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We want to explore, by numerical lattice simulations, the non-perturbative physics of quantum field theories formulated in 5 and 6 dimensions.The possibility that the world has more dimensions than (3+1)d space and time was considered by Kaluza and Klein almost a century ago.Nowadays, extra dimensions are being studied by a large fraction ofthe theoretical physics community: string theory and the AdS-CFT correspondance are prominent examples. Several BSM (Beyond the StandardModel) scenarios, currently tested against experimental data at CERN'sLHC, also assume extra dimensions. Yet in these many applications, theeffects of extra dimensions are treated perturbatively, at the classicalor at most the 1-loop level.We are interested in performing first non-perturbative studies of someparticularly interesting proposals involving 5 and 6 space-time dimensions.In 5 dimensions, with the extra dimension curled up in a tiny circle,the non-contractible closed loop of gauge field along the extra dimensioncan play the role of a Higgs field in the effective 4-dimensional theory.This powerful and popular idea of ``Gauge-Higgs Unification'' needs to be tested non-perturbatively. In particular, the dynamical breaking of the gauge symmetry from SU(3) to SU(2)xU(1) is only supported by perturbative arguments, while the 4d physics is non-perturbative. Similarly, the finite-temperature properties of the model will differ from the usual ones because of the extra dimension.In 6 dimensions one can define an additional chirality, because of the two extra dimensions. It has been claimed that the appropriately compactified 4d theory contains fermions having only left-handed (or right-handed) couplings to the gauge field. If that would be true, it would enable lattice simulations of the Standard Model and solve a decades-old problem. We want to understand the transcription of this mechanism on the lattice, and the reason for its failure if such is the case.Similarly, one can introduce a flux through the 2 extra dimensions.Depending on the fermion content of the theory, a non-zero flux may beenergetically favored, which would affect the fermion masses of theeffective 4d theory, and perhaps shed new light on the mass hierarchyamong the 3 quark families observed in the Standard Model.