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We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the cotangent microbundle category, and they admit a total symbol calculus in terms of symplectic micromorphisms enhanced with half-density germs. This new operator category encompasses the semi-classical pseudo-differential calculus and offers a functorial framework for the semi-classical analysis of the Schrodinger equation. We also comment on applications to classical and quantum mechanics as well as to a functorial and geometrical approach to the quantization of Poisson manifolds.
We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the us
Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $gamma$ topology defined by the author, to $bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is
We say that a subset of a symplectic manifold is symplectically (neighbourhood) excisable if its complement is symplectomorphic to the ambient manifold, (through a symplectomorphism that can be chosen to be the identity outside an arbitrarily small n
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surf
We construct a new family of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. We prove various structural properties of the capacities and discuss the connections with the equivariant L-i