In this paper, we study the space of translational limits T (M) of a surface M properly embedded in R3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface Σ ∈ T (M) the set T (Σ) ⊂ T(M). Among various dynamics type results we prove that surfaces in minimal T -invariant sets of T (M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T -invariant set in T (M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.

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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.

We give a survey on Mœglin's construction of representations in the Arthur packets for p-adic quasisplit symplectic and orthogonal groups. The emphasis is on comparing Mœglin's parametrization of elements in the Arthur packets with that of Arthur.

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