Séminaire de Géométrie Tropicale
Mercredi 12 mars, École Polytechnique, CMLS, salle Laurent Schwartz
11h00:
Antonio Lerario
(Institut Camille Jordan, Lyon)
A random intersection of real quadrics is usually connected
Résumé :
Betti numbers b_2i(C) of a complete intersection C in complex projective space distributes in the range 0,...,dim(C) in a "delta shaped" function:
they are all "ones" except in the middle dimension where all the topology is concentrated
(the sum of all Betti numbers is represented by the "integral" of this function).
Looking at the real analogue of this picture the only thing we can say is that the sum ("integral") of all the Betti numbers of the real part R is bounded by that of its complex counterpart C.
Despite this high level of freedom over the reals, we can still ask what is the typical behavior of this picture. In other words, picking a random real complete intersection, what do we expect its Betti numbers to be? And how do we expect them to distribute in the range 0,...,dim(R)?
In this talk I will combine techniques from algebraic topology (spectral sequences) and probability theory (random matrices) to give an answer to these questions in the case of an intersection of random quadrics. I will show that, in a sense, the real picture is on average the "square root" of the complex one.
The statement in the title will be a special case.
(This is joint work with E. Lundberg)
14h: Jean-Philippe Monnier
(Université d'Angers)
Systèmes linéaires très spéciaux sur les courbes algébriques réelles
Résumé :
Pour une courbe algébrique réelle donnée, on étudie les systèmes linéaires spéciaux appelés "très spéciaux", la dimension de ces systèmes linéaires ne satisfait pas une inégalité de type "Clifford". On détermine ces systèmes linéaires très spéciaux lorsque la partie réelle de la courbe a peu de composantes connexes et aussi lorsque la gonalité de la courbe est petite.
Tropicalization of the moduli space of stable maps
Résumé :
I will begin by an introduction to Berkovich spaces and to the context of global tropical geometry. A basic fact of global tropical geometry is that analytic curves in the analytic space map to tropical curves in the Clemens polytope. This fact can be proved either using the functorial property of tropicalization by formal models or using quantifier eliminations from model theory. I will explain the latter approach, which will then be generalized to study the tropicalization of the moduli space of stable maps. I will prove that the tropicalization of the moduli space of stable maps of bounded genus and degree is a compact finite polyhedral complex. The proof has four major ingredients:
(i) the k-analytic/tropical Kähler structures,
(ii) the continuity theorem via formal models and balancing conditions,
(iii) the non-Archimedean analytic Gromov compactness theorem,
(iv) the model theory of rigid subanalytic sets develop by Lipshitz.
The talk is based on Section 7 of my preprint arXiv:1401.6452.