Quantum K-theory is as quantum cohomology a generalisation of the classical coho- mology algebra of a variety X . In this talk I will explain the connection between the geometry of the moduli space of stable maps, in particular rational connectedness properties, and the computation of structure constants for X a rational homogeneous space. This is based on a joint work with A. Buch, P.-E. Chaput and L. Mihalcea
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

The topic concerns relations among the kappa classes in the tautological ring of the moduli space of genus g curves. After a discussion of classical constructions in Wick form, we derive an explicit set of relations obtained from the virtual geometry of the moduli space of stable quotients. In a series of steps, the stable quotient relations are transformed to simpler and simpler forms. Our nal result establishes a previously conjectural set of tautological relations proposed a decade ago by Faber{Zagier.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

In this talk I will explain the idea of the Laplace transform that connects a counting problem in the A-model side with a recursion formula based on complex analysis in the B-model side, using a concrete example. The talk is based on my joint work with Chapman, Penkava and Safnuk.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric analysis, metric geometry, topology

We will give an introduction to Donaldson-Thomas theory and some basic tools and computations. In the last lecture, we hope to explain some aspects of the proof of the GW/DT correspondence for toric threefolds.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

The Christodoulou Klainerman proof of existence of asymptotically simple space-­times shows that it is reasonable to consider the scattering of characteristic data for the Einstein field equations from past null infinity to that on future null infinity in a neighbourhood of Minkowski space. In this talk I present new explicit perturbative formulae for this scattering for general data to arbitrary order. Unlike previous such formulae, these new formulae are not chiral, and naturally respect the real structures and may therefore be more amenable to analysis. This is based on joint work with Yvonne Geyer and Arthur Lipstein and with David Skinner.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

Out of the quantum product of a projective smooth variety, we can construct a vector bundle with a at connection and a pairing, these data are called quantum D-modules. In a recent paper of Iritani, he gives an explicit presentation of this D module for toric varieties. In this talk, we will consider a hypersurface in a toric variety and we give a link between the quantum D module of the ambient variety and the one of the hypersurface. Moreover, in the toric case, we will give a presentation of these D-modules. This is a common work with Thierry Mignon (Montpellier).
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

The Satake isomorphism identi es the irreducible representations of a semisimple algebraic group with the intersection cohomologies of the Schubert varieties in the ane Grassmannian of the Langlands dual group. In the very special case where the Schubert varieties are smooth, one gets an identi cation between the so-called minuscule representations and the cohomology of the so-called minuscule homogeneous spaces. I will explain how this extends to quantum cohomology.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory