Deformation Theory of infinite-type hyperbolic manifolds

Hyperbolic geometry, and its connection to 3-dimensional geometry, have been a key topic in contemporary mathematics, leading for instance to the resolution of the Poincaŕe Conjecture (2006) and to the Fields medals awarded to Thurston (1982), McMullen (1998), Perelman (2006, declined) and Mirzakhani (2014).

The project will enter the unexplored territory that opens when removing this fundamental hypothesis.

Specifically, the PI plans to attack the very challenging problem of understanding the space of hyperbolic metrics on 3-manifolds that have a non-finitely generated fundamental group.

The project lies at the intersection between the study of the topology and geometry of hyperbolic 3-manifolds and is well-suited to the complementary expertise of the PI and his supervisor, professor Schlenker, the PI being an expert of the topology of infinite-type 3-manifolds and the supervisor being an expert in hyperbolic geometry.

An essential aspect of this research program is understanding the interplay between the topology and the geometry of infinite-type hyperbolic manifolds with the goal to borrow insights from each side to address issues in the other.

One of the first objectives is to understand, by looking at topological properties, how much of the rich theory of the finite-type setting extends to the case where the fundamental group is not finitely generated.

The second objective is more geometric and plans to study infinite-type 3-manifolds by seeing them as geometric ‘limits’ of finite-type hyperbolic 3-manifolds and looking at which geometric, or topological, aspects survive in the limit.

The second part of the project will involve, under the direction of professor Krasnov, is to investigate the AdS-CFT correspondence, an important conjectural relationship linking quantum gravity (formulated as M-theory) in M and conformal field theories (CFT) in the boundary of M, using tools from hyperbolic geometry, e.g. renormalised volume.