In this talk, I will present recent joint work with S.~Matsumura, J.~Wang, and X.~Wu on the structure of compact Kähler manifolds whose anti-canonical bundle is nef. We establish a general structure theorem in the Kähler setting, showing that X admits a locally trivial fibration whose fibers are rationally connected and whose base has vanishing first Chern class. Our approach extends the method of Cao–Höring from the projective to the Kähler case, requiring new tools such as a flatness criterion for pseudo-effective sheaves and a refined analysis of direct image sheaves equipped with singular Hermitian metrics. I will also discuss the application, about the generalization of the Beauville–Bogomolov decomposition.

An important family of solutions to the Einstein vacuum equations is given by the Kerr metrics, which describe rotating black holes. In this talk, I will present some important geometric properties of these spacetimes relevant to the study of classical field equations such as the scalar waves, electromagnetism and linearized gravity. As observed by Teukolsky, by exploiting a special algebraic property of the spacetime, it is possible to decouple certain components of the fields from the rest of the system, leading to the so-called Teukolsky equation. Solutions of this equation can then be analyzed to recover information about the full system.

Mori dream spaces are a special kind of varieties introduced by Hu and Keel in 2000 that enjoy very good properties with respect to the minimal model program. In this talk we explore when blowups of P^3 along smooth curves are Mori dream spaces, generalizing an early example of A. Küronya.  This is joint work with Sokratis Zikas.

Geometric methods in computational complexity

Le problème de Kodaira demande si toute variété compacte kählérienne admet une déformation (arbitrairement petite) vers une variété projective. Nous présenterons des résultats positifs de ce problème pour certaines variétés de dimension 3 fibrées par des surfaces c_1-triviales. A un biméromorphisme près, ces variétés recouvrent les solides kählériens de dimension de Kodaira 1.

Dans cet exposé je présenterai une démarche poursuivie récemment par Gigli et Mantegazza pour décrire le flot de Ricci uniquement à  partir de l’aspect “espace métrique” des variétés Riemanniennes mises en jeu. L’objectif en est d’obtenir une reformulation permettant au flot de Ricci de s’appliquer à  des espaces métriques. Les outils en sont la diffusion de la chaleur et le transport optimal. Je présenterai le résultat d’investigations menées en commun avec Matthias Erbar (Univ. Bonn) concernant quelques espaces métriques emblématiques.

The quantum cohomology ring of a flag variety encodes the Gromov-Witten invariants that count curves meeting Schubert varieties in general position. When infinitely many such curves exist, the arithmetic genus of the corresponding family of curves is called a K-theoretic Gromov-Witten invariant. The quantum K-theory ring of
Givental is a generalization of the quantum cohomology ring that encodes the K-theoretic Gromov-Witten invariants. While little is known about the quantum K-theory of general flag varieties, the (small) quantum K-theory of cominuscule flag varieties has been studied in a series of papers with Chaput, Mihalcea, and Perrin. I will speak about geometric and combinatorial aspects of this work.

Le problème central de l’exposé sera d’étudier une propriété d’hyperbolicité dans une classe particulière de groupes grâce à  une action sur un complexe cubique CAT(0). Plus précisément, nous nous intéresserons à  la question suivante : quand un groupe de diagrammes est-il acylindriquement hyperbolique ? La majeure partie de l’exposé consistera à  introduire les différentes définitions et motivations relatives à  ce problème. Le temps restant sera consacré à  l’énoncé des résultats principaux et aux techniques permettant de les démontrer.

The “classical equals quantum” theorem states that any equivariant Gromov-Witten invariant (3 point, genus zero) of a Grassmann variety can be expressed as a triple intersection of Schubert classes on a two-step partial flag variety. An equivariant triple intersection on a two-step flag variety can in turn be expressed as a sum over puzzles that generalizes both Knutson and Tao’s puzzle rule for Grassmannians and the cohomological puzzle rule for two-step flag varieties. These results together give a Littlewood-Richardson rule for the equivariant quantum cohomology of Grassmannians. I will speak about geometric and combinatorial aspects of this story, which is based on papers with Kresch, Purbhoo, Mihalcea, and Tamvakis.