Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov's compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin's representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

We provide characterizations of $p\mspace {1mu}$ -nilpotency for fusion systems and $p\mspace {1mu}$ -local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a finite dimensional variational principle.

We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S^3∖K. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of SL(2,C) Chern-Simons theory, one can define perturbative invariants of knot K in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of K and K. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.

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