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Cremona conference- Basel 2016
Basel, Switzerland, September 5-16, 2016,
A Cremona transformation is a birational transformation of the plane (or more generally of the n-dimensional space). It corresponds to a map given by rational functions that admits an inverse of the same type. These maps have been a subject of study since centuries: the first appearance are in the ancient Greek: inversions and Moebius transformations.
The subject has then been really much developped by the italian school in the nineteenth century and then never ceased to interest mathematicians until now. It has also many applications and relations with other topics of mathematics, like number theory, group theory , dynamical systems or hyperbolic geometry.
The aim of this conference was to bring together a maximum of people having worked on this during the last decade, and to let young people discover it.
Speakers
In the first week (5-9 September 2016), there were mini-courses introducing the subject, given by
Alberto CALABRI
Serge CANTAT
Igor DOLGACHEV
Stéphane LAMY
Yuri PROKHOROV
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Ferrara
Rennes
Ann Arbor
Toulouse
Moscow
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In the second week (12-16 September 2016), we had research talks. Speakers were:
Eric BEDFORD
Ivan CHELTSOV
Ciro CILIBERTO
Julie DÉSERTI
Jeffrey DILLER
Igor DOLGACHEV
Charles FAVRE
Jean-Philippe FURTER
Marat GIZATULLIN
Massimiliano MELLA
Keiji OGUISO
Ivan PAN
Vladimir POPOV
Yuri PROKHOROV
Francesco RUSSO
Nicholas SHEPHERD-BARRON
Aron SIMIS
Junyi XIE
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Indiana
Edinburgh
Roma
Paris
Notre Dame
Ann Arbor
Paris
La Rochelle
Samara
Ferrara
Tokyo
Montevideo
Moscow
Moscow
Catania
London
Recife
Toulouse
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We will also have a special historical talk on September 12, given by
We also had poster sessions and question sessions, in order to encourage discussions between the participants and to promote all recent results on the subject.
Schedule If not indicated: In Basel, Alte Universität, Rheinsprung 9
Basel 4051
Monday
September 5
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Tuesday
September 6
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Wednesday
September 7 |
Thursday
September 8 |
Friday
September 9 |
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9h00-10h00
Calabri
10h30-11h15
Cantat
11h30-12h30
Lamy |
9h00-10h00
Cantat
10h30-11h15
Calabri
11h30-12h30
Lamy |
9h00-10h00
Dolgachev
10h30-11h30
Lamy
11h45-12h45
Prokhorov |
9h00-10h00
Dolgachev
10h30-11h15
Lamy
11h30-12h30
Prokhorov |
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13h30 welcome
14h00-15h00
Calabri
15h30-16h30
Cantat
Spiegelgasse 5
5th floor:
17h00-18h00
exercise session
Welcome Apéritif
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14h00-15h00
Calabri
15h30-16h30
Cantat
Spiegelgasse 5
5th floor:
17h00-18h00
exercise session
Poster session
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social activities
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14h00-14h45
Dolgachev
15h00-15h45
Prokhorov
Apéritif
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14h00-15h00
Dolgachev
15h30-16h30
Prokhorov
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Monday
September 12
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Tuesday
September 13
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Wednesday
September 14 |
Thursday
September 15 |
Friday
September 16 |
10h30
welcome
11h00-12h00
Furter
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9h30-10h30
Déserti
11h00-12h00
Ciliberto |
9h-10h
Shepherd-Barron
10h30-11h30
Popov
11h40-12h40
Prokhorov |
9h30-10h30
Gizatullin
11h00-12h00
Mella |
9h30-10h30
Diller
11h00-12h00
Bedford |
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14h00-15h00
Dolgachev
15h30-16h30
Pan
Kollegienhaus
Hörsaal 118:
17h00-18h00
Kraft
Historical talk
Welcome
Apéritif
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14h00-15h00
Favre
15h30-16h30
Simis
17h00-18h00
Oguiso
Poster session
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social activities
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14h00-15h00
Russo
15h30-16h30
Xie
17h00-18h00
Cheltsov
Question session
Social dinner
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Mini-courses
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Mini-Course 1 - Alberto CALABRI - Introduction to plane Cremona maps |
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The following subjects will be explained: fundamental points and exceptional curves of a plane Cremona map, examples like quadratic and De Jonquières maps, properties like Noether's equations and inequality, factorization of maps and proofs of Noether-Castelnuovo theorem, Cremona equivalence of plane curves.
