**11:00**Stéphane Nonnenmacher (Orsay)

**Quantum maps on Kaehler manifolds: toy models for quantum chaos.**

*Abstract*: A positive holomorphic line bundle over a compact complex manifold M equips the latter with a symplectic form: the resulting Kähler manifold becomes a compact "phase space" hosting classical mechanics. The holomorphic sections of the N-th power of this line bundle build up a "quantum" Hilbert space associated with this phase space, such that classical observables are mapped to quantum observables through the Toeplitz quantization. A symplectic diffeomorphism K: M -> M is said to be quantizable when one can associate to it a sequence of unitary operators (U_N(K)) satisfying a form of quantum-classical correspondence in the limit N>>1 (semiclassical limit). Such a sequence is called a "quantum map" in the physics literature. Such quantum maps have been a fruitful playing ground in the field of "quantum chaos", namely in situations where K displays chaotic dynamical properties. I will describe some results obtained for such "quantum chaotic maps", essentially in the case where M is the 2-dimensional torus. .

**14:15**Dietmar Salamon (ETH Zurich)

**Complex structures, moment maps, and the Ricci form.**

*Abstract:*In this talk I will explain how the the Ricci form appears as a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil—Petersson symplectic form on the Teichmueller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kaehler cone. In the Fano case this is closely related to Donaldson's work on Berndtsson convexity and the Ding functional. This talk is based on joint work with Oscar Garcia-Prada and Samuel Trautwein.

**16:00**Vera Vertesi (Strasbourg).

**Additivity of the minimal genus for tight contact structures.**

*Abstract:*A corollary of Haken’s Lemma from 1968 gives that the minimal genus of Heegaard splittings of 3-manifolds is additive under connected sum. In this talk I generalise this result for contact 3-manifolds. As a consequence of Giroux's 2002 paper about open book decompositions, (closed, connected) contact 3-manifolds have contact Heegaard splittings i.e. contact 3-manifolds decompose as the union of two contact handlebodies, these are neighbourhoods of Legendrian graphs. There is a simple homological counterexample of Ozbagci, showing that the minimal genus for these contact Heegaard splittings is not additive in general. After introducing the basic notions, I describe the main idea of the proof, that the minimal genus for contact Heegaard splittings of tight contact structures is additive under connected sum.

Prochaines séances: à l'automne !

**Autre activité symplectique à Paris:**

Séminaire Nantes-Orsay