## Algebraic differential geometry seminar

Title

**
Period maps, spectral numbers and Stokes matrices of isolated
hypersurface singularities.
**

Date

July 6 (Thu), 15:00-17:00, 2017

July 7 (Fri), 15:00-17:00, 2017

Room

RIMS, Rm 111

Speaker

Claus Hertling氏 (University of Mannheim)

Abstract

Holomorphic function germs with isolated singularities have
been studied since the end of the 1960ies. A lot is known about their
topology, their Milnor lattices, their Gauss-Manin connections and
induced Hodge structures, and the behaviour of these data in families
of functions. But also a lot is still not well understood, basic
properties of the integral monodromy and Seifert form, a mysteriously
well working interplay between lattice data and period maps for
families, the spectral numbers.

The talk will give an introduction to the playing characters in the
theory of isolated hypersurface singularities and will present a
bouquet of old and new conjectures and related results. The first talk
will focus on single singularities and mu-constant families, the
second talk on universal unfoldings and Stokes structures of generic
members.

For unmarked as well as marked singularities, moduli spaces, period
maps, Torelli conjectures and Torelli results will be presented. An
old conjecture on the integral monodromy of quasihomogeneous
singularities will be recalled. Results and conjectures on the
spectral numbers will be given. This will be useful for a conjectural
characterization of the Stokes matrices and Coxeter-Dynkin diagrams of
singularities.

Title

**
Topological recursion, WKB analysis and Painlevé equations
**

Date

June 22 (Thu), 13:30-15:30, 2017

Room

RIMS, Rm 006

Speaker

岩木 耕平 氏(名古屋大学)

Abstract

B. Eynard と N. Orantin が導入した位相的漸化式 (topological recursion) は, 与えられた代数曲線からある種の不変量の族を機能的に定めるアルゴリズムである. 例えば Gromov-Witten 不変量や Hurwitz 数などを含む様々な幾何学的不変量, さらには KdV 方程式の解 (タウ函数) のような可積分系の対象もこの枠組みから 現れることが知られている. 講演では位相的漸化式の入門から始め, WKB 解析や Painleve 方程式との関係について講演者の結果について説明したい.

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.