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Topological aspects of physical systems, including the called topological states of matter, have become hot topics in the frontiers of physics in recent years. Here I would like to present a mathematically "popular" talk for professional physicists for a highlight or overview of how one can systematize knowledge of topological aspects of quantum field theories. Our starting points are Descent Equations and Gauge Structure in Configuration Space in Field Theory. (The audience needs only to know the meaning of "differential forms".)

We derive the constitutive relations of first order charged hydrodynamics for theories with Lifshitz scaling and broken parity in 2+1 and 3+1 spacetime dimensions. In addition to the anomalous (in 3+1) or Hall (in 2+1) transport of relativistic hydrodynamics, there is an additional non-dissipative transport allowed by the absence of boost invariance. We analyze the non-relativistic limit and use a phenomenological model of a strange metal

to argue that these effects can be measured in principle by using electromagnetic fields with non-zero gradients.

I will first briefly report the current status on the stability problem of global AdS space under gravitational self-interaction. I will then present evidence that the possibility of a blackhole-forming instability is strongly connected to phase-locked cascade, which is different from the usual energy cascade in turbulent flow.

Using holography, we study the evolution of a spatially homogeneous, far from equilibrium, strongly coupled N=4 supersymmetric Yang-Mills plasma with a non-zero charge density or a background magnetic field. This gauge theory problem corresponds, in the dual gravity description, to an initial value problem in Einstein-Maxwell theory with homogeneous but anisotropic initial conditions. We explore the dependence of the equilibration process on different aspects of the initial departure from equilibrium and, while controlling for these dependencies, examine how the equilibration dynamics are affected by the presence of a non-vanishing charge density or an external magnetic field. The equilibration dynamics are remarkably insensitive to the addition of even large chemical potentials or magnetic fields, the equilibration time is set primarily by the form of the initial departure from equilibrium. For initial deviations from equilibrium which are well localized in scale, we formulate a simple model for equilibration times which agrees quite well with our results.