Abelian Tits Sets

• Bernhard MÜHLHERR

Room: 11 place Marcelin-Berthelot, 75005 Paris A Tits set is a pair (G,X) consisting of a group G and a conjugacy class X of subgroups satisfying certain conditions. It is called Abelian if the elements of X are Abelian and it is called a Moufang set if any two elements of X intersect trivially. Moufang sets were introduced by Tits in the 1990s in order to extend the Moufang property to buildings of rank one. By the classification of Moufang polygons of Tits and Weiss, each Moufang building of rank at least two is associated with a simple algebraic group over a field or a variation thereof. It is an open question whether this is also true for Moufang sets. Examples of Tits sets can be constructed from spherical Moufang buildings by means of a Tits index. The Tits sets obtained in this fashion are called of index type. Our main result asserts that each Abelian Tits set is of index type. As a corollary, we deduce that there is a natural correspondence between indecomposable Abelian Tits sets that are not Moufang sets (up to isomorphism) and simple Jordan algebras of finite capacity that are not division algebras (up to isotopy). This is joint work with Paulien Jansen.

Jacques Tits’ Work Related to the Theory of Finite (Simple) Groups

• Gernot STROTH

Room: 11 place Marcelin-Berthelot, 75005 Paris The influence of the classification of the buildings of spherical type and rank at least three on the classification of the finite simple groups and a very personal choice of some of the results of Jacques Tits on multiple transitive groups and sporadic simple groups.

The Category of Representations of p-adic Groups and Hecke Algebras

• Jessica FINTZEN

Room: 11 place Marcelin-Berthelot, 75005 Paris An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will give an overview of what we know about the structure of the Bernstein blocks. In particular, I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

Tits-Kantor-Koecher Lie Algebras and 5 x 5-Gradings

• Tom DE MEDTS

Room: 11 place Marcelin-Berthelot, 75005 Paris The classical Tits-Kantor-Koecher construction produces a Lie algebra starting from a Jordan algebra; the resulting Lie algebra is 3-graded. This can be generalized to other algebraic structures as input, giving rise to 5-graded Lie algebras. From a completely different point of view, Tits and Weiss have developed algebraic structures that parametrize spherical buildings associated with isotropic simple linear algebraic groups; the most complicated of those are the “quadrangular algebras” introduced by Weiss. We combine both ideas, and we show that Lie algebras equipped with two different 5-gradings give rise, under some natural conditions, to quadrangular algebras. This is based on joint work with Jeroen Meulewaeter.

Valeurs de caractères

• Jean-Pierre SERRE

Room: 11 place Marcelin-Berthelot, 75005 Paris

Compactifications de variétés de caractères et actions sur des immeubles affines

• Anne PARREAU

Room: 11 place Marcelin-Berthelot, 75005 Paris

Rational Points On Linear Algebraic Groups

• Zev ROSENGARTEN

Room: 11 place Marcelin-Berthelot, 75005 Paris In 1984, Oesterlé proved that a wound unipotent group of dimension strictly less than p –1 over a global function field of characteristic p has only finitely many rational points. The bound p –1 is sharp, as one can construct wound unipotent groups of dimension p –1 which are unirational and therefore have Zariski dense sets of rational points. Oesterle posed the natural question: Must a wound unipotent group over a global function field which admits infinitely many rational points admit a nontrivial unirational subgroup? One can of course formulate the question for arbitrary linear algebraic groups (though the wound unipotent case turns out to be the crucial one). In this talk, I will discuss a proof of the affirmative answer to Oesterlé's question (in this slightly greater generality).

Projection du film de Jean-François Dars et Anne Papillault « A Jacques Tits » Room: 11 place Marcelin-Berthelot, 75005 Paris

Simple Modules for Algebraic Groups

• Michael BATE

Room: 11 place Marcelin-Berthelot, 75005 Paris I will speak about joint work with David Stewart, giving a classification by highest weight of simple modules for smooth connected algebraic groups over a field, and also describe some of the properties of those modules. Much of the work here shows the fundamental influence of Tits: our work relies on work of Conrad-Gabber-Prasad classifying pseudo-reductive groups, which in turn (pseudo-)completed a program initiated by Tits in the 1990s. Further, in moving from the split to the non-split case, and in calculating the endomorphisms of the simple modules, we also mimic work of Tits from the 1970s for reductive groups.

Higher Tate Traces and Classification of Chow Motives

• Anne QUÉGUINER-MATHIEU

Room: 11 place Marcelin-Berthelot, 75005 Paris This talk is based on a joint work with Charles De Clercq. The main result is a classification theorem for some Chow motives with finite coefficients, which applies, notably, to motives of projective homogeneous varieties under some semi-simple algebraic groups. The result uses a new invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. If time permits, the notion of critical variety, related to the Tits index of the underlying algebraic group, will be presented, as an application of our result.

On the Tits Alternative and its Refinements

• Emmanuel BREUILLARD

Room: 11 place Marcelin-Berthelot, 75005 Paris I shall present Tits' original statement and method of proof of his celebrated alternative for finitely generated linear groups and present various refinements that have been obtained over the years as well as open problems. In particular, I shall discuss the notion of strongly dense free subgroup appearing in recent joint work with Guralnick and Larsen on this topic as well as connections to the study of word maps and expander graphs.

Tits Alternative for the Cremona Group

• Anne LONJOU

Room: 11 place Marcelin-Berthelot, 75005 Paris A group G satisfies the Tits alternative if any subgroup contains either a non-Abelian free subgroup or a solvable group of finite index. This alternative has been proved by Jacques Tits for the linear groups (up to restricting to finitely generated subgroups when the field is of positive characteristic). Since then, this alternative has been proved for many other groups. In particular, Serge Cantat (in the case of finitely generated subgroups) and Christian Urech proved that the Cremona group, namely the group of birational transformations of the projective plane, satisfies the Tits alternative. After an introduction to the Cremona group, the goal of this talk is to give their proof's strategy of this important result.

Simple Algebraic Groups among Locally Compact Groups

• Pierre-Emmanuel CAPRACE

Room: 11 place Marcelin-Berthelot, 75005 Paris The real simple Lie groups and the simple algebraic groups over non-Archimedean local fields are ubiquitous in mathematics. For that reason, they are probably the most remarkable members of the class of non-discrete simple locally compact groups. But what makes them so special within that class? What are their characteristic properties as locally compact groups? The goal of this talk is to present an overview of attempts at answering that question, by adopting algebraic, geometric, and representation-theoretic points of view.