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.

Organizer T. Mochizuki

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.

Organizer T. Mochizuki

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.

Organizer T. Mochizuki

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.

Organizer T. Mochizuki

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