We shall briefly present in very elementary terms the `games' of Steiner and Poncelet, amusing mathematical solitaires of the XIX Century, also related to elliptic billiards. We shall recall that the finiteness of the game is related to torsion in tori or elliptic curves.

# Minerva Lectures

For lecture information, contact Fernando CodÃ Marques.

##### Lecture 1: The games of Steiner and Poncelet and algebraic group schemes

Scuola Normale Superiore-Pisa

##### Lecture 2: Torsion values for sections in abelian schemes and the Betti map

We shall consider variations in the games, related to the so-called `Betti-map', which we shall describe. We shall also illustrate some links of the Betti map with several other contexts and state some theorems on torsion values, both of existence type and finiteness type (obtained mainly in joint work with David Masser).

Scuola Normale Superiore-Pisa

##### Lecture 3: Ambients for the Betti map and the question of its rank

In this last lecture we shall consider in more detail some of the mentioned contexts involving the Betti map. We shall also discuss in short some recent work with Yves AndrÃ© and Pietro Corvaja, extending what comes from Manin's theorem of the kernel.

Scuola Normale Superiore-Pisa