To every right-angled Coxeter G group belongs a unique countable Tits building B(G) with infinite residues. Using a suitable language, we study the first order theory of B(G). It has a nice axiomatization, is omega-stable, equational and has trivial forking. It is not n-ample, when n is the number of generators of G. (Joint work with A. Baudisch and A. Martín Pizarro)
We give a new proof of the André-Oort conjecture under the generalised Riemann hypothesis. In fact, we generalise the strategy pioneered by Edixhoven, and implemented by Klingler and Yafaev, to all special subvarieties. Thus, we remove ergodic theory from the proof of Klingler, Ullmo and Yafaev and replace it with tools from algebraic geometry. Our key ingredient is a lower bound for the degrees of strongly special subvarieties coming from Prasad's volume formula for S-arithmetic quotients of semisimple groups.
In a series of papers by Alon, Conlon, Fox, Gromov, Naor, Pach, Pinchasi, Radoi, Sharir, Sudakov, Lafforgue, Suk and others it is demonstrated that families of graphs with the edge relation given by a semialgebraic relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces modulo a small mistake (for example the incidence relation between points and lines on the real plane, or higher dimensional analogues). We show that in fact the whole theory can be developed for families […]
Une théorie est NIP si et seulement si certains espaces de types sont des `compacts de Rosenthal' : des objets étudiés en topologie et théorie descriptive des ensembles. Grâce à cette observation, on peut appliquer des résultats de topologie générale pour obtenir de nouveaux (et d'anciens) théorèmes sur les théories NIP. Je parlerai en particulier de conséquences concernant les types invariants.
Motivés par des questions topologiques, nous présentons plusieurs problèmes sur les intersections singulières de sous-groupes algébriques et de sous-variétés dans les tores multiplicatifs. Ces questions sont étroitement liées aux conjectures de Zilber-Pink. L'heuristique sous-jacente est que, dans ces conjectures, la singularité des intersections peut compenser une décrémentation de la codimension des sous-groupes considérés.Il s'agit d'un travail en commun avec J. Marché.
(Work in progress, together with Yimu Yin)One tool to describe singularities of (e.g. algebraic or analytic) subsets X of R^n or C^n are stratifications: a partition of X into finitely many ?Roestrata?R such that any two points x, y in X within the same stratum have the ?Roesame type of neighbourhood?R. The most classical stratifications are Whitney stratifications, which classify neighbourhoods up to homeomorphism. The strongest known stratifications are Mostowski's bi-Lipschitz stratifications, which classify neighbourhoods up to a bi-Lipschitz map. I will present a new way of obtaining such bi-Lipschitz […]
