Special Lecture: Mostow rigidity and hyperbolic 3-manifolds


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April 28, 2023 10:20 am - 10:50 am
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA

Benjy Firester - Harvard Class of 2023

My thesis develops the theory of hyperbolic manifolds and explains two proofs of the foundational Mostow rigidity theorem. Great math theorems bridge fields, enabling us to transfer tools from one to another, exhibiting overarching themes that unify mathematics. Mostow rigidity is such a theorem. It proves the uniqueness of geometric structures on spaces of dimension at least three, demonstrating a deep connection between topology and geometry. Most 3-dimensional spaces are hyperbolic, meaning “negatively curved" like a hyperboloid. Understanding hyperbolic spaces is important to topology, algebra, dynamics, and more. This theory is critical to the geometrization theorem, perhaps the most celebrated result in geometry. Mostow rigidity shows that any homotopy equivalence between two hyperbolic manifolds, a topological relationship, can be uniquely deformed to an isometry, a geometric one. Mostow’s own proof uses quasi-conformal theory to improve an initial function only preserves the topological structure into one that preserves the geometric structure, linking the two notions of equivalence. The second proof defines a topological quantity capturing a manifold’s complexity, and computes the volume for hyperbolic manifolds. The algebraic realization of a hyperbolic manifold encodes the geometric data of which simplices (higher-dimensional triangles) have maximal volume, which are rigid above dimension two.