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6th Swiss-French workshop on algebraic geometry
Charmey (near Gruyères, Fribourg, Switzerland), January 9-13, 2017
The workshop will be held in Charmey from January 9 to 13, 2017.
Mini-courses
In the morning, there will be three mini-courses of 5 hours (3 times one hour each day).
Daniele FAENZI (Dijon)
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Vector bundles, construction and classification |
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Gaël REMOND (Grenoble) |
Mordell's conjecture and some generalisations |
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Ronan TERPEREAU (Dijon)
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Structure and linear representations of algebraic groups |
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In the afternoon, we will have research talks of 50 minutes.
Schedule Talks: In Charmey, ( "Centre Les Dents Vertes, Viva-Gruyère" )
Monday
January 9
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Tuesday
January 10
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Wednesday
January 11 |
Thursday
January 12 |
Friday
January 13 |
12h30 welcome
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breakfast
8h45-9h45
mini-course 1
10h15-11h15
mini-course 2
11h45-12h45
mini-course 3
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breakfast
8h45-9h45
mini-course 1
10h15-11h15
mini-course 2
11h45-12h45
mini-course 3
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breakfast
8h45-9h45
mini-course 1
10h15-11h15
mini-course 2
11h45-12h45
mini-course 3
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breakfast
8h45-9h45
mini-course 1
10h-11h
mini-course 2
11h15-12h15
mini-course 3
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| lunch |
lunch |
lunch |
lunch |
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14h30-15h30
mini-course 1
16h00-17h00
mini-course 2
17h30-18h30
mini-course 3
dinner
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time for discussion / enjoying the mountain
17h20-18h10
Elek
18h30-19h20
van Santen
dinner
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time for discussion / enjoying the mountain
17h20-18h10
Dill
18h30-19h20
Canci
dinner
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time for discussion / enjoying the mountain
17h20-18h10
Vargas De León
18h30-19h20
Poloni
dinner
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Mini-courses - titles and abstracts
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Daniele FAENZI - Vector bundles, construction and classification |
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The course is intended as an introduction to some topics
related to vector bundles on algebraic varieties, mainly complex
projective manifolds.
We will first review some basic notions such as
locally free sheaves, line bundles and the Picard group,
Grassmannians and operations on vector bundles.
Then we will focus shortly on characteristic classes (notably Chern
classes) and related topics.
Next, we will consider the problem of
classification of vector bundles, introduce some splitting results
and say some words on stable bundles.
Finally (and hopefully) we will focus on a special class of bundles,
namely Ulrich bundles, and give a short survey of their properties.
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Gaël REMOND - Mordell's conjecture and some generalisations |
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Mordell's conjecture (1922) is the statement, proven by Faltings in 1983,
that a curve of genus at least two over a number field has only finitely
many rational points. A second proof by Vojta (1991) has led to several
generalisations, the most famous being the so-called Mordell-Lang theorem
on subvarieties of abelian varieties, also due to Faltings. The aim of
this course is to present these results and discuss the strategy of proofs.
The main tool is the notion of height, which plays a central role in
Diophantine geometry and is particularly useful for finiteness statements.
The lectures will include an introduction to heights at a basic level
allowing to formulate the key height inequality, called (generalised)
Vojta inequality, and see how it implies the result on rational points.
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Ronan TERPEREAU - Structure and linear representations of algebraic groups |
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In this lecture we will prove the Chevalley's structure theorem: any connected algebraic group over an algebraically closed field is an extension of an abelian variety by a connected linear algebraic group, and these are unique. Moreover, we will see applications of the Chevalley's structure theorem (e.g. any algebraic subgroup of the Cremona group is linear).
Along the way we will review important facts regarding the structure of algebraic groups, algebraic group actions on algebraic varieties, and linear representations of algebraic groups. We will also present examples of algebraic groups that appear naturally in algebraic geometry.
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Talks - titles and abstracts
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Jung Kyu CANCI - Periodic points for rational functions and integral points of varieties |
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I will present a joint work with S. Vishkautsan where we provide an explicit bound on the number of rational periodic points of a rational function of degree at least 2, where everything is defined over a given number field. Our result is obtained by applying some theorems about integral points of certain varieties.
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Gabriel DILL - The André-Pink-Zannier conjecture for curves |
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In the spirit of the Mordell-Lang conjecture, we consider the intersection of a curve in a family of abelian varieties with the images of a finite-rank subgroup of a fixed abelian variety A0 under all isogenies between A0 and some member of the family. After excluding certain degenerate cases, we can prove that this intersection is finite. We describe the strategy of the proof, due to Zannier, and present some of the techniques we used. If time permits, we give a glimpse of our current work on replacing the curve by a subvariety of arbitrary dimension.
