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The canonical example of an arithmetic lattice is the modular group PSL(2, Z), whose deep connections with geometry and number theory (among many other areas) have been of profound interest for well over a century. Geometric invariants of the modular surface—the quotient of the complex upper half-plane by PSL(2, Z)—are typically paired with objects of equally deep interest in number theory. For example, its volume in its metric of constant curvature ­-1 is naturally related to a special value of the Riemann zeta function, and the lengths of its closed geodesics are intimately related to class numbers and regulators of real quadratic fields. More generally, rigidity phenomena (Weil, Mostow, etc.) imply that similar connections exist between number theory and the geometry of higher-dimensional hyperbolic manifolds.

The primary focus during the workshop will be to introduce the participants to problems at the interface of geometry and number theory that are currently attracting significant interest, and provide them with the tools necessary to make progress on some open questions. General areas to be discussed include the Laplace eigenvalue spectra and geodesic length spectra of hyperbolic 2- and 3-manifolds, growth of the systole of a hyperbolic manifold, and the ‘realization problem’ for trace fields of hyperbolic 3-manifolds. The number theoretic techniques that we will use to address these problems make use of quaternion algebras over number fields, Mahler measures of algebraic integers, and classical results from multiplicative number theory. The ultimate goal of this workshop is to start a dialogue between young mathematicians from different backgrounds that will lead to new and long-lasting collaborations between fields that have a great deal to say to one another. No background in either subject is expected.