L3

 We consider the emergence of classical correlations in macroscopic quantum systems, and its connection to monogamy relations for violation of Bell-type inequalities. We work within the framework of Abramsky and Brandenburger [1], which provides a unified treatment of non-locality and contextuality in the general setting of no-signalling empirical models. General measurement scenarios are represented by simplicial complexes that capture the notion of compatibility of measurements. Monogamy and locality/noncontextuality of macroscopic correlations are revealed by our analysis as two sides of the same coin: macroscopic correlations are obtained by averaging along a symmetry (group action) on the simplicial complex of measurements, while monogamy relations are exactly the inequalities that are invariant with respect to that symmetry. Our results exhibit a structural reason for monogamy relations and consequently for the classicality of macroscopic correlations in the case of multipartite scenarios, shedding light on and generalising the results in [2,3].More specifically, we show that, however entangled the microscopic state of the system, and provided the number of particles in each site is large enough (with respect to the number of allowed measurements), only classical (local realistic) correlations will be observed macroscopically. The result depends only on the compatibility structure of the measurements (the simplicial complex), hence it applies generally to any no-signalling empirical model. The macroscopic correlations can be defined on the quotient of the simplicial complex by the symmetry that lumps together like microscopic measurements into macroscopic measurements. Given enough microscopic particles, the resulting complex satisfies a structural condition due to Vorob'ev [4] that is necessary and sufficient for any probabilistic model to be classical.  The generality of our scheme suggests a number of promising directions. In particular, they can be applied in more general scenarios to yield monogamy relations for contextuality inequalities and to study macroscopic non-contextuality.

[1] Samson Abramsky and Adam Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, New Journal of Physics 13 (2011), no. 113036.
[2] MarcinPawłowski and Caslav Brukner, Monogamy of Bell’s inequality violations in nonsignaling theories, Phys. Rev. Lett. 102 (2009), no. 3, 030403.
[3] R. Ramanathan, T. Paterek, A. Kay, P. Kurzynski, and D. Kaszlikowski, Local realism of macroscopic correlations, Phys. Rev. Lett. 107 (2011), no. 6, 060405.
[4] N.N.Vorob’ev, Consistent families of measures and their extensions, Theory of Probability and its Applications VII (1962), no. 2, 147–163, (translated by N. Greenleaf, Russian original published in Teoriya Veroyatnostei i ee Primeneniya).

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a

cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

In 2005, Bonk and Kleiner showed that a hyperbolic group admits a

quasi-isometrically embedded copy of the hyperbolic plane if and only if the

group is not virtually free. This answered a question of Papasoglu. I will

discuss a generalisation of their result to certain relatively hyperbolic

groups (joint work with Alessandro Sisto). Key tools involved are new

existence results for quasi-circles, and a better understanding of the

geometry of boundaries of relatively hyperbolic groups.

I will show how to associate a quaternion algebra to a hyperbolic 3-manifold. I will then go through some examples and applications of this theory