Workshop SINTROP

This is the third meeting of the ANR group Aspects tropicaux des singularités (SINTROP).
The workshop will take place at the Centre de Mathématiques Laurent Schwartz, in the Campus of École Polytechnique, from Wednesday, March 20th to Friday, March 22nd, 2024.

Organizing committee: Lorenzo Fantini, Charles Favre.

Speakers

• Omid Amini
• Vlerë Mehmeti

• André Belotto da Silva
• Erwan Brugallé
• Marco Maculan
• Martin Ulirsch

Schedule

Abstracts

Minicourses

• Omid Amini (École polytechnique)
  "Hybrid multiscale geometry, higher rank valuation theory, and applications"
  show abstract
  Hybrid varieties are multiscale geometric objects with both complex and polyhedral components. They arise as multiscale limits of families of complex algebraic varieties. Geometry in the hybrid setting combines tropical and complex geometries, and provides a common generalization of both. In these lectures I will provide an introduction to hybrid geometry, present several examples of applications to asymptotic complex geometry, and discuss interesting features of underlying higher rank valuation theory.
• Vlerë Mehmeti (Sorbonne Université)
  "Berkovich spaces, Schottky groups, and the Hausdorff dimension of their limit sets"
  show abstract
  Koebe proved a uniformization theorem for closed Riemann surfaces (not to be confused with the classification of their universal covers), which was extended to the non-Archimedean setting by Mumford. It gives rise to so-called Schottky groups, which are free and finitely generated subgroups of PGL_2 acting on the projective line. In these lectures, using Berkovich spaces, I will give an introduction to Schottky groups over any complete valued field. I will then present a result on the variation of certain associated invariants, and discuss some consequences on their asymptotic behavior over the complex numbers. This is joint work with Nguyen-Bac Dang.

Talks

• André Belotto da Silva (Université Paris Cité)
  "Partial desingularization preserving normal crossings singularities"
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  Can we understand the nature of the singularities that have to be admitted after a blow-up sequence that preserves the normal crossing locus of an algebraic (or complex-analytic) variety X? For example, every surface can be transformed by blowings-up preserving normal crossings to a surface with at most additional Whitney umbrella singularities. We will discuss general conjectures, and solutions for X of dimension up to four. The techniques involve circulant matrices, elementary Galois theory and Newton-Puiseux expansion in several variables. Work in collaboration with Edward Bierstone and Ramon Ronzon Lavie.
• Erwan Brugallé (Université de Nantes)
  "Mr Jourdain compte"
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  En interprétant 1 comme l'unique forme quadratique complexe z->z^2, certains dénombrements classiques (ie à valeurs dans N) acquièrent un sens lorsqu'on remplace le corps des nombres complexes par un corps quelconque k. Le résultat du dénombrement est alors une forme quadratique sur k plutôt qu'un nombre entier. Mon but dans cet exposé est de partager avec les membres de l'ANR ce que je retire pour l'instant de ma récente incursion dans ce monde en ce qui concerne l'énumération des courbes rationnelles dans les surfaces algébriques. Si tout se passe bien, j'aborderai d'un travail en commun avec Kirsten Wickelgren sur une version quadratique de la formule d'Abramovich-Bertram. N'étant pas encore très au point avec le haut niveau de technicité requis par la géométrie algébrique véritable (et non réelle), mon exposé sera informel et souffrira probablement d'un manque de rigueur.
• Marco Maculan (Sorbonne Université)
  "Variétés affines vs. espaces Stein en géométrie complexe et rigide"
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  Le théorème GAGA de Serre affirme que, sur une variété projective complexe, les objets holomorphes (fonctions, fibrés vectoriels et leurs sections, etc.) sont en fait algébriques. Sans compacité ceci n’est pas vrai. Mais on peut alors se poser des questions plus faibles : par exemple, une variété qu’on plonge holomorphiquement dans C^n est-elle nécessairement affine ? Un exemple classique de Serre montre que ce n’est pas le cas. Dans cet exposé je discuterai l’analogue non-archimédien de cette question. De manière un peu surprenante, la situation est—mauvaise blague—beaucoup plus rigide.
• Martin Ulirsch (Goethe-Universität Frankfurt am Main)
  "Logarithmic concavity of bimatroids"
  show abstract
  Bimatroids are a matroidal generalization of the collection of all regular/non-singular minors of a given matrix. In this talk I will explain how to use the theory of Lorentzian polynomials to prove natural log-concavity results for bimatroids. This includes a new proof of the log-concavity of morphisms of matroids and, in particular, of Mason's conjecture. Using our framework, we can also prove a conjecture by Eur and Huh on volume polynomials for morphisms of nullity one. Here (singular) matroid Schubert varieties play a central role. Given time, I will explain how to extend this story to valuative matroids, leading e.g. to a valuative generalization of Mason's conjecture. This talk is based on joint work with F. Röhrle.

Practical informations

If you are interested in attending the workshop you can get in touch with Lorenzo Fantini and/or Charles Favre.

The schedule has been prepared in such a way that the participants can travel to Paris on Wednesday and leave on Friday. A social dinner will take place on Thursday evening at Living Room in Palaiseau.

Information on how to get to École Polytechnique and to CMLS can be found here.

This workshop is founded by the ANR SINTROP.