Emmanuel Breuillard

The strong uniform Tits alternative : The goal of this mini-course will be to discuss the strong uniform Tits alternative for subgroups of GL_d(K) over an arbitrary field. More specifically I will describe the links between the uniformity in this theorem and the Lehmer problem for the variety fo representations of a semisimple algebraic group. In this problem equidistribution properties of points of small Weil height play a key role. Finally I will attempt to describe some applications, mostly those dealing with the spectrum of the Laplace operator on finite groups in relation with the recent works of Helfgott and Bourgain-Gamburd.