Periodic Reporting for period 1 - GOAT (Groups Of Algebraic Transformations)
Okres sprawozdawczy: 2023-01-01 do 2025-06-30
Geometric shapes can often be described by equations, and the simplest equations are polynomial equations. The study of geometric shapes described by such polynomial equations is the domain of algebraic geometry.
Given a geometric object, we need to study the group of all transformations of this object that preserves its fundamental geometric characteristics; in Euclidean geometry, this is known as the group of isometries.
These groups contain valuable information about the geometric object, the comparison between such objects, or the comparison between different geometries on the same underlying topological shape.
In algebraic geometry, two groups naturally emerge: the group of automorphisms and the group of birational transformations.
Dynamical systems are a branch of mathematics whose initial aim was to describe the long-term evolution of certain mechanical systems, such as the movement of planets.
When we retain only the first significant terms in the equations describing some of these systems, we can obtain algebraic transformations of algebraic varieties;
when the system is reversible, the transformations obtained are birational transformations or automorphisms. The theory of dynamic systems then joins algebraic geometry.
This project lies at the intersection of the two themes just mentioned: dynamical systems and algebraic geometry.
Its main aims are to study groups of algebraic transformations and analyze their dynamical properties.
On the one hand, the aim is to study the algebraic properties of groups of birational transformations, comparing them with linear groups.
On the other, we want to describe the dynamics of certain birational actions that arise when we seek to understand the moduli space of certain projective surfaces
or the variety of all representations of a finite-type group in low dimension.
Given a geometric object, we need to study the group of all transformations of this object that preserves its fundamental geometric characteristics; in Euclidean geometry, this is known as the group of isometries.
These groups contain valuable information about the geometric object, the comparison between such objects, or the comparison between different geometries on the same underlying topological shape.
In algebraic geometry, two groups naturally emerge: the group of automorphisms and the group of birational transformations.
Dynamical systems are a branch of mathematics whose initial aim was to describe the long-term evolution of certain mechanical systems, such as the movement of planets.
When we retain only the first significant terms in the equations describing some of these systems, we can obtain algebraic transformations of algebraic varieties;
when the system is reversible, the transformations obtained are birational transformations or automorphisms. The theory of dynamic systems then joins algebraic geometry.
This project lies at the intersection of the two themes just mentioned: dynamical systems and algebraic geometry.
Its main aims are to study groups of algebraic transformations and analyze their dynamical properties.
On the one hand, the aim is to study the algebraic properties of groups of birational transformations, comparing them with linear groups.
On the other, we want to describe the dynamics of certain birational actions that arise when we seek to understand the moduli space of certain projective surfaces
or the variety of all representations of a finite-type group in low dimension.
So far, the project has brought together experts in the field on a regular basis, and welcomed young researchers on thesis or post-doctorate contracts.
This has made it possible to treat the first significant cases concerning “dynamics on varieties of representations”, and to extend a theorem specific to linear groups to all groups of birational transformations of projective surfaces.
Surprisingly, only a small part of this theorem ceases to be valid when transposed to the birational case.
In parallel, new results have been obtained concerning the dynamics of automorphismes of surfaces over ultra metric fields, a subject which is somewhat in its infancy.
This has made it possible to treat the first significant cases concerning “dynamics on varieties of representations”, and to extend a theorem specific to linear groups to all groups of birational transformations of projective surfaces.
Surprisingly, only a small part of this theorem ceases to be valid when transposed to the birational case.
In parallel, new results have been obtained concerning the dynamics of automorphismes of surfaces over ultra metric fields, a subject which is somewhat in its infancy.
Even though interesting achievements have been obtained, the main goals have not been reached yet, and the results obtained so far only give us a better idea of the direction in which to pursue our efforts.