Colloquia in the current semester
| [colloque] Tue 19.02.2013, Phys 2.52 at 17h15 | more |
Robert Meyerhoff (Boston College): Understanding 3-Manifolds
A 3-manifold is a space that locally looks like 3-dimensional space.
3-manifolds exist in great variety and abundance, and the collection of
all 3-manifolds is bewildering and hard to understand. After a discussion
of general properties of 3-manifolds an approach to organizing 3-manifolds
will be presented where the organizing theme involves the geometric construction
known as Dehn Surgery.
| [colloque] Tue 5.3.2013, Phys 2.52 at 17h15 | more |
Viktor Schroeder (Zürich): Moebius Geometry on the boundary of negatively curved spaces
There is a deep and well known relationship between the
geometry of the classical hyperbolic space and the Moebius geometry of
its boundary at infinity. We generalize this relation to more general
negatively curved spaces. In particular we are interested in a
characterization of the boundaries of rank one symmetric
spaces in terms of Moebius geometry. This is joined work with Sergei
Buyalo and Thomas Foertsch.
| [colloque] Tue 19.03.2013, Phys 2.52 at 17h15 | more |
Claude Marion (Fribourg): Finite simple quotients of triangle groups
This talk is about the $(a, b, c)$-generation problem for finite simple groups,
where we say that a finite group is an $(a,b,c)$-group if it is a homomorphic image of the
triangle group $$T = T_{a,b,c} = \langle x, y, z : x^a = y^b = z^c = xyz = 1\rangle.$$ Typically, given $T$
(or more generally a Fuchsian group $\Gamma$) and a finite (simple) group $G_0$, one investigates
the following deterministic and probabilistic questions:
\begin{enumerate}[a)]
\item is there an epimorphism in ${\rm Hom}(\Gamma, G_0)$?
\item in the case $G_0$ is an $(a, b, c)$-group, what is the abundance of epimorphisms in
${\rm Hom}(\Gamma,G_0)$?
\end{enumerate}
We first give a short survey of some results in this area where two main methods have been applied: either explicit or probabilistic ones. As a consequence, given a
simple algebraic group $G$ defined over an algebraically closed field of prime characteristic
p, we call $(a,b,c)$ rigid for $G$ if the sum of the dimensions of the subvarieties of $G$
of elements of orders dividing respectively $a$, $b$ and $c$ is equal to $2 \dim G$, and we
conjecture that in this case there are only finitely many integers $r$ such that the finite
group $G_0 = G(p^r)$ of Lie type is a $(a, b,c)$-group. We discuss this conjecture and present a third method we recently developed with Larsen and Lubotzky to study the $(a,b,c)$-generation problem for finite (simple) groups using deformation theory.
This new approach gives some systematic explanation of when finite simple groups of Lie type are quotients of a given $T$.
| [colloque] Tue 9.4.2013, Phys 2.52 at 17h15 | more |
Roman Sauer (Karlsruhe): Limits of the homology of finite covers
We study the homology of finite covers of a fixed compact space. Do the Betti numbers and sizes of the torsion submodule of the homology (when suitably normalized) have a limit when the degree of the covers goes to infinity? If yes, what is the meaning of the limit?
| [colloque] Tue 16.04.2013, Phys 2.52 at 17h15 | more |
Alain-Sol Sznitman (ETHZ): Random interlacements and the local geometry of random walks
Random interlacements offer a model for the local structure
left by random walks on large recurrent graphs, which are locally
transient. They have been the object of intensive research over the
recent years and we will discuss some of these developments in this
talk, which is aimed at a non-specialist audience.
| [colloque] Monday! 22.04.2013, Phys 2.52 at 17h15 | more |
Ara Basmajian (CUNY z.Zt. Fribourg): Geodesics on Hyperbolic Surfaces
Associated to any geometric structure are a number of geometrically significant invariants: for example systole, the eigenvalue spectrum of the Laplacian, the length spectrum,... One goal is to make transparent these invariants and their relations. This talk will be an introduction to hyperbolic geometry and the study of some of its invariants. Our focus will be on the closed geodesics of a surface with a hyperbolic geometry.
| [colloque] Tue 30.4.2013, Phys 2.52 at 17h15 | more |
Vincent Bansaye (Ecole Polytechnique Paris): Evénements rares pour des processus de branchement en environnement aléatoire
Les processus de branchement visent à modéliser la reproduction d'une population sans interactions (en particulier sans compétition, prédation). Dans un environnement fixe et avec des générations discrètes, le modèle historique intégrant un aléa dans les événements de reproduction est le modèle de Galton Watson.
Nous considérerons un modèle prenant en compte une stochasticité environnementale, en plus de cette stochasticité démographique. La façon a priori la plus simple de le faire est de tirer aléatoirement la loi de reproduction à chaque génération. Ces processus sont bien étudiés depuis les années 70s, en particulier avec les premiers travaux de Athreya et Karlin.
