Emerging Topics on Scalar Curvature and Convergence

Last Fall the Emerging Topics on Scalar Curvature and Convergence groupdiscussed and devised a number of conjectures concerning the intrinsic flat limits of sequences of manifolds with lower bounds on their scalar curvature. Many of thesewere almost rigidity conjectures that a sequence of M_j almost satisfying a variety of hypothesesmust converge in the intrinsic flat sense to a limit space M_\infty which is rigidly determinedby actually satisfying those hypotheses. A first step towards proving such a theoremif the sequence has volume and diameter from above is to apply Wenger's CompactnessTheorem to produce a limit space. However this limit could be the 0 space. In this talk, Iwill present a variety of theorems proven in S-Wenger, Portegies-S, Matveev-Portegies, Perales,Huang-Lee-S, Allen-S, Perales-Nunez, Cabrera-Ketterer-Perales, and Allen-Bryden which can be applied to prove the limit space is nonzero. For the most part these theorems are hidden within papers proving almost rigidity theorems.