A Homological Approach to Belief Propagation and Bethe Approximations

We introduce a differential complex of local observables given a decomposition of a global set of random variables into subsets. Its boundary operator allows us to define a transport equation equivalent to Belief Propagation. This definition reveals a set of conserved quantities under Belief Propagation and gives new insight on the relationship of its equilibria with the critical points of Bethe free energy.Comment: 14 pages, submitted for the 2019 Geometric Science of Information colloquiu

Chirality in photonic systems

The optical modes of photonic structures are the so-called TE and TM modes which bring intrinsic spin-orbit coupling and chirality to these systems. This, combined with the unique flexibility of design of the photonic potential, and the possibility to mix photon states with excitonic resonances, sensitive to magnetic field and interactions, allows to achieve many phenomena, often analogous to other solid state systems. In this contribution, we review in a qualitative and comprehensive way several of these realizations, namely the optical spin Hall effect, the creation of spin currents protected by a non-trivial geometry, Berry curvature for photons, and the photonic/polaritonic topological insulator