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.[-]

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 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).[-]

Starting with the Gauss-Bonnet formula : rigidity phenomena on bounded symmetric domains Ngaiming MOK (The University of Hong Kong, Hong Kong) Let $E$ be a compact Riemann surface of genus 1, and $Z$ be a compact Riemann surface of genus $≥ 2$. Then, every holomorphic map $f : E → Z$ is constant, as can be proven by contradiction by pulling back a nontrivial holomorphic differential on $Z$ which necessarily vanishes at some point. A metric version of the proof using the Gauss-Bonnet formula is more flexible, and a variation of the proof based on a Chern integral gives a Hermitian metric rigidity theorem, first established by the author in 1987 in the case of compact quotients $X\left\lceil := \Omega/\right\lceil$ of irreducible bounded symmetric domains $\mathrm{X}_{Γ} := \Omega/Γ$ of rank $≥ 2$ and then extended in the finite-volume case by To in 1989, which gives rigidity results on holomorphic maps from $X\lceil$ to Kähler manifolds of nonpositive holomorphic bisectional curvature, and geometric superrigidity results in the special cases of $Γ\G/K$ for $G/K$ of Hermitian type and of rank $≥ 2$ and for cocompact lattices $Γ ⊂ G$ via the use of harmonic maps and the $∂∂$-Bochner-Kodaira formula of Siu's in 1980. The Hermitian metric rigidity theorem was the starting point of the author's investigation on rigidityphenomena mostly on bounded symmetric domains $\Omega$ irreducible of rank $≥ 2$, but also, in the presence of irreducible lattices Γ ⊂ G := Aut0(Ω), on reducible $Omega$, and, for certain problems also on the rank-1 cases of n-dimensional complex unit balls Bn. The proof of Hermitian metric rigidity serves both (I)as a prototype for metric rigidity theorems and (II) as a source for proving rigidity results or making conjectures on rigidity phenomena for holomorphic maps. For type-I results the author will explain (1) the finiteness theorem on Mordell-Weil groups of universal polarized Abelian varieties over functionfields of Shimura varieties, established by Mok (1991) and by Mok-To (1993), (2) a Finsler metric rigidity theorem of the author's (2004) for quotients $XΓ := Ω/Γ$ of bounded symmetric domains Ω of rank $\ge2$ by irreducible lattices and a recent application by He-Liu-Mok (2024) proving the triviality of the spectral base when $XΓ$ is compact, (3) a rigidity result of Clozel-Ullmo (2003) characterizing commutants of certain Hecke correspondences on irreducible bounded symmetric domains Ω of rank $\ge 2$ via a reduction to a characterization of holomorphic isometries and the proof of Hermitian metric rigidity. For type-II results the author will focus on irreducible bounded symmetric domains Ω of rank $\ge2$ and explain (4) the rigidity results of Mok-Tsai (1992) on the characterization of realizations of Ω as convex domains in Euclidean spaces, (5) its ramification to a rigidity result of Tsai's (1994) on proper holomorphic maps in the equal rank case, (6) a theorem of Mok-Wong (2023) characterizing Γ-equivariant holomorphic maps into arbitrary bounded domains inducing isomorphisms on fundamental groups, and (7) a semi-rigidity theorem of Kim-Mok-Seo (2025) on proper holomorphic maps between irreducible bounded symmetric domains of rank $\ge2$ in the non-equirank case. Through Hermitian metric rigidity the author wishes to highlight the fact that complex differential geometry links up with many research areas of mathematics, as illustrated for instance by the aforementioned results (6) of Mok-Wong in which harmonic analysis meets ergodic theory and Kähler geometry, and (7) of Kim-Mok-Seo on proper holomorphic maps in which techniques of several complex variables cross-fertilize with those in $CR$ geometry and the geometric theory of varieties of minimal rational tangents ($VMRTs$).[-]

Given a computer algebra problem described by polynomials with rational coefficients, I will present various tools that help measuring the cost of solving it, where cost means giving bounds for the degrees and heights (i.e. bit-sizes) of the output in terms of those of the input data. I will detail an arithmetic Bézout inequality and give some applications to zero-dimensional polynomial systems. I will also speak about the Nullstellensatz and Perron's theorem for implicitization, if time permits.[-]

Given a computer algebra problem described by polynomials with rational coefficients, I will present various tools that help measuring the cost of solving it, where cost means giving bounds for the degrees and heights (i.e. bit-sizes) of the output in terms of those of the input data. I will detail an arithmetic Bézout inequality and give some applications to zero-dimensional polynomial systems. I will also speak about the Nullstellensatz and Perron's theorem for implicitization, if time permits.[-]