In my talk, I will discuss an application of the theory of motives to transcendence theory, concentrating on the formal aspects. The Period Conjecture predicts that all relations between period numbers are induced by properties of the category of motives. It is a theorem for motives of points and curves, but wide open in general. The Period Conjecture also implies fullness of the Hodge-de Rham realization on Nori motives. If time permits, I will discuss how this generalizes (conjecturally) to triangulated motives and thus to motivic cohomology.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

Kummerfourfolds, a generalization of K3 surfaces (and, in some sense, of abelian surfaces), belong to the class of Hyperk¨ahler manifolds, which exhibit rich but intricate geometry. In this talk, we explore the projective duality of certain special Kummer fourfolds and explain how O'Grady's theory of theta groups can be used to derive their equations. This work, carried out in collaboration with Agostini, Beri, and Rios-Ortiz, contributes to a broader framework of classical results involving moduli spaces of sheaves on curves and embeddings of abelian surfaces.[-]

The preceding Etats de la recherche on this topic happened in Rennes, 2006, and the organizers of the present edition asked me to make the bridge between these two sessions. My 2006 talks were devoted to the theory of heights and equidistribution theorems for algebraic dynamical systems. I will start from there by presenting the framework allowed by Arakelov geometry, and explaining the recent manuscript of X. Yuan and S.-W. Zhang who provide a birational perspective to these concepts. The theory is a bit complex and technical but I will try to emphasize the parallel between those ideas and the ones that lie at the ground of pluripotential theory in complex analysis, or in the theory of b-divisors in algebraic geometry.[-]

In these lectures, we will examine a series of conjectures about the geometry of preperiodic points for endomorphisms of $ \mathbb{P}^{N}$. Lecture 1 will focus on the Dynamical ManinMumford Conjecture (DMM), formulated by Shouwu Zhang in the 1990s as an extension of the well-known Manin-Mumford Conjecture (which investigated the geometry of torsion points in abelian varieties and was proved in the early 1980s by Raynaud). The DMM aims to classify the subvarieties of $\mathbb{P}^{N}$ containing a Zariski-dense set of preperiodic points. Lectures 2 and 3 will be devoted to conjectures that treat families of maps on $\mathbb{P}^{N}$. One conjecture in particular was inspired by the recently-proved ”Relative Manin-Mumford” theorem of Gao-Habegger for abelian varieties, but the dynamical version turns out to be closely related to the study of dynamical stability and to contain many previously-existing questions/conjectures/results about moduli spaces of maps on $\mathbb{P}^{N}$. These lectures are based on joint work with Myrto Mavraki.[-]

The group Aut($\mathbb{A}^{n}$) of polynomial automorphisms of the a ne space is an interesting huge group, and a slightly simpler group is its subgroup Aut($\mathbb{A}^{n}$) of tame automorphisms. Natural questions about these groups include:– does they admit normal subgroups beside the obvious subgroup of automorphisms with Jacobian 1?– do they satisfy a Tits alternative?– what are the possible dynamical degrees of their elements? The method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we plan to introduce in detail. The lectures will focus on the following three cases: the group Aut($\mathbb{A}^{2}$) = Tame($\mathbb{A}^{2}$) following Chapter 7 of my book in preparation, then the group Tame($\mathbb{A}^{3}$) (Lamy, LamyPrzytycki, Blanc-Van Santen), and finally the group Tame(Q($\mathbb{A}^{4}$)) of tame automorphisms of $\mathbb{A}^{4}$ preserving a nondegenerate quadratic form (Bisi-Furter-Lamy, Martin, Dang).[-]

The group of tame automorphisms. The group Aut($\mathbb{A}^{n}$) of polynomial automorphisms of the a ne space is an interesting huge group, and a slightly simpler group is its subgroup Aut($\mathbb{A}^{n}$) of tame automorphisms. Natural questions about these groups include:– does they admit normal subgroups beside the obvious subgroup of automorphisms with Jacobian 1?– do they satisfy a Tits alternative?– what are the possible dynamical degrees of their elements? The method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we plan to introduce in detail. The lectures will focus on the following three cases: the group Aut($\mathbb{A}^{2}$) = Tame($\mathbb{A}^{2}$) following Chapter 7 of my book in preparation, then the group Tame($\mathbb{A}^{3}$) (Lamy, LamyPrzytycki, Blanc-Van Santen), and finally the group Tame(Q($\mathbb{A}^{4}$)) of tame automorphisms of $\mathbb{A}^{4}$ preserving a nondegenerate quadratic form (Bisi-Furter-Lamy, Martin, Dang).[-]

The KSBA moduli space of stable pairs ($\mathrm{X}, \mathrm{B}$), introduced by Kollár-Shepherd-Barron, and Alexeev, is a natural generalization of the moduli space of stable curves for higher dimensional varieties. This moduli space is described concretely only in a handful of situations. For instance, if $\mathrm{X}$ is a toric variety and $\mathrm{B}=\mathrm{D}+\varepsilon\mathrm{C}_{}^{}$, where D is the toric boundary divisor and $\mathrm{C}$ is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. More generally,for stable pairs of the form$\left( \mathrm{{X,D}+\varepsilon\mathrm{C}} \right)$ with $\left( \mathrm{X,D} \right)$ a log Calabi–Yau variety and C an ample divisor, it was conjectured by Hacking–Keel–Yu that the KSBA moduli space is still toric, up to passing to a finite cover. In joint work with Alexeev and Bousseau, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log geometry and mirror symmetry.[-]