Mathematics Colloquia and Seminars

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The Cheeger constant of a finite-volume Riemannian manifold is the infimum of isoperimetric ratios over smooth, top-dimensional submanifolds with boundary, where the top-dimensional volume of these submanifold is constrained to less than or equal to half the volume of the manifold. Aside from its clear relationship to the isoperimetric problem, a significant motivation for studying the Cheeger constant specifically is its relationship to spectral geometry. Specifically, independent work of Cheeger and Buser give explicit quadratic bounds in the Cheeger constant for the first positive eigenvalue of the Laplace-Beltrami operator of the manifold from below and above, respectively. I plan to describe forthcoming work with Grant Lakeland and Holger Then, where we explicitly compute the Cheeger constant of hyperbolic orbisurfaces, motivated by questions of Belilopetsky about hyperbolic reflection groups. In order to complete these computations, I will discuss how we applied work of Hass-Morgan and Adams-Morgan which provided key insights, ideas, and information regarding isoperimetric minimizers of these surfaces.