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Mini-Course 2 - Serge CANTAT - Examples of birational transformations |
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One example of birational transformation will be given at each course. We will look at the dynamics, the sequence of degrees of iterates, the possibility to conjugate it to an automorphism or not,...
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Mini-Course 3 - Stéphane LAMY - Polynomial automorphisms |
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Polynomial automophisms of the affine plane (or space) can be view as birational maps.
This subgroup is sufficiently rich to share many properties with the full Cremona group, but at the same time many specific combinatorial tools are available for its study.
In particular in these lectures I plan to cover the following topics:
- Almagamated product structure of Aut(𝔸2) and action on the related Bass-Serre tree.
- Linearisation of finite subgroups and other consequences (description of algebraic subgroups, Tits alternative...).
- Action on (a subtree of) the tree of valuations (following Jonsson-Favre), comparison with the Bass-serre tree (they are not the same!)
- If time permits, a glimpse at possible generalisations to Tame(𝔸3) (or even Tame(𝔸n)...), of both the Bass-Serre point of view, and the valuative point of view.
Notes of the course
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Mini-Course 4 - Igor DOLGACHEV - Dynamical degrees and Salem numbers |
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I will discuss the notions of topological entropy and the equivalent notion of a dynamical degree of a birational automorphism of an algebraic surface. I will consider examples of computations of the dynamical degrees of automorphisms of Coble rational surfaces:blow-ups of at least ten points in the projective plane with no effective anticanonical divisor but with an effective bi-anticanonical
divisor. The automorphism groups of such surfaces realize many interesting infinite discrete subgroups of the plane Cremona group.
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Mini-Course 5 - Yuri PROKHOROV - Automorphism groups of Fano threefolds |
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I am going to speak about automorphisms of Fano varieties (with applications to finite subgroups of Cremona groups). I will start with simple examples of "interesting" automorphism groups of Fano varieties in dimension 2 and 3. Then I will explain the classification of smooth Fano 3-folds according to Iskovskikh and Mukai , and
outline a recipe of describing automorphisms in terms of Hilbert schemes of lines, conics etc.
Finally, if time permits, I can say something about singular Fano 3-folds. |
Exercises
We had exercise sessions on Monday and Tuesday, at 5pm.
List of exercises
Historical talk
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Hanspeter KRAFT - The Cremona Group, a Historical Survey |
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In two fundamental papers from 1863 and 1865 Luigi Cremona introduced and studied birational transformations of the projective plane. The group of all these transformations was later called Cremona Group. Because of its strong interaction with the geometry of rational surfaces--e.g., the automorphism group of every rational surface is contained in the Cremona Group-it became a central object in classical algebraic geometry.
What do we know today about this group, after 150 years of research? Many interesting results can be found in the classical literature, although some of the proofs appear to be incomplete. We will also see that several basic questions have only been answered in the last 20 years.
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Research talks
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Eric BEDFORD - Real surface automorphisms of maximal entropy |
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We discuss rational surface automorphisms that preserve the real points. When these maps have maximal entropy, the dynamics of the complex map is closely related to its restriction to the real points. We can use complex techniques to get a closer understanding of the real dynamics.
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Ivan CHELTSOV - Cylinders in del Pezzo surfaces |
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For an ample divisor H on a variety V, an H-polar cylinder
in V is an open ruled affine subset whose complement is a support of an
effective ℚ-divisor that is ℚ-rationally equivalent to H.
In the case when V is a Fano variety and H is its anticanonical divisor,
this notion links together affine, birational and Kahler geometries.
In my talk I will show how to prove existence and non-existence of H-polar cylinders
in smooth and mildly singular del Pezzo surfaces for different ample divisors H.
As an application, I will answer an old question of Zaidenberg and Flenner
about additive group actions on the cubic Fermat affine threefold cone.
This is a joint work with Park and Won.
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Ciro CILIBERTO - Contractible curves on a rational surface |
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Let (S,D) be a pair with S a smooth, irreducible, projective, surface
and D an effective, reduced divisor on S. The pair (S,D) is said to
be contractible if there is a birational map φ: S- - >
S' with S' smooth such that φ*(D)=0.
The contractibility problem consists in finding necessary and
sufficient conditions for pairs (S,D) to be contractible.
The contractibility problem is somehow trivial, unless S is rational, in
which case it has its roots in the study of the Cremona geometry
of the complex projective plane.
A result by Kumar-Murthy (1982) solves the problem if D is irreducible.
In the reducible case, the only known general result so far was due to
Iitaka (1983-88).