Plus récemment, une étude des événements rares associés à ces processus a été réalisée. Elle permet de comparer l'effet de la stochasticité démographique et environnementale. Par exemple, si un événement exceptionnel est observé, est-il du à des reproductions exceptionnelles ou un environnement exceptionnel ?
Nous explorerons en particulier le cas où le processus prend des valeurs positives (bornées) et les questions de grandes déviations.
| [colloque] Tue 07.05.2013, Phys 2.52 at 17h15 | more |
Serguei Ivashkovich (Lille-1 / MPIM Bonn): Banach analytic sets and non-linear versions of the E. Levi extension theorem
We shall prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite-dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite-dimensional) analytic family of complex curves and its domain of existence is a "pinched domain" filled in by this analytic family.
A version of this statement on projective surfaces will be also presented.
| [colloque] Tue 14.05.2013, Phys 2.52 at 17h15 | more |
Rodrigo Platte (ASU): Mapped polynomial methods for approximation on equispaced points
The recovery of a function from a finite set of its values is a common problem in scientific computing. It is required, for instance, in the reconstruction of surfaces from data collected by 3D scanners, and is one of the main underlying problems in the numerical solution of partial differential equations. This talk focuses on the special case of approximating functions from values sampled at equidistant points.
It is known that polynomial interpolants of smooth functions at equally spaced points do not necessarily converge, even if the function is analytic. Instead one may see wild oscillations near the endpoints, an effect known as the Runge phenomenon. Associated with this phenomenon is the exponential growth of the condition number of the approximation process. Several other methods have been proposed for recovering smooth functions from uniform data, such as polynomial least-squares, rational interpolation, and radial basis functions, to name but a few. It is now known that these methods cannot converge at geometric (exponential) rates and remain stable for large data sets. In practice, however, some methods perform remarkably well. This talk will focus on mapped polynomial approximations and how they relate to Fourier continuation and radial basis function approximation.
| [colloque] Tue 21.05.2013, Phys 2.52 at 17h15 | more |
Ezio Venturino (Turin): Some recent mathematical applications in biology and ecology
with relevant economic impact
We will present recent research directions pursued together with several
collaborators to understand natural situations via mathematical tools.
a) exploiting the biological control of pests in fruit orchards
This line of investigations, carried out with the biologists
colleagues in Torino, considers models for spiders as generalist predators
of pests in agroecosystems.
b) modeling the interplay of herbivores and vegetation in regional parks;
Mathematical models are used here for the management of resources in public
parks, to assess the policy decisions available to administrators.
c) transmissible disease containment in farms.
The long lasting collaboration with field veterinarians aims at lessening
the consequences of an epidemic in the Cuneo province, one of the hog most
densely populated areas in the country.
| [colloque] Tue 28.05.2013, Phys 2.52 at 17h15 | more |
Stefan Bechtluft-Sachs (Maynooth): A minimal model for the rational homotopy type of a Riemannian manifold
The rational homotopy type of a manifold $X$ (or any topological space) discards all torsion from the homotopy (or homology) groups of $X$.
Its dominating role for partial differential equations stems from its close relation to differential forms which was discovered in the pioneering work
of Sullivan and Quillen in the early 70's. In fact, the rational homotpy type of a simply connected compact manifold is completely
determined by a suitable differentially graded algebra (DGA) of differential forms. This minimal model however is not natural.
On a Riemannian manifold, the metric determines such an DGA which partly remedies the non-functoriality of the minimal model.
We will construct this metric minimal model and discuss some applictions in the calculus of variations.
In part this is a generalization of investigations of geometrically formal manifolds by Kotschick and Terzic.
The metric minimal model of such spaces is just the algebra of harmonic forms.
| [colloque] Tue 4.6.2013, Phys 2.52 at 17h15 | more |
Peter Dragnev (Indiana-Purdue University, Fort Wayne): Characterizing stationary logarithmic points
The product of all $N(N - 1)/2$ possible distances for a collection of $N$ points on the circle is maximized when the points are (up to rotation) the $N$--th roots of unity.
There is an elegant elementary proof of this fact. In higher dimensions the problem becomes much more complicated. For example, if the points are restricted to the unit sphere in 3--space, the result is known
for $N = 1-6$, and 12. We will derive a characterization theorem for the stationary points in $d$--space and illustrate it with a couple of examples of optimal configurations that are new in the literature.
Other talks and events in the current semester
| [mathematikon] Mon 15.04.2013, Phys. 2.52 at 17h15 | more |
Matthieu Jacquemet : Napoléon aurait-il pu gagner à Waterloo ?
18 Juin 1815 : Napoléon est défait à Waterloo par une armée coalisée.
Son abdication quatre jours plus tard met fin à une épopée qui aura marqué durablement la France et l'Europe.
Dans cet exposé, nous allons voir une application un peu exotique de la théorie des jeux. Après une introduction aux concepts de base du domaine, nous allons voir comment on peut les appliquer (grossièrement) dans le cas d'une modélisation de campagne militaire.
La base de l'exposé est l'article "Retour à Waterloo - Histoire militaire et théorie des jeux" de Philippe Mongin (2008).
