Mauricio Garay

A rigidity theorem for Lagrangian deformations

We consider deformations of singular Lagrangian varieties in symplectic manifolds. We prove that a Lagrangian deformation of a Lagrangian complete intersection is analytically rigid provided that this is the case infinitesimally. This result is given as a consequence of the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations.

On the topology of Lagrangian Milnor fibres

,analogous to the Koszul-Chevalley complex in the the-ory of Lie algebras,was introduced. The first cohomology vector space of this complexcomputes theinfinitesimal deformationoftheLagrangian variety,likeforthecaseofLiealgebras.In [4],a relative version of this complexwas introduced for singular Lagrangianfibrations. Under some transversality conditions,it was proved in that paper that thehigher direct image sheaves are coherent. In this paper,we prove that they are actuallysheavesof

Resurgent deformation quantisation

We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. The algebra would be large enough to capture quantum effects that escape ordinary formal deformation quantisation.

Analytic geometry and semi-classical analysis

We study deformation theory for quantum integrable systems and prove several theorems concerning the Gevrey convergence and the unicity of perturbative expansions.

Finiteness and constructibility in local analytic geometry

We use Kiehl-Verdiers and Houzels finiteness theorems in the setting of local analytic geometry, and the Whitney-Thom theory of stratified spaces, to prove that fibrewise constructible complex of sheaves have coherent direct images. We review the notions of stratification theory which are needed.

ON SIMPLE FAMILIES OF FUNCTIONS AND THEIR LEGENDRIAN MAPPINGS

We study germs of

Classical and quantum integrability

n

An isochore versal deformation theorem

-parameter families of functions, that is, function-germs of the type

Stable moment mappings and singular lagrangian Fibrations

f : (\mathbb{R}^n \times \mathbb{R}, 0) \to (\mathbb{R}, 0)

PERTURBATIVE EXPANSIONS IN QUANTUM MECHANICS

defined on the total space of the trivial bundle