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We study the Newton polygon jumping locus of a Mumford family in char p. Our main result says that, under a mild assumption on p, the jumping locus consists of only supersingular points and its cardinality is equal to (p^r − 1)(g − 1), where r is the degree of the defining field of the base curve of a Mumford family in char p and g is the genus of the curve. The underlying technique is the p-adic Hodge theory.

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$ -forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the NavierStokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the NavierStokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate (1+ t) 1 4, it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of (1+ t) 1 4. Thus, this reciprocal order of decay rates for the time

This dissertation splits into two distinct halves. The first half is devoted to the study of the holography of higher spin gauge theory in AdS_3. We present a conjecture that the holographic dual of W_N minimal model in a 't Hooft-like large N limit is an unusual ``semi-local" higher spin gauge theory on AdS_3×S^1. At each point on the S^1 lives a copy of three-dimensional Vasiliev theory, that contains an infinite tower of higher spin gauge fields coupled to a single massive complex scalar propagating in AdS_3. The Vasiliev theories at different points on the S^1 are correlated only through the AdS_3 boundary conditions on the massive scalars. All but one single tower of higher spin symmetries are broken by the boundary conditions. This conjecture is checked by comparing tree-level two- and three-point functions, and also one-loop partition functions on both side of the duality. The second half focuses on the holography of higher spin gauge theory in AdS_4. We demonstrate that a supersymmetric and parity violating version of Vasiliev's higher spin gauge theory in AdS_4 admits boundary conditions that preserve N=0,1,2,3,4 or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan-Paton and N=6 boundary condition is holographically dual to the 2+1 dimensional U(N)_k×U(M)_{−k} ABJ theory in the limit of large N,k and finite M. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t'Hooft coupling is M/N. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS_4×CP^3, and that the non-Abelian Vasiliev theory at strong bulk 't Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev's higher spin particles held together by U(M) gauge interactions.