Tag - Hyperbolic geometry

Dalimil Mazac: Sphere Packings, Spectral Gaps and the Conformal Bootstrap

8th November, 2023

I will discuss infinite-dimensional linear programs producing bounds on the spectral gap in various settings. This includes new bounds on the spectral gap of hyperbolic manifolds as well as the Cohn-Elkies bound on the density of sphere packings. The bounds allow us to essentially determine the complete set of spectral gaps achieved by hyperbolic 2-orbifolds. The linear programs involved have been the subject of intense study by mathematical physicists in the context of the conformal bootstrap.

I will review the method of analytic extremal functionals, introduced by the speaker to prove sharp bounds in the conformal bootstrap. When used within the Cohn-Elkies linear program, this method reproduces the groundbreaking solution of Viazovska et al of the sphere packing problem in dimensions 8 and 24, as well as the interpolation basis used in the proof of universal optimality of the E8 and Leech lattice. The connections covered in this talk offer a broader framework for studying optimality in infinite-dimensional linear programs.

Or Landesberg: Non-Rigidity of Horocycle Orbit Closures in Geometrically Infinite Surfaces

31st January, 2023

Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider ℤ-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.

Uri Bader: Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity

6th June, 2021

Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which is naturally a subgroup of the rational valued invertible matrices in some dimension, GLn(ℚ). When the fundamental group essentially consists of the integer points of some algebraic subgroup of GLn we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years. In my talk I will survey the subject and focus on the relation between the geometric, algebraic and arithmetic concepts