In this paper, we establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh--Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and the Cauchy formula (1882) for the magnetic field. This enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and on the other hand, it allows us to define a special good unknown of the magnetic field to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh--Taylor sign condition required by Morando, Tarkhinin and Trebeschi (\emph{J. Differential Equations} \textbf{258} (2015), no. 7, 2531--2571; \emph{Arch. Ration. Mech. Anal.} \textbf{228} (2018), no. 2, 697--742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is {\it no loss of derivatives} in our well-posedness theory. The solution is constructed as {\it the inviscid limit} of solutions to well-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.

Three dimensional topologically massive gravity (TMG) with a negative cosmological constant -\ell^{-2} and positive Newton constant G admits an AdS_3 vacuum solution for any value of the graviton mass \mu. These are all known to be perturbatively unstable except at the recently explored chiral point \mu\ell=1. However we show herein that for every value of \mu\ell< 3 there are two other (potentially stable) vacuum solutions given by SL(2,R)x U(1)-invariant warped AdS_3 geometries, with a timelike or spacelike U(1) isometry. Critical behavior occurs at \mu\ell=3, where the warping transitions from a stretching to a squashing, and there are a pair of warped solutions with a null U(1) isometry. For \mu\ell>3, there are known warped black hole solutions which are asymptotic to warped AdS_3. We show that these black holes are discrete quotients of warped AdS_3 just as BTZ black holes are discrete quotients of ordinary AdS_3. Moreover new solutions of this type, relevant to any theory with warped AdS_3 solutions, are exhibited. Finally we note that the black hole thermodynamics is consistent with the hypothesis that, for \mu\ell>3, the warped AdS_3 ground state of TMG is holographically dual to a 2D boundary CFT with central charges c_R={15(\mu\ell)^2+81\over G\mu((\mu\ell)^2+27)} and c_L={12 \mu\ell^2\over G((\mu\ell)^2+27)}.

We show that the Kazhdan-Lusztig category KL_k of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1)ˆ has vertex algebraic braided tensor supercategory structure, and that its full subcategory O_k^fin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)ˆ-module in KL_k has a projective cover in O_k^fin, and we determine all fusion rules involving simple and projective objects in O_k^fin. Then using Knizhnik-Zamolodchikov equations, we prove that KL_k and O_k^fin are rigid. As an application of the tensor supercategory structure on O_k^fin, we study certain module categories for the affine Lie superalgebra sl(2|1)ˆ at levels 1 and −1/2. In particular, we obtain a tensor category of sl(2|1)ˆ-modules at level −1/2 that includes relaxed highest-weight modules and their images under spectral flow.

We propose to study the Hessian metric of given functional in the space of probability space embedded with L^2–Wasserstein (optimal transport) metric. We name it transport Hessian metric, which contains and extends the classical L^2–Wasserstein metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and math physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton’s flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of negative Boltzmann-Shannon entropy forms the Shallow water’s equation; The transport Hessian gradient flow of Fisher information forms the heat equation. Several examples and closed-form solutions of finite-dimensional transport Hessian metrics and dynamics are presented.

In this expository note we discuss some arithmetic aspects of the mirror symmetry for plane cubic curves. We also explain how the Picard-Fuchs equation can be used to reveal part of these arithmetic properties. The application of Picard-Fuchs equations in studying the genus zero Gromov-Witten invariants of more general Calabi-Yau varieties and the Weil-Petersson geometry on their moduli spaces will also be discussed.