(Continued) Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few: - How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V? - How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V? - Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space? Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces. In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.
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