Séminaire de Géométrie Tropicale
Mercredi 16 avril, École Polytechnique, CMLS, salle Laurent Schwartz
Distance to the discriminant
Résumé :
We will study algebraic hyper-surfaces on the real unit sphere S of dimension n-1 (given
by an homogeneous polynomial of degree d in n variables)
with the view point rarely exploited of Euclidian geometry using Bombieri's scalar product and norm.
We will first show some remarkable properties of this scalar product, for instance
a combinatoric formula for the scalar product of two products of linear-forms which allow to give
a (new ?) proof of the invariance of Bombieri's norm by composition with the orthogonal group.
These properties yield a simple formula for the distance of an algebraic hyper-surface to the
"real discriminant" (the set of hyper-surfaces with a real singularity on the sphere).
This property can be further simplified when the hyper-surface has extremal Betti numbers. In this case we have
dist({x in S |P(x)=0}, Delta) = min_{x critical point of P on S} |P(x)|
Finally, we will show that extremal hyper-surfaces that maximize the distance to the discriminant are
very remarkable objects.
We will illustrate the talk showing all extremal sextics curves far from the discriminant and
obtained by numerical optimisation (hence no garanty).
The Tropical Martin boundary
Résumé :
In classical or probabilistic potential theory, one describes the set of
positive harmonic functions on a space using the Martin boundary.
An analogue of this boundary exists in the tropical world, and is in fact
a generalisation of the horofunction boundary of metric spaces.
I will describe what is known about this boundary and give some of its
applications, focusing on a particular example of metric space: Teichmuller
space.
15h30: Kristin Shaw
(University of Toronto)
Tropical surfaces of class VII
Résumé :
In Kodaira's original classification of compact complex surfaces, the surfaces of class VII satisfy b_1=1 and are therefore non-algebraic. We will present some examples of analogues of abstract tropical surfaces by examining their tropical (p,q)-homology and Picard groups.