Séminaire de Systèmes Dynamiques
# Journée d'équipe DGC

→
Europe/Paris

207 (Bat 1R2)
### 207

#### Bat 1R2

Description

**10h coffee **

**10h20-10h50**:**Athmane Hachim**,*Dynamics of Kleinian groups and Halphen systems*

Halphen vector fields and Halphen equations in

**C**^{3 }appear in numerous problems in mathematics and mathematical physics, one of the most know Halphen system, the so-called Darboux-Halphen, was discovered by Darboux in his study of orthogonal systems (1878) and was solved by Halphen(1881). Recently, it has been shown that Halphen systems are in one-to-one correspondence with Kleinian groups and their dynamics on compact complex surfaces. We study these new dynamical systems to describe the dynamics of Halphen equations.**11h-11h30**:**Jinxing Xu,***Monodromy representations from hyperplane arrangements*

Given a hyperplane arrangement in general position, we can construct the double cover of the projective space ramified along this hyperplane arrangement. As the hyperplane arrangement varies, we get a family of projective varieties which generalizes the family of hyperelliptic curves. I will discuss the monodromy representations of this family, from both the complex and mod l side. This is based on a joint work with Xiaopeng Xia.

**11h40-12h10**:**Abdelouahab Khelifati**,*Notions of hyperbolicity in complex geometry*

Hyperbolicity is an important concept in the theory of complex manoifolds which allows us to characterize their geometric and topological properties. During this talk, we will present classical notions of hyperbolicity such as Kobayashi/Brody-hyperbolicity and Kähler hyperbolicity (Gromov) as well as some recently developed notions of hyperbolicity, such as balanced hyperbolicity and divisorial hyperbolicity. We will give the intuition from a differential and geometric point of view behind each of these notions and we will mention a deformation result for one of these notions of hyperbolicity which we wish to obtain for the others.

**12h10 lunch at esplanade**

**14h-14h30**:**Shengxuan Zhou,***On the Gromov-Hausdorff limits of tori with Ricci conditions*

In [BNS23], Bruè-Naber-Semola pose a question: Is the Gromov-Hausdorff limit of a sequence of Riemannian tori with bounded diameter and Ricci curvature lower bound a topological torus? In the 3-dimension case, they give an affirmative answer. In this talk, we will explain why the answer to this question is negative in higher dimensions [Z23], and consider the problem under additional conditions, such as bounded two-sided Ricci curvature or under Kähler conditions.

Reference:

[BNS23] Bruè-Naber-Semola, Stability of Tori under Lower Sectional Curvature, preprint, accepted for publication by Geom. Topol. (2023).

[Z23] S. Zhou, On the Gromov-Hausdorff limits of Tori with Ricci conditions, preprint, arXiv: 2309.10997.**14h40-15h10**:**Pouriya Torkinejad-Ziarati,**

*Cyclicity in Poletsky-Stessin Weighted Bergman Spaces*

Here we use Poletsky-Stessin definition of Weighted Bergman Spaces on Hyperconvex domains and extend the definition to more general indices to possibly resemble Dirichlet-type spaces there. Our aim is to study the cyclicity phenomenon with respect to shift operator on these spaces and we will give a sufficient condition for non-cyclicity there.

**15h20-15h50**:**Caio Silva de Souza,***Valuation Theory: History and Key Concepts*

In this talk, we will briefly discuss the historical development of Valuation Theory and its main motivations. We will also introduce the concept of key polynomials and their relation to the defect of a valued field extension.

**16h-16h30**:**Chia-Yu Chang***Subrank and border subrank of tensors*

I will talk about the motivation of studying subrank and border subrank of a tensor, and then I will list some results that we know about those measures.

16h30 : Refreshments

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Organized by

Pascale Roesch