## Algebraic differential geometry seminar

Title

**
Differential equations and algebraic points on transcendental varieties
**

Date

April 6 (Thu), 13:30-15:00, 2017

Room

RIMS, Rm 110

Speaker

Gal Binyamini氏 (Weizmann Institute)

Abstract

The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate for a very wide class of transcendental sets. This result plays a key role in Pila-Zannier's proof of the Manin-Mumford conjecture, Pila's proof of the Andre-Oort conjecture for modular curves, Masser-Zannier's work on torsion anomalous points in elliptic families, and many more recent developments. I will briefly sketch the Pila-Wilkie theorem and the way it enters into the arithmetic applications. I will then discuss recent work on an effective form of the Pila-Wilkie theorem (for certain sets) which leads to effective versions of many of the applications. I will also discuss a joint work with Dmitry Novikov on sharpening the asymptotic from sub-polynomial to poly-logarithmic for certain structures, leading to a proof of the restricted Wilkie conjecture. The structure of the systems of differential equations satisfied by various transcendental functions play the main role for both of these directions.

Title

**
Filtered holonomic D-modules in dimension one
**

Date

December 1 (Thu), 15:00-17:00, 2016

December 2 (Fri), 15:00-17:00, 2016

Room

RIMS, Rm 110

Speaker

Claude Sabbah氏

Abstract

Holonomic D-modules on the affine complex line offer a simple prototype
of various properties also occurring in higher dimensions. We will focus
on filtered holonomic D-modules from various points of view.

In the first lecture, we start from a filtered regular holonomic
D-module M underlying a Hodge module, and we explain the construction an
some properties of the associated Deligne filtration on the D-module
obtained from M by applying an exponential twist. This is strongly
related to considering the Laplace transform of M. We will give
motivations for considering such a filtration.

In the second lecture, we focus on rigid irreducible holonomic D-modules
on the affine line. Generic (possibly confluent) hypergeometric
differential equations give naturally rise to examples of such objects.
After having explained the Katz algorithm (respectively the
Arinkin-Deligne algorithm) for reducing the regular (respectively
possibly irregular) such D-modules to ones having generic rank one, we
will consider the behaviour of Hodge (respectively Deligne) filtrations
along the algorithm (joint work with M. Dettweiler) and we will explain
some examples.