Master DS – Page 2

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Forum Etudiants-Entreprises – 22 novembre 2018

Les étudiant(e)s du master DS seront au forum étudiants-entreprises le 22 novembre, à l’ISTIA. Plus de 60 entreprises présentes.

Rentrée M1 DS

La rentrée du M1 DS aura lieu le lundi 3 septembre à 9h. La semaine du 3/09 au 7/09 sera consacrée à différentes remises à niveau en informatique.

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Séminaire des doctorant.es
The complex Hessian equation generalizes the complex Monge--Ampère equation, a central tool in complex geometry. In open subsets of $\mathbb{C}^n$, its "admissible" solutions are given by $m$-subharmonic functions, while on compact Hermitian manifolds, the corresponding notion becomes $(\omega, m)$-subharmonic functions, where $\omega$ is a Hermitian metric. I’ll begin with a brief review of elementary symmetric polynomials, then introduce $m$-subharmonic functions and the complex Hessian equation on $\mathbb{C}^n$, which is associated with these polynomials. We'll mainly focus on the local setting--- that is, within open subsets of $\mathbb{C}^n$. If time permits, I’ll also discuss the case of compact complex manifolds and some geometric applications.

Séminaires systèmes dynamiques et géométrie
For a given smooth manifold $M$ the Teichmüller space $\mathcal{T}(M)$ is a topological space parametrising all complex structures on $M$, up to diffeomorphisms smoothly isotopic to the identity. The Kuranishi space of a complex manifold is an analytic space encoding its small deformations. Catanese posed the question when the Teichmüller space, or at least some of its connected components, can be endowed with a natural complex structure coming from the Kuranishi spaces of the complex manifolds parametrised by $\mathcal{T}(M)$. Unfortunately, there are very few known examples where this occurs. In this talk, I will discuss this question for nilmanifolds, that is, $M= \Gamma \backslash G$ is the compact quotient of a simply connected nilpotent Lie group $G$ by a discrete subgroup $\Gamma$. Such manifolds admit a Kähler structure only if $M$ is a torus, which is one of the few examples where $\mathcal{T}(M)$ is known to admit connected components carrying a natural complex structure. Nevertheless, we will see that similar results still hold for several classes of nilmanifolds.

Séminaire de topologie et géométrie algébriques
Le Combinatorial Nullstellensatz est un théorème de Noga Alon généralisant aux polynômes à plusieurs variables l'idée qu'un polynôme de degré $d$ ne peut avoir $d+1$ racines. S'il n'a été isolé et publié qu'en 1999, certaines de ses applications l'avaient précédé. C'est la multitude de ses applications combinatoires qui ont mis en valeur ce résultat algébrique subtil mais élémentaire. Dans un premier temps, je préciserai plusieurs énoncés équivalents du Combinatorial Nullstellensatz, qui justifieront son appelation algébrique et donnerai une ébauche de preuve. Ensuite, je développerai autant que possible le vaste éventail combinatoire de ses applications, en géométrie discrète, en théorie des graphes et plus particulièrement en combinatoire additive.

Séminaire des doctorant.es
A germ of a complex analytic set X at the origin of C^n is, roughly speaking, the zero locus of a finite collection of convergent power series in n complex variables f_1,...,f_k, defined in a neighborhood of the origin in C^n. When the Jacobian matrix at the origin of the map x --> (f_1(x),...,f_k(x)) has maximal rank, the implicit function theorem applies. In this case, X is locally biholomorphic (i.e., complex diffeomorphic) to C^{n-k}. However, if the Jacobian does not have maximal rank at the origin, we say that the origin is a singular point of X. This leads to a natural, though vague, question: What does a germ of a complex analytic set look like near a singular point? Topologically, this question has been answered: we can describe the local homeomorphism type (also called the topological type) of a complex analytic germ using what is known as the conical structure theorem. However, the classification up to biholomorphism—that is, the analytic type—remains completely out of reach, even in the case of complex curves. In this talk, I will introduce the notion of the Lipschitz type of a complex analytic set, which lies between the analytic and topological types. I will give an overview of this area of geometry, present some recent results, and—if time permits—discuss some ideas behind the proofs

Séminaire des doctorant.es