ON ALGEBRAIC GROUP ACTIONS
Conférence en l'honneur de Thierry Vust
Genève, 17-19 Février 2010

La conférence aura lieu à Genève du 17 au 19 février 2010.
Le sujet sera actions (rationelles ou birégulières) de groupes algébriques sur des variétés algébriques.
Quelques exposés porteront également sur d'autres sujets.

Arnaud BEAUVILLE Michel BRION Serge CANTAT Pierre DE LA HARPE Hanspeter KRAFT Domingo LUNA Lucy MOSER-JAUSLIN Raghavan NARASIMHAN Ivan PAN Gerald SCHWARZ Andras SZENES

Nice Grenoble Rennes Genève Basel Grenoble Dijon Chicago Montevideo Brandeis (Boston) Genève

Arnaud BEAUVILLE - Finite subgroups of PGL(2,K)
	We describe the conjugacy classes of finite subgroups of PGL (2,K), for an arbitrary field K.
Michel BRION - Homogeneous bundles over abelian varieties
	Given an algebraic variety X equipped with the action of an algebraic group G, a principal bundle over X is homogeneous if it is isomorphic to all of its pull-backs under elements of G. Generalizing results of Miyanishi and Mukai about homogeneous vector bundles, we describe the structure of homogeneous principal bundles over an abelian variety. We reduce their classification to (difficult) problems of linear algebra.
Serge CANTAT - Recent results on the Cremona Group
	The Cremona group is the group of birational transformations of the (complex) projective plane. I shall describe how hyperbolic geometry and geometric group theory can be used to study the algebraic structure of this group.
Pierre DE LA HARPE - Dénombrements d'actions de groupes sur les ensembles finis et sur les espaces vectoriels
	L'exposé décrira deux programmes de comptage de certaines actions de certains groupes. D'abord, compter les actions par permutations sur les ensembles finis, ce qui revient à compter les sous-groupes d'indice fini selon leurs indices. Ensuite, compter les représentations linéaires irréductibles selon leurs dimensions. Les comptages du premier type sont notamment motivés par des questions de géométrie riemannienne. L'intérêt pour les comptages du second type est plus récent (Witten, Lubotzky, ...) et sera l'occasion d'énoncer un résultat obtenu en collaboration avec Laurent Bartholdi.
Hanspeter KRAFT - The linearization problem: old and new
	The linearization problem asks if an action of a reductive algebraic group on complex affine n-space An is equivalent to a linear representation. For A2 this is indeed the case, due to the structure of the automorphism group of A2 as an amalgamated product. However, it does not hold in dimension >2 The first counterexamples were given by G. W. Schwarz in 1989; they initiated an interesting development. We will describe some highlights, some open problems and some recent developments.
Domingo LUNA - Examples of pseudo-spherical subgroups
	Let G be a (complex) semi-simple group, B a Borel subgroup of G , and U the unipotent radical of B . An algebraic subgroup H of G is called "spherical", if B has an open orbit in G/H ; H spherical implies dim(H) ≥ dim(U) . In my talk, I will discuss examples of H's that are not spherical, although dim(H) ≥dim(U).
Lucy MOSER-JAUSLIN - Isomorphism classes of certain hypersurfaces in complex affine four-space
	I will discuss work done in collaboration with A. Dubouloz and P.M. Poloni on a set of hypersurfaces of 4 having a non-trivial Makar-Limanov invariant. More precisely, we are interested in hypersurfaces defined by an equation of the form xdy+r(x,z,t)=0, where d ≥ 2, and r(0,z,t) satisfies some additional properties. Among these varieties, one finds the Koras-Russell threefolds, which are contractible smooth affine threefolds, endowed with a hyperbolic action of *. They are, in this sense, very similar to affine three-space. We study properties of the automophism groups of these threefolds, and also give some partial results on isomorphisms between such varieties. This study allows us to determine some surprising results about automorphisms of Koras-Russell threefolds.
Raghavan NARASIMHAN - Riemann's Lectures of 1858/59 on the Hypergeometric Series
	This course of Riemann was written down in "Gabelsberger Stenographie" by Wilhem von Bezold, and came to the attention of the Berlin mathematicians in the 1890's. The aim of this talk is to point out how Riemann had anticipated the ideas of L. Fuchs and H.A. Schwarz and to say a few words about how he proposed studying what we now call finite dimensional local systems of germs of holomorphic functions on 1\ S, where S is a finite set.
Ivan PAN - On Cremona transformations of 3 with minimal length
	Cremona transformations of the dimension 3 complex projective space may be factorized as a product of elementary links (i.e, elementary birational maps between Mori Fiber Spaces). We classify all Cremona transformations which factorize as a product of m links (without flips) for m ≤ 2.
Gerald SCHWARZ - Characteristic invariants of reductive groups
	Let G be a reductive complex group and V a finite dimensional G-module. Associated to G there are various invariant objects: orbits, fibers of the quotient mapping, invariant polynomial functions, etc. Following Raïs we say that an object is characteristic if the subgroup of GL(V) preserving it is G or at least has identity component contained in G. We discuss some examples of characteristic objects with special attention to orbits which are characteristic. For many V it turns out that all nonzero orbits are characteristic.
Andras SZENES - Quantization of symplectic manifolds and the combinatorics of partition functions
	On a polarized compact symplectic manifold endowed with an action of a compact Lie group, in analogy with geometric invariant theory, one can define the space of invariant functions of degree k. A central statement in symplectic geometry, the quantization commutes with reduction hypothesis, is equivalent to saying that the dimension of these invariant functions depends polynomially on k. This statement was proved by Meinrenken and Sjamaar under positivity conditions. In joint work with Mich�le Vergne, we found a new proof of this polynomiality, which is much less technical than the earlier proofs. In this talk, I will explain the basic ideas of this proof, in which a prominent role is played by the theory of partition functions.

