# Programme

Monday | Tuesday | Wednesday | Thursday | Friday | |

10:00-11:00 | Wilkie | Wilkie | Wilkie | Pila | Habegger |

11:00-11:30 | Break | ||||

11:30-12:30 | Wilkie | Pila | Habegger | Habegger | Free |

12:30-13:30 | Lunch | ||||

13:30-14:30 | Pila | Wilkie | Pila | Pila | Habegger |

14:30-15:30 | Habegger | Masser | Jones | Yafaev | Finish |

15:30-16:00 | Break | ||||

16:00-17:00 | Tutorial | Tutorial | Tutorial | Tutorial |

- Rational points on definable sets (Alex Wilkie)

Abstract - Lecture Notes - Functional transcendence via o-minimality (Jonathan Pila)

Abstract - Diophantine Properties of Torsion Points and Special Points (Philipp Habegger)

Abstract

**Guest Lectures**

**David Masser**

Relative Manin-Mumford for abelian varieties

**Abstract:**

With an eye or two towards applications to Pell's equation and Davenport's work on integration of algebraic functions, Umberto Zannier and I have recently characterised torsion points on a fixed algebraic curve in a fixed abelian scheme of dimension bigger than one (when all is defined over the algebraic numbers): there are at most finitely many provided the natural obstacles are absent. I sketch the proof as well as the applications.

**Andrei Yafaev
**Hyperbolic Ax-Lindemann-Weierstrass theorem

**Abstract:**

This is a joint work with B. Klingler and E. Ullmo. We generalise the definability result of Peterzil and Starchenko to general Shimura varieties and prove the general hyperbolic Ax-Lindemann-Weierstrass theorem.

**Gareth Jones**

Integer valued definable functions

**Wilkie has conjectured that for sets definable in the real field with exponentiation, the bound in the Pila-Wilkie theorem can be improved to a power of log H. At present this is only known for curves and certain surfaces. An easy consequence of the result for curves is that a definable analytic function on the real line that takes integer values at all natural numbers either grows faster than exp(x^epsilon) for some positive epsilon or is a polynomial. This can be thought of as a (rather weak) analogue of Polya's theorem on integer valued entire functions. I'll discuss the above and then show how some simple o-minimality arguments can be used to extend this result to functions of several variables.**

Abstract:

Abstract: