Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle braiding statistics, chiral central charge and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3+1D and it has been shown that loop like excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and loop like excitation in 3+1D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

Published in 1999, Christodoulou proved that the naked singularities of a self-gravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical symmetry. In this paper, appropriate a priori estimates for the solution are obtained. These estimates are more relaxed but sufficient for giving another robust argument in proving the instability, in particular not by contradiction. In a companion paper, we are able to prove certain instability theorems of the spherically symmetric naked singularities of a scalar field under gravitational perturbations without symmetries. The argument given in this paper plays a central role.

We classify three fold isolated quotient Gorenstein singularity C^ 3/G . These singularities are rigid, ie there is no non-trivial deformation, and we conjecture that they define 4d C^ 3/G SCFTs which do not have a Coulomb branch.

In this continuation of [L-Y1],[LLSY],[L-Y2], and [L-Y3](arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], arXiv: 0901.0342 [math. AG], arXiv: 0907.0268 [math. AG]), we study D-branes in a target-space with a fixed B -field background B along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of B -field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to B , are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of B -field to both the D-brane world-volume and the superstring target-space (-time) B . This brings the notion of twisted B -modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled" Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3](arXiv: hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately" Dijkgraaf-Holland-Sukowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [DHSV](arXiv: 0709.4446 [hep-th]) and [DHS](arXiv: 0810.4157 [hep-th]) of Dijkgraaf, Hollands, Sukowski, and Vafa.

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), the force acting on the particles is not normalized, and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space-inhomogeneous extension of (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012). However, the modified normalization of the force gives rise to different convection speeds, and the resulting model may lose its hyperbolicity in some regions of the state space.