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

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Let k be a commutative Noetherian ring and C−− be a locally finite k-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of C−− are super finitely presented (that is, they have projective resolutions each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when k is a field of characteristic 0, we obtain another upper bound for homological degrees of finitely generated FI-modules.

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No abstract uploaded!