Séminaires du LAREMA

Séminaire des doctorant.es

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 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.

Séminaires systèmes dynamiques et géométrie

Séminaire de topologie et géométrie algébriques
Le résumé : Une des questions importantes concernant une surface lisse complexe de P^4 est le calcul de son irrégularité. Dans cet exposé nous parlerons de ce problème en supposant que la surface est contenue dans une hypersurface de degré plus petit ou égal à 4. On montre que les fibrés elliptiques en droites et, respectivement, en coniques sont les seules surfaces irrégulières contenues dans une hypersurface cubique et, respectivement, quartique (ayant seulement des points doubles ordinaires). L'outil technique principal de ce calcul sera le complexe de Koszul associé à la section globale du fibré conormal (tordu) de la surface, section induite par l’hypersurface.

Séminaire des doctorant.es
In this seminar, we introduce a new approach to associating a semigroup with a polynomial in k[x1,..., xe][y], which is a generalization of associating a semigroup with a polynomial in k[[x1,...,xe]][y] discuses in [1], where k is an algebraically closed field of characteristic zero. This construction is motivated by recent results and forms the central contribution of our (unpublished) paper. Our main theorem establishes that every prepared polynomial is birationally equivalent to a quasi-ordinary polynomial. To ensure the accessibility of the main result, we begin by reviewing foundational definitions in algebraic geometry relevant to our work. Particular attention is given to the concept of polynomials with one place at infinity. We provide a precise definition and examples to clarify this notion. We then define the semigroup associated with a polynomial whose coefficients are in the ring of power series. This semigroup will be shown to relate closely to the semigroup associated with the meromorphic series expansion of the main polynomial f. Through this connection, we explicitly compute the semigroup associated with f. And the goal of associating a semigroup to polynomial is to classify curves with only one place at infinity using their associated invariants, particularly the Milnor number and the Turina number. Finally One goal of associating a semigroup to a hypersurface in k n, is to use its arithmetic to study non-elementary automorphisms such as the morphism proposed by Nagata in the 1970s. This approach will rely on the theory of quasi-ordinary polynomials.

Séminaires systèmes dynamiques et géométrie
We give an overview of para-complex geometry and study the Boothby-Wang fibration over para-Hermitian symmetric spaces. We remark that in contrast to the Hermitian setting the center of the isotropy group of a simple para-Hermitian symmetric space G/H can be either one- or two-dimensional, and prove that the associated metric is not necessarily the G-invariant extension of the Killing form of G. Using the Boothby-Wang fibration and the classification of semisimple para-Hermitian symmetric spaces, we explicitly construct semisimple para-Sasakian $\phi$-symmetric spaces fibering over semisimple para-Hermitian symmetric spaces.