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\ \ In 1939 Rademacher derived a conditionally convergent series expression for the modular j-invariant, and used this expression---the first Rademacher sum---to verify its modular invariance. We may attach Rademacher sums to other discrete groups of isometries of the hyperbolic plane, and we may ask how the automorphy of the resulting functions reflects the geometry of the group in question.

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\ \ In the case of a group that defines a genus zero quotient of the hyperbolic plane the relationship is particularly striking. On the other hand, of the common features of the groups that arise in monstrous moonshine, the genus zero property is perhaps the most elusive. We will illustrate how Rademacher sums elucidate this phenomena by using them to formulate a characterization of the discrete groups of monstrous moonshine.

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\ \ A physical interpretation of the Rademacher sums comes into view when we consider black holes in the context of three dimensional quantum gravity. This observation, together with the application of Rademacher sums to moonshine, amounts to a new connection between moonshine, number theory and physics, and furnishes applications in all three fields.

Many problems from combinatorics, number theory, quantum field theory and topology lead to power series of a special kind called q-hypergeometric series. Sometimes, like in the famous Rogers-Ramanujan identities, these q-series turn out to be modular functions or modular forms. A beautiful conjecture of W. Nahm, inspired by quantum theory, relates this phenomenon to algebraic K-theory.

In a different direction, quantum invariants of knots and 3-manifolds also sometimes seem to have modular or near-modular properties, leading to new objects called "quantum modular forms".

A key birational invariant of a compact complex manifold is its "canonical ring."

The ring of modular forms in one or more variables is an example of a canonical ring. Recent developments in higher dimensional algebraic geometry imply that the canonical ring is always finitely generated:this is a long-awaited major foundational result in algebraic geometry.

In this talk I define all the terms and discuss the result, some applications, and a recent remarkable direct proof by Lazic.

Speakers include:

* David Abrahams (Manchester, UK); * Stuart Antman (Maryland, USA); * Martine Ben Amar (Ecole Normale Supérieure, France); * Mary Boyce (MIT, USA); * John Hutchinson (Harvard, USA); * Nadia Lapusta (Caltech, USA); * John Maddocks (Lausanne, Switzerland); * Stefan Mueller (Bonn, Germany); * Christoph Ortner (Oxford, UK); * Ares Rosakis (Caltech, USA); * Hanus Seiner (Academy of Sciences, Czech Republic); * Eran Sharon (Hebrew University, Israel); * Lev Truskinovsky (Lab de Mécanique des Solids, France); * John Willis (Cambridge, UK).

Speakers include:

* Graeme Ackland (School of Physics and Astronomy, Edinburgh) * Andrea Braides (Rome II) * Thierry Bodineau (École Normale Supérieure, Paris) * Matthew Dobson (Minneapolis) * Laurent Dupuy (CEA, Saclay) * Ryan Elliott (Minneapolis) * Roman Kotecky (Warwick) * Carlos Mora-Corral (BCAM, Bilbao) * Stefano Olla (CEREMADE, Paris-Dauphine) * Bernd Schmidt (TU Munich) * Lev Truskinovsky (École Polytechnique, Palaiseau) * Min Zhou (Georgia Tech, Atlanta)