Exposé: À Uni-Mail, boulevard du Pont-d'Arve 40, 1205 Genève, Switzerland, dans la salle S130 (au sous-sol)

Mercredi   17 Février	Jeudi   18 Février	Vendredi   19 Février
	8h45-9h35  Domingo LUNA  Examples of pseudo-spherical subgroups 9h50-10h40  Pierre DE LA HARPE  Dénombrements d'actions de groupes sur les ensembles finis et sur les espaces vectoriels  pause café 11h10-12h  Michel BRION  Homogeneous bundles over abelian varieties	9h50-10h40  Lucy MOSER-JAUSLIN  Isomorphism classes of certain hypersurfaces in complex affine four-space  pause café 11h10-12h  Hanspeter KRAFT  The linearization problem: old and new
13h15-14h  Accueil des participants en face de la salle S130 (sous-sol d'Uni-Mail) 14h-14h50  Serge CANTAT  Recent results on the Cremona Group 15h10-16h  Raghavan NARASIMHAN  Riemann's Lectures of 1858/59 on the Hypergeometric Series  pause café 16h30-17h20  Ivan PAN  On Cremona transformations of 3 with minimal length	14h-14h50  Gerald SCHWARZ  Characteristic invariants of reductive groups 15h10-16h  Andras SZENES  Quantization of symplectic manifolds and the combinatorics of partition functions    pause café 16h30-17h20  Arnaud BEAUVILLE  Finite subgroups of PGL(2,K)
Apéritif	Repas officiel

Si vous arrivez à Genève en train, prenez le tram 15 de "Gare Cornavin" (la gare principale de Genève) à "Uni-Mail" (direction: "Palettes"). Durée: 10 minutes.
Si vous arrivez à Genève en avion, n'oubliez pas de prendre un ticket gratuit avant de sortir du hall à bagages, prenez un train en direction de Genève (tous les trains y vont et c'est le prochain arrêt) et prenez le tram 15 (voir ci-dessus). Le billet est valable une heure pour le train ET le bus. Durée: 25 minutes.

Pour vous inscrire, merci d'envoyer un e-mail à l'organisateur avec votre nom, grade, et institution. C'est une conférence sans frais d'inscription.

Nous prions les participants (pas les conférenciers invités) de réserver leur hôtel eux-mêmes.
Voici une longue liste d'hôtels de Genève.

Klaus Altmann (Berlin), Daniel Amiguet (Renens), Guillaume Batog (Nancy), Tanja Becker (Mainz), Pierre-Emmanuel Chaput (Nantes), Adrien Dubouloz (Dijon), Cyrille Extermann (Genève), Shaula Fiorelli-Vilmart (Genève), André Haefliger (Genève), Jean-Claude Hausmann (Genève), Yi-Ning Hsiao (Genève), Jean-Louis Kozsul (Grenoble), Stéphane Lamy (Lyon/Warwick), Henri-Michel Maire (Genève), Nadia Morsli (Lyon), Afsaneh Mehran (Genève), Tatiana Nagnibeda (Genève), Manuel Ojanguren (EPFL), Pierre-Marie Poloni (Basel), Jose Ribeiro (Genè), Jean-Pierre Serre (Paris), John Steinig (Genève), Alain Valette (Neuchâtel), Donna Testermann (EPFL), Stéphane Vénereau (Basel), Claude Weber (Genève)

Nous remercions chaleureusement le support de:
Fonds national Suisse de la recherche scientifique
Fondation "L'Enseignement mathématique"
Société mathématique Suisse
Université de Genève, Faculté des sciences et Section de mathématiques
Cercle mathématique romand