We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special. This result fits into the context of Zilber-Pink conjecture and partially generalizes a result of Faltings. Moreover by using the polyhedral reduction theory we give a new proof of a result of Bertrand.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, and J. Pitts, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$of positive Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.

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This note is based on a series of four lectures the author gave at University of Montpellier, September 28-30, 2015. He started with the notion of mass in general relativity, gave a brief review of some known constructions of quasi-local mass, and then discussed the new quasi-local energy and quasi-local mass which Shing-Tung Yau and the author introduced in 2009. At the end, the proof of the positivity of quasi-local energy was sketched and a stability theorem of critical points of quasi-local energy by Po-Ning Chen, Shing-Tung Yau, and the author was discussed