In this talk I will, focus on the reducible case, I will discuss some
examples and open problems and I will mention two recent results, in
collaboration with A. Calabri.
The first says that (S,D) is contractible if D is reduced and
connected and κ(S,D)=-∞ (which extends both Kumar-Murthy's and
Iitaka's theorems).
The second, which relies on Miyanishi-Tsunoda's Theory of Peeling, says
that (S,D) is contractible if κ(S,D)=-∞, unless (S,D) is a
logarithmic del Pezzo surface of rank 1 and one of the following occurs
(1) (S,D) has shrinkable boundary, (2) if (S',D') is an
almost minimal model of (S,D), then a connected component of
D' (and only one) is a non-admissible fork.
If time allows, I will discuss similar (widely open) questions in higher
dimensions.
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Julie DÉSERTI - Birational maps preserving the contact structure on ℙ3ℂ
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We study the group of polynomial automorphisms of ℂ3 (resp. birational self-maps of ℙ3ℂ) that preserve the contact structure defined by the Darboux 1-form ω=z0dz1+dz2. This study is motivated by a conjecture due to Klein which says that the group of contact maps is generated by the Legendre involution (z1,z0,-z2-z0z1) and what we will call the Klein group.
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Jeffrey DILLER - Rotation numbers and algebraic stability |
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Plane birational maps that preserve the form dx dy / xy naturally induce certain piecewise linear circle diffeomorphisms. Ghys and Sergiescu showed by rather elaborate means that these diffeomorphisms all have rational rotation numbers. Their result was given simpler and more direct proofs later by Liousse and then by Calegari. Here I will explain how a theorem about algebraic stability for birational maps not only implies the result about rotation numbers but, as Favre pointed out, leads to a new direct and simple proof of the fact.
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Igor DOLGACHEV - Automorphisms of Coble surfaces |
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A Coble surface is a rational surface with empty anti-canonical linear system but non-empty anti-bicanonical linear system. They are cousins of Enriques surfaces and their group of automorphism is ofter similar. I will survey known results about automorphisms of Coble surfaces, and, in particular, its relationship with the open problem of finite generation of the group of automorphisms of an algebraic variety.
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Charles FAVRE - Dynamical Manin-Mumford problem for plane automorphisms |
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Joint work with R. Dujardin
The dynamical Manin-Mumford problem asks whether a subvariety containing a Zariski dense set of preperiodic points is itself preperiodic. We shall discuss the status of this problem in the context of plane automorphisms.
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Jean-Philippe FURTER - Length in the Cremona group |
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By a famous result of Noether and Castelnuovo, each plane Cremona transformation admits a decomposition
f = a1 ○ b1 ○ ... ○ ak ○ bk ○ ak+1,
where the elements ai are automorphisms of the projective plane and the elements bi are Jonquières transformations. The length of f is defined as the least possible integer k one can get in such a decomposition.
In this joint work with Jérémy Blanc, we explain how to compute this length and give a few properties that it satisfies.
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Marat GIZATULLIN - Homogeneous real semialgebraic open sets |
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The transformation group under consideration consists of real birational transformations of
the ambient space such that they preserve a fixed open subset and do not have real fundamental points or non-empty parts of real principal subvarieties inside the subset. In contrast with the complex case, there are unexpected examples of homogeneous open semialgebraic subsets whose borders are irreducible irrational (over the complex field) real algebraic hypersurfaces.
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Massimiliano MELLA - Waring problem and Cremona Transformations |
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Waring problem is about additive decompositions of homogeneous polynomials. I am interested in understanding when such decomposition is unique, this is called a canonical form.
The problem of canonical forms is related to the existence of Cremona transformation associated to linear sytstem with assigned double points.
I will report on the state of the art and give some new contribution that confirm the conjecture that there are very few canonical forms.
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Keiji OGUISO - Higher dimensional projective manifolds with primitive automorphisms of positive entropy |
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We first remark that, in any dimension>1, there is an abelian variety with primitive biregular automorphisms of positive entropy. As will be explained in my talk, primitivity of automorphism is a kind of irreducibility in birational geometry. We then show that there are smooth complex projective, hyperkähler fourfolds, Calabi-Yau fourfolds and especially rational fourfolds, with primitive biregular automrphisms of positive entropy. The last automorphisms are regarded as an infinite series of new Cremona transformations of ℙ4 with complex dynamical origin. If time will be allowed, we also discuss some other candiadtes of rational or rationally connected manifolds of this kind.
