Algèbre et géométrie

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 […]