We study smooth hyperbolic systems with singularities and their SRB measures. Here we assume that the singularities are submanifolds, the hyperbolicity is uniform aside from the singularities, and one-sided derivatives exist on the singularities. We prove that the ergodic SRB measures exist, are finitely many, and mixing SRB measures enjoy exponential decay of correlations and a central limit t...

The paper deals with the propagation of singularities of semi-convex functions. We obtain lower bounds on the degree of the singularities and on the size of the singular set in a neighborhood of a singular point. These results apply to viscosity solutions of HamiltonJacobi-Bellman equations. In particular, they provide sufficient conditions for the propagation of singularities, depending only o...

An analytic construction of compact Calabi-Yau manifolds with del Pezzo singularities is found. We present complete intersection CY manifolds for all del Pezzo singularities and study the complex deformations of these singularities. An example of the quintic CY manifold with del Pezzo 6 singularity and some number of conifold singularities is studied in details. The possibilities for the ’geome...

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. Stability conditions for these moduli problems depend on two parameters describing allowable singularities. At the extreme values of the parameters, we obtain the stacks TAn and TDn of stable limits of An and Dn singularities, as well as the quotients of the miniversal deformation spaces of ...

We propose a class of spectral singularities emerging from the coincidence of two independent singularities with highly directional responses. These spectral singularities result from resonance trapping induced by the interplay between parity-time symmetry and Fano resonances. At these singularities, while the system is reciprocal in terms of a finite transmission, a simultaneous infinite refle...

Parabolic geometric flows are smoothing for short time however, over long time, singularities typically unavoidable, can be very nasty and may impossible to classify. The idea of [CM6] here is that, by bringing in the dynamical properties flow, we obtain also large generic initial conditions. When combined with [CM1], this shows, an important special case, simplest possible. question dynamics a...