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Ivan PAN - On the de Jonquières type maps |
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A de Jonquières type map is a Cremona transformation of ℙn for which there exist a positive integer m and points o,o'∈ℙn, such that a general m-plane passing through o is transformed onto a m-plane passing throught o'. Several problems involving Cremona transformations are related to de Jonquières type maps. In this talk we describe properties of de Jonquières type maps and discuss criteria for decide whether a Cremona transformation is of this form.
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Vladimir POPOV - Around the Bass' Triangulability Problem |
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The talk is aimed to elaborate on the Bass' Triangulability Problem concerning rational triangulability of unipotent algebraic subgroups of the affine Cremona groups.
The following topics will be discussed: a rational triangulability criterion;
the existence of rationally non-triangulable connected solvable affine algebraic subgroups of the Cremona groups; stable rational triangulability of such subgroups (in particular, the affirmative answer to the Bass' Triangulability Problem in the stable range); a general theorem on invariant subfields of purely transcendental field extensions used for proving these results; a general construction of all rationally triangulable subgroups of the Cremona groups; application: the classification of all rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.
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Yuri PROKHOROV - p-subgroups of the space Cremona group |
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We discuss two questions about finite p-subgroups
of the space Cremona group posed by J.-P. Serre.
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Francesco RUSSO - Every cubic fourfold in C14 is rational |
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We shall report on joint work with Michele Bolognesi on the irreducible divisor C14 inside the moduli space of smooth cubic hypersurfaces in ℙ5. A general point of C14 is, by definition, a smooth cubic fourfold containing a smooth quartic rational normal scroll (or, equivalently, a smooth quintic del Pezzo surfaces) so that it is rational. We shall prove that every cubic fourfold contained in C14 is rational.
In passing we shall review and put in modern terms some ideas of Fano, yielding a geometric insight to some known results on cubic fourfolds, e.g. the Beauville-Donagi isomorphism or, it time allows, some derived category statements
on cubics in C14.
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Nicholas SHEPHERD-BARRON - Exceptional groups and del Pezzo surfaces |
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We show that simultaneous log resolutions of simply elliptic singularities can be constructed in terms of exceptional groups; this is analogous to what Brieskorn, Grothendieck, Slodowy and Springer did for simple singularities. In particular, there is a direct geometrical path from exceptional groups to del Pezzo surfaces which permits a description of unipotent singularities in low characteristic.
This is joint work with Ian Grojnowski.
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Aron SIMIS - Plane fat points of sub-homaloidal type |
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This is joint work with Zaqueu Ramos. One aims at the ideal theoretic and homological properties of a class of plane fat ideals, based on general points, such that their second symbolic powers are fat ideals having virtual multiplicities of proper homaloidal types. For this purpose one carries a detailed examination of their linear systems at the initial degree, a good deal of the results depending on the method of applying the classical arithmetic quadratic transformations of Hudson-Nagata (called Cremona equivalence by same authors). A subsidiary guide to understand these ideals through their initial linear systems has been supplied by questions of birationality with source ℙ2 and target higher dimensional spaces. This leads, in particular, to the retrieval of birational maps studied by Geramita-Gimigliano-Pitteloud, including a few of the celebrated Bordiga-White parameterizations.
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Junyi XIE - Algebraic actions of discrete groups: the p-adic method |
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With Serge Cantat, we study groups of automorphisms and birational transformations of quasi-projective varieties by p-adic methods. For instance, we show that if SLn(ℤ) acts faithfully on a complex quasi-projective variety X by birational transformations, then dim(X) ≥ n-1 and X is rational if dim(X) = n-1.
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Poster sessions
All participants were welcome to present a poster during the conference. There were two poster sessions, one in each week with an apéritif. The aim was to present to people the recent work made on Cremona transformations and encourage the discussion around the topic.
Question sessions
We had a question session, were open questions asked by the participants were registered. These are available here.
List of questions
Literature
People that were interested to read before the conference could have a look at the following surveys on Cremona transformations:
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Cremona transformations in plane and space. |
Hilda. P. Hudson.
Cambridge University Press, 1927. |
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Geometry of the plane Cremona maps |
Maria Alberich-Carramiñana.
Lecture Notes in Mathematics, 1769, Springer-Verlag, Berlin, xvi+257, 2002. |
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Lectures on Cremona transformations |
Igor Dolgachev.
(2010-11, updated in 2016) |
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Some properties of the Cremona group |
Julie Déserti.
