Title

トーリック束の量子コホモロジー

Date

April 30 (Tue), 13:00--14:30, 2024

Place

Room 011, RIMS

Speaker

厚東裕紀氏（京都大学）

Abstract

非特異複素射影多様体Xの量子コホモロジーQH*(X)は種数0のGromov-Witten不変量を用いて定義される，コホモロジー環の積構造の変形族である．QH*(X)を調べることはミラー対称性の文脈で重要であり，また双有理幾何や導来圏の構造と深く関わることが期待されている．ファイバー束のコホモロジー環と底空間/ファイバーのコホモロジー環との関係はよく理解されているが，QH*に関する同様の現象は一般の状況ではよく知られていない．BrownおよびCoates-Givental-Tsengは，直線束の直和のトーラスによるGIT商として得られる ファイバーがトーリック多様体であるファイバー束のGromov-Witten不変量を調べて，全空間と底空間/ファイバーの同変量子コホモロジーの関係を明らかにした． 本講演では，より一般のトーリック束E→Bに対してQH*(E)とQH*(B)の関係を解説する．

Title

A^{(1)*}_2型曲面の接続のモジュライによる実現

Date

April 17 (Wed), 13:30--15:00, 2024

Place

Room 206, RIMS

Speaker

松本 孝文 氏(京大数理研)

Abstract

坂井理論とは、曲面論からPainleve・離散Painleve方程式を導出する方法である。坂井理論に現れる曲面はaffineルート系を用いて分類される。各曲面の族にはaffine Weyl群の作用が自然に入り、離散Painleve方程式は平行移動の作用が作る離散力学系として定式化される。差分Painleve方程式の場合、有理射影曲面から反標準因子を除いた空間が、ある有理型接続のモジュライ空間として実現されることが知られている。ここで、有理型接続のモジュライ空間の自然なコンパクト化によって曲面全体が導出できないだろうか、という問いが生じる。本講演ではA^{(1)*}_2型の場合に曲面全体がphi-接続のモジュライ空間として実現できることを説明する。特に、A^{(1)*}_2型曲面の各点とphi-接続との具体的な対応について紹介する。また、affine Weyl群の実現についてもわかっている範囲で紹介したい。

Title

On the WKB approximation and the Riemann-Hilbert problem of meromorphic linear systems with Poncaré rank 1.

Date

August 18 (Friday), 10:00--12:00, 2023

Place

Room 006, RIMS

Speaker

Xiaomeng Xu (Peking University)

Abstract

This talk first gives an introduction to the Stokes matrices of meromorphic linear systems with Poncaré rank 1. It then express explicitly the Stokes matrices via the leading asymptotics of solutions of the associated isomonodromy equation at a critical point. Such an expression of Stokes matrices can be understood as a perfect analytic model from various perspective. In particular, the expression is used to study the WKB approximation of the Stokes matrices, and its relation with Cauchy interlace inequalities, the spectral network and so on. If time allows, the talk discusses the WKB approximation and Stokes matrices of a quantum meromorphic linear systems with Poncaré rank 1, with a relation to quantum groups and crystal basis.

Title

Wall-crossing for framed quiver moduli

Date

May 12 (Thu), 17:00-18:30, 2022

Room

Zoomによるオンラインセミナーです.

Speaker

大川 領 氏(大阪公立大学数学研究所)

Abstract

We try to formulate wall-crossing formula of framed quiver moduli, based on the works by Takuro Mochizuki about moduli of parabolic sheaves on algebraic surfaces, and Nakajima-Yoshioka about moduli of framed perverse coherent sheaves on blow-up $\hat{\PP}^{2}$.

Title

Stokes filtered sheaves and differential-difference modules

Date

September 10 (Thu), 13:30-15:30, 2020

Room

Zoomによるオンラインセミナーです.

Speaker

社本 陽太 氏(東京大学)

Abstract

P. Deligneは, 複素領域における線形微分方程式の不確定特異点に おけるStokes現象を内在的に記述するため, Stokesフィルター付き局所系の概念を導入しました. 本講演では, 微分差分方程式に対して類似の構造を与えることを目的として導入した, Stokesフィルター付き擬局所系の概念と, その具体例について説明します.

Title

Sheaves and symplectic geometry of cotangent bundles

Date

February 19 (Mon), 10:00-12:00, 15:00-17:00, 2018
February 20 (Tue), 10:00-12:00, 15:00-17:00, 2018

Room

RIMS, Rm 110

Speaker

Stéphane Guillermou氏 (Université Grenoble Alpes Institut Fourier)

Abstract

We will explain how the microlocal theory of sheaves, introduced by Kashiwara and Schapira in the 80's, is used in symplectic geometry (mainly of cotangent bundles), since works of Nadler-Zaslow and Tamarkin in 2008. In particular we will sketch a proof of the following result: a compact exact Lagrangian submanifold of a cotangent bundle $T^*M$ is homotopically equivalent to the base $M$, through the natural projection from $T^*M$ to $M$.
The lectures will be a more detailed exposition of the topics introduced in the first talk (Feb. 2):
-- (constructible) sheaves and microsupport; operations
-- sheaves associated with a Hamiltonian isotopy of a cotangent bundle; first applications (maybe the non squeezing lemma)
-- Sato's microlocalization and the muhom functor
-- study of compact exact Lagrangian submanifolds of a cotangent bundle.

Title

Sheaves and symplectic geometry of cotangent bundles

Date

February 2 (Fri), 15:00-17:00, 2018

Room

RIMS, Rm 110

Speaker

Stéphane Guillermou氏 (Université Grenoble Alpes Institut Fourier)

Abstract

We will explain how the microlocal theory of sheaves, introduced by Kashiwara and Schapira in the 80's, is used in symplectic geometry (mainly of cotangent bundles), since works of Nadler-Zaslow and Tamarkin in 2008. In particular we will sketch a proof of the following result: a compact exact Lagrangian submanifold of a cotangent bundle $T^*M$ is homotopically equivalent to the base $M$, through the natural projection from $T^*M$ to $M$.
In the course of the talk we will quickly recall the main notions of the microlocal theory of sheaves and give examples.

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.