Jérémy Blanc - Universität Basel - Mathematik
Borel Seminar 2013
Les Diablerets (Switzerland), June 16-21, 2013
The Summer school was supported by the SWISS DOCTORAL PROGRAM IN MATHEMATICS (part of the CUSO) and by the ERC TROPGEO and was be held in June from June 16 to 21, 2013.
Mini-courses
There was four mini-courses of introduction to tropical geometry and real algebraic geometry, and four colloquium-style talks on subjects related:
Mini-courses on tropical geometry
I. ITENBERG (Paris) Tropical aspects of real algebraic geometry
K. SHAW (Toronto) Tropical curves in surfaces
 
Mini-courses on real algebraic geometry
D. GAYET / J. WELSCHINGER (Lyon) Betti numbers of random real hypersurfaces
J. MONNIER (Angers) Regulous functions
 
Colloquiums
N. A'CAMPO (Basel) How to detect a singularity?
P. BIRAN (Zurich) Lagrangian topology, cobordisms and the Fukaya category
B. COLBOIS (Neuchâtel) Some aspects of Hilbert's geometry
F. MANGOLTE (Angers) There is exactly one rational algebraic model for each nonorientable topological surface

Schedule
Talks: In Les Diablerets, ("Hôtel les Sources, chemin du Vernex, CH- 1865 Les Diablerets" )
Sunday  
June 16 		Monday  
June 17 		Tuesday  
June 18 		Wednesday  
June 19		 Thursday  
June 20		 Friday  
June 21
   breakfast

9h15-10h30 
 mini-course
Itenberg

coffee break

11h00-12h15 
 mini-course
Monnier		  breakfast

9h15-10h30 
 mini-course
 Gayet / Welschinger

coffee break

11h00-12h15 
 mini-course
Itenberg		  breakfast

9h15-10h30 
 mini-course
Shaw

coffee break

11h00-12h15 
 mini-course
 Gayet / Welschinger		  breakfast

9h15-10h30 
 mini-course
Itenberg

coffee break

11h00-12h15 
 mini-course
 Gayet / Welschinger		  breakfast

9h15-10h30 
 mini-course
Shaw

coffee break

11h00-12h00 
Colloquium
 Mangolte
   lunch  lunch  lunch  lunch  lunch








 dinner

 16h45-17h45 
Colloquium
 Colbois

18h-19h15 
 mini-course
Shaw

 dinner

 16h45-17h45 
Colloquium
 A'Campo

18h-19h15 
 mini-course
Monnier

 dinner

 walk in
 the mountains




 dinner

 16h45-17h45 
Colloquium
 Biran

18h-19h15 
 mini-course
Monnier

 dinner
Mini-courses - titles and abstracts
D. GAYET / J. WELSCHINGER - Betti numbers of random real hypersurfaces
We will explain how to bound from above the Betti numbers of the real locus of a random real hypersurface in a real projective manifold, the hypersurface being taken in the linear system of a big tensor power of an ample line bundle over the manifold. The proof involves peak sections of Hörmander and the result the expected determinant of random symmetric matrices.
 
Ilia ITENBERG - Tropical aspects of real algebraic geometry
The purpose of the course is to give an introduction to topology of real algebraic varieties making an emphasis on connections between real and tropical geometries.
In particular, we will discuss tropical constructions of real algebraic varieties and the tropical approach to real enumerative problems.
 
J.-P. MONNIER - Regulous functions
Let k be an integer. We study the ring of rational functions admitting an extension of class Ck to the real affine space. These functions are called k-regulous functions. We establish several properties of this ring. In particular, we prove a strong Nullstellensatz. We also compare the k-regulous topology with the arc-symetric topology and the constructible topology.
  Joint work with J. Huisman, F. Mangolte and G. Fichou.
 
K. SHAW - Tropical curves in surfaces
This will be an introduction to the study of tropical geometry in dimension two. The tropical geometry has been widely studied in the case of curves. I will present tools for studying tropical surfaces and curves contained in them, such as intersection theory, birational transformations, tropical sums of surfaces and (p,q) homology.
Talk 1 Tropical cycles, divisors and modifications, matroidal fans and compactifications, degree of cycles.
Talk 2 Intersecting tropical cycles in matroidal fans, and local approximation of tropical curves in surfaces.
Talk 3 Adjunction formula, Noether's formula and Hirzebruch signature formula in special cases.
 
Registration
SEND an email to Jeremy Blanc unibas ch if you want to participate, or if you need any information.
DEADLINE: April 1st, 2013.
For members of the swiss doctoral program, the participation is free. For other participants, the price of the hotel is 152CHF per person per day will full board in single room (or 130 CHF in double room), so respectively 675CHF / 760 CHF for the stay.
Some support for a few participants will be provided by the ERC grant TROPGEO.
The deadline for the support was 15th of March.
List of participants
Members of the Swiss doctoral program:
Ivan Bazhov (Geneva)
Noémie Combe (Geneva)
Peter Feller (Bern)
Johannes Josi (Bern)
Nikita Kalinin (Geneva)
Natalia Kolokolnikova (Geneva)
Lionel Lang (Geneva)
Paul-Henry Leemann (Geneva)
Matthias Leuenberger (Bern)
Filip Misev (Bern)
Maria Fernanda Robayo (Basel)
Alexandre Ramos Peon (Bern)
Andriy Regeta (Basel)
Immanuel Stampfli (Basel)
Mikhail Shkolnikov (Geneva)
Susanna Zimmermann (Basel)

Participants supported by ERC TropGeo (for lodging/meals) :
Johan Bjorklund (Geneva)
Arne Buchholz (Saarland)
Rémi Crétois (Geneva)
Yaniv Ganor (Tel Aviv)
Andreas Gross (Kaiserslautern)
Christian Jürgens (Saarland)
Maksim Karev (Geneva)
Simon Hampe (Saarland)
Arthur Renaudineau (Paris 6)
Arnaud Tomasini (Strasbourg)
Anna Lena Winstel (Kaiserslautern)

Other participants :
Farhad Babaee (Bordeaux)
Jens Forsgård (Stockholm)
Antonio Lerario (Purdue)
Pavel Mnev (St. Petersburg)
Organisers
Jérémy Blanc (Basel)
Grigory Mikhalkin (Geneva)