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Balazs ELEK - Kazhdan-Lusztig atlases on Toric surfaces |
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A Bruhat atlas, introduced by He, Knutson and Lu, on a stratified variety is a way of modeling the stratification locally on the stratification of Schubert cells by opposite Schubert varieties. He, Knutson and Lu described Bruhat atlases on many interesting varieties, including partial flag varieties and on wonderful compactifications of groups. We will discuss some results toward a classification of varieties with Bruhat atlases, focusing on the 2-dimensional toric case. In this case, the answer may be stated in terms of the moment polygon of the toric surface, which one should first slice up, then put toppings on, much like one would do while preparing a pizza.
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Pierre-Marie POLONI - The Jonquières subgroup is a Borel subgroup (joint work with Jean-Philippe Furter) |
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The Jonquières subgroup is the group Bn of triangular polynomial automorphisms of the complex affine n-space. By analogy with linear groups, many authors refer to it as the triangular "Borel" subgroup of Aut(𝔸n). In this talk, we will show that the quotation marks could be dropped. Indeed, we will prove that Bn is maximal among all (connected) solvable subgroups of Aut(𝔸n), thus a Borel subgroup of Aut(𝔸n), when the latter is viewed as an ind-group.
We will also consider the following question: Are Borel subgroups of Aut(𝔸n) all conjugate?
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Immanuel VAN SANTEN - Uniqueness of Embeddings of the Affine Line into Algebraic Groups |
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This is joint work with Peter Feller (Max Planck Institute for Mathematics, Bonn).
We consider embeddings X ⟶ Y of affine varieties and
study them up to automorphisms of Y. After recalling some classical results
and examples in case Y is the affine space ℂn, we focus on the main
result we present in this talk: Let Y be the underlying variety of
a connected algebraic group G. If the character group of G is
trivial and dim(G) ≠3, then all embeddings ℂ ⟶ Y
are the same up to automorphisms of Y. We give
some techniques used in the proof of this result and describe the main strategy.
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Alejandro José VARGAS DE LEON - Chip-firing and a Riemann-Roch Formula for Graphs |
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We present an interaction between algebraic geometry and combinatorics developed by Matthew Baker and Serguei Norine. Certain chip-firing game played on a graph is described, and the idea is to introduce a language inspired by linear divisors. Encoding the distribution of chips as a linear divisor and the firing of chips inducing a linear equivalence suggests several analogies and conjectures, leading up to a Riemann-Roch formula for graphs. The exposition also describes algorithms of practical importance for the game, and at the end we state several open questions.
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How to come
The address is VIVA GRUYERE Charmey, Rte des Arses 4, 1637 Charmey
The journey to Charmey is 2h10 from Geneva, 2h30 from Basel/Zurich, 1h30 from Lausanne. See timetables on www.cff.ch, the bus stop is "Charmey (Gruyère), Le Chêne". The place is very close to the bus stop ( map).
Participants
Rémi Bignalet-Cazalet (Dijon)
Arthur Bik (Bern)
Jérémy Blanc (Basel)
Réda Boumasmoud (EPFL)
Alberto Calabri (Ferrara)
Jung Kyu Canci (Basel)
Julie Déserti (Paris)
Gabriel Dill (Basel)
Adrien Dubouloz (Dijon)
Balazs Elek (Cornell)
Daniele Faenzi (Dijon)
Andrea Fanelli (Düsseldorf)
Enrica Floris (Basel)
Linda Frey (Basel)
Jean-Philippe Furter (La Rochelle)
Philipp Habegger (Basel)
Mattias Hemmig (Basel)
Stéphane Lamy (Toulouse)
Anne Lonjou (Toulouse)
Lucy Moser-Jauslin (Dijon)
Karol Palka (Warsaw)
Tomasz Pełka (Warsaw)
Pierre-Marie Poloni (Bern)
Gaël Rémond (Grenoble)
Julia Schneider (Basel)
Ronan Terpereau (Dijon)
Christian Urech (Basel/Rennes)
Immanuel van Santen (Hamburg)
Alejandro José Vargas De León (Bern)
Francesco Veneziano (Basel)
Marius Vuille (EPFL)
Susanna Zimmermann (Toulouse)
Send an email to susannamaria  zimmermann  gmail com if you would like to participate.
Organiser:
Susanna Zimmermann (Toulouse) with the help of
Adrien Dubouloz (Dijon)
Philipp Habegger (Basel)
Jérémy Blanc (Basel)
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