L2

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)