We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient
numerical methods for higher order Sobolev type metrics is an extremely desirable goal.
We address the following problem: given two smooth densities on a manifold, find an optimal diffeomorphism that transforms one density into the other. Our framework builds on connections between the Fisher-Rao information metric on the space of proba
bility densities and right-invariant metrics on the infinite-dimensional manifold of diffeomorphisms. This optimal information transport, and modifications thereof, allows us to construct numerical algorithms for density matching. The algorithms are inherently more efficient than those based on optimal mass transport or diffeomorphic registration. Our methods have applications in medical image registration, texture mapping, image morphing, non-uniform random sampling, and mesh adaptivity. Some of these applications are illustrated in examples.
On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.
We study classically scale invariant models in which the Standard Model Higgs mass term is replaced in the Lagrangian by a Higgs portal coupling to a complex scalar field of a dark sector. We focus on models that are weakly coupled with the quartic s
calar couplings nearly vanishing at the Planck scale. The dark sector contains fermions and scalars charged under dark SU(2) x U(1) gauge interactions. Radiative breaking of the dark gauge group triggers electroweak symmetry breaking through the Higgs portal coupling. Requiring both a Higgs boson mass of 125.5 GeV and stability of the Higgs potential up to the Planck scale implies that the radiative breaking of the dark gauge group occurs at the TeV scale. We present a particular model which features a long-range abelian dark force. The dominant dark matter component is neutral dark fermions, with the correct thermal relic abundance, and in reach of future direct detection experiments. The model also has lighter stable dark fermions charged under the dark force, with observable effects on galactic-scale structure. Collider signatures include a dark sector scalar boson with mass < 250 GeV that decays through mixing with the Higgs boson, and can be detected at the LHC. The Higgs boson, as well as the new scalar, may have significant invisible decays into dark sector particles.
We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms.
We find isometries (called $R$-transforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.
We study extensions of the standard model by one generation of vector-like leptons with non-standard hypercharges, which allow for a sizable modification of the h -> gamma gamma decay rate for new lepton masses in the 300 GeV - 1 TeV range. We analyz
e vaccum stability implications for different hypercharges. Effects in h -> Z gamma are typically much smaller than in h -> gamma gamma, but distinct among the considered hypercharge assignments. Non-standard hypercharges constrain or entirely forbid possible mixing operators with standard model leptons. As a consequence, the leading contributions to the experimentally strongly constrained electric dipole moments of standard model fermions are only generated at the two loop level by the new CP violating sources of the considered setups. We derive the bounds from dipole moments, electro-weak precision observables and lepton flavor violating processes, and discuss their implications. Finally, we examine the production and decay channels of the vector-like leptons at the LHC, and find that signatures with multiple light leptons or taus are already probing interesting regions of parameter space.