Ensaios Matemáticos, 21. Sociedade Brasileira de Matemàtica, Rio de Janeiro, 2012. ii+188 pp. |
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The Cremona group in two variables |
Serge Cantat. Proceedings of the sixth European Congress of Math., pages 211-225 (Europ. Math. Soc., 2013) |
A list (maybe not exhaustive, but quite large) of articles on Cremona transformations is available on the webpage of Julie Déserti: click here to see the list. Links to the full-text of the articles are given for most of the texts.
Pictures
Pictures of the conference
Participants (95)
Annette A'CAMPO
Norbert A'CAMPO
Jacques ALEV
Dan AVRITZER
Eric BEDFORD
Mohamed BENZERGA
Hannah BERGNER
Marcello BERNARDARA
Rémi BIGNALET-CAZALET
Michel Arthur BIK
Cinzia BISI
Jérémy BLANC
Alberto CALABRI
Jung Kyu CANCI
Serge CANTAT
Ivan CHELTSOV
Ciro CILIBERTO
Nguyen-Bac DANG
Lucas DAS DORES
Matthew DAWES
Julie DESERTI
Gabriel DILL
Jeffrey DILLER
Igor DOLGACHEV
Jan DRAISMA
Adrien DUBOULOZ
Alexander DUNCAN
Sara DURIGHETTO
Eric EDO
Andrea FANELLI
Charles FAVRE
Nestor FERNANDEZ VARGAS
Enrica FLORIS
Kento FUJITA
Mihai FULGER
Jean-Philippe FURTER
Maksymilian GRAB
Philipp HABEGGER
Isac HEDÉN
Mattias HEMMIG
Michel HILSUM
Kyusik HONG
Andrés JARAMILLO PUENTES
Ewan JOHNSTONE
Johannes JOSI
Masayuki KAWAKITA
Hanspeter KRAFT
Igor KRYLOV
Stéphane LAMY
Kwangwoo LEE
Konstantin LOGINOV
Anne LONJOU
Frédéric MANGOLTE
Matilde MANZAROLI
Alexandre MARTIN
Jesus MARTINEZ GARCIA
Massimiliano MELLA
Grigory MIKHALKIN
Olivier MILA
Dmitry MINEYEV
Filip MISEV
Shoetsu OGATA
Keiji OGUISO
Ashraf OWIS
Karol PALKA
Ivan PAN
Tomasz PEŁKA
Alexander PEREPECHKO
Pierre-Marie POLONI
Vladimir POPOV
Yuri PROKHOROV
Raeez RAEEZ LORGAT
Alexandre RAMOS PEON
Andriy REGETA
Matteo RUGGIERO
Francesco RUSSO
Julia SCHNEIDER
Ursina SCHWEIZER
Nicholas SHEPHERD-BARRON
Constantin SHRAMOV
Aron SIMIS
Immanuel STAMPFLI
Ronan TERPEREAU
Mahdi TEYMURI GARAKANI
Takato UEHARA
Christian URECH
Anna USS
Francesco VENEZIANO
Thierry VUST
Joonyeong WON
Alex WRIGHT
Junyi XIE
Yahor YASINSKI
Mikhail ZAIDENBERG
Susanna ZIMMERMANN
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Basel
Basel
Reims
Belo Horizonte
Indiana
Angers
Freiburg (D)
Toulouse
Dijon
Bern
Ferrara
Basel
Ferrara
Basel
Rennes
Edinburgh
Roma
Paris
Liverpool
Bath
Paris
Basel
Notre Dame
Michigan
Bern
Dijon
S. Carolina
Ferrara
Nouméa
Basel/Düsseldorf
Paris
Rennes
Basel
Kyoto
EPFL
La Rochelle
Warsaw
Basel
Kyoto
Basel
Paris
Seoul
Paris
Liverpool
Geneva
Kyoto
Basel
Edinburgh
Toulouse
Seoul
Moscow
Toulouse
Angers
Paris
Vienna
Bonn
Ferrara
Geneva
Bern
Moscow
Bern
Tohoku
Tokyo
Cairo
Warsaw
Montevideo
Warsaw
Moscow
Bern
Moscow
Moscow
Boston
Bern
Grenoble
Paris
Catania
Basel
Basel
London
Moscow
Recife
Hamburg
Dijon
Tehran
Saga
Basel
Warsaw
Basel
Geneva
Seoul
Stanford
Toulouse
Moscow
Grenoble
Basel
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(the third column is 1-2 if the participant come on both weeks, or 1, respectively 2, if he/she comes the first, respectively the second week)
We had some partial support for the lodging.
Organisers
Jérémy Blanc, Mattias Hemmig, Christian Urech, Susanna Zimmermann
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