O-minimality and Diophantine Geometry

Date: 26 - 30 September 2022

Location: University of Oxford

The last ten years have seen a flourishing interaction between o-minimality, a part of model theory, and diophantine geometry. The catalyst for this interaction was Pila’s proof in 2011 of the André-Oort conjecture for products of modular curves. This applied the Pila-Wilkie counting theorem, using a method introduced by Pila and Zannier, in combination with results in function transcendence. Pila introduced further ideas to prove these transcendence results using o-minimality.

Since then there have been many further developments. Many of these have taken place in the framework of the Zilber-Pink conjecture, a conjecture on unlikely intersections in mixed Shimura varieties, which implies the André-Oort conjecture and a great deal more. Work by various researchers has shown that the Pila-Zannier method can also apply to these unlikely intersection problems, and now both o-minimality and functional transcendence are important tools in several areas of diophantine geometry.

More recently, Bakker, Brunebarbe, Klingler and Tsimerman have applied ideas from o-minimality, in the study of Hodge theory. Their work further develops applications of o-minimality to functional transcendence. This in turn has applications back to diophantine geometry. For instance, the general Ax-Schanuel result established by Bakker and Tsimerman for period maps plays a key role in the work of Lawrence and Venkatesh.

This workshop, organised in honour of Jonathan Pila’s 60th birthday, brings together leading experts in these various developments. The speakers include a blend of established experts and promising junior researchers, as well as people from nearby areas in which o-minimality or functional transcendence are applied.

Speakers:  Enrico Bombieri (IAS) tbc, Laura Capuano (Roma Tre), Gabriel Dill (Hannover), Ziyang Gao (Leibniz Hannover),  Philipp Hagbegger (Basel) tbc, Annette Huber (Freiburg), David Masser (Basel), Kobi Peterzil (Haifa), Marta Pieropan (Utrecht), Jonathan Pila (Oxford), Peter Sarnak (Princeton and IAS), Tom Scanlon (Berkeley), Harry Schmidt (Basel), Margaret Thomas (Purdue), Jacob Tsimerman (Toronto), Umberto Zanier (Pisa)

Participation in the workshop is by invitation, but a limited number of additional places is available. To register for the Clay Research Conference or to register interest in a workshop, email Naomi Kraker. Students please also provide a letter of reference from your supervisor.

Graduate students interested in these topics may also be interested in the CMI-HIMR Summer School on Unlikely Intersections in Diophantine Geometry.

Getting to Oxford