimits, piecewise polynomials, and Thom complexes.

[Invited by Prof. Dessai]

**[colloque] Tue 20.05.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Juan Carlos ALVAREZ (Lille): There is always something new around Hilbert's fourth problem

Hilbert's fourth problem is the simplest inverse problem in variational calculus : determine all metrics on the plane for which straight lines are geodesics. If we generalize the statement to include non-symmetric metrics, then all that has been achieved is the construction of a few examples. In this talk I will show that while an integral-geometric solution such as exists in the symmetric case is not possible, it is easy to construct a great many examples of such metrics and that the general solution may be just around the corner. This is work in progress with Gautier Berck.

[Invited by Dr. G. Berck / Prof. Bernig]

**[colloque] Tue 27.05.08, Phys 2.52 at 17h15** | more |

### Prof. Dr. Bernard DACOROGNA (EPFL): Sur l'équation de rappel en géométrie différentielle: un point de vue analytique

Abstract in PDF is
here.

[Invited by Prof. Hungerbühler, PD Dr. F. Meylan]