パンルヴェ方程式の完全WKB解析において、90年代に河合らによって構成された形式解や形式的変換論がある。その解析的意味付けについて説明する。また、それらを用いて、接続問題について可能な限り議論する。

qパンルヴェ方程式について、付随するq線形差分方程式やワイル群対称性の観点などから捉え直す。qミドルコンボルーションとの関係についても述べたい。

TBA

非特異射影曲線上の放物Higgs束や放物接続のモジュライ空間は、古典的なパンルべ方程式を含む可積分系の初期値空間として重要な研究対象である。このモジュライ空間は代数多様体であり適当なパラメータを固定したときに、自然な代数的シンプレクテック構造を持つ。この講演では、モジュライ空間の標準座標（ダルブー座標）を与える見かけの特異点理論について説明する。また、パンルべ方程式を含むモノドロミー保存変形の微分方程式について考察する。この講演の内容の一部は光明新氏、Frank Loray氏, Szilard Szabo氏との共著論文「arXiv:2309.05012」による。

It is conjectured that a relationship exists between the accessory parameters of (confluent) Heun's equation and the classical limit of conformal blocks. 

In this talk, we propose a method to obtain a formal power series expansion of the accessory parameter with respect to the time variable by considering deformations of the Heun equation that preserve a certain Voros period.

We will then provide a computational verification of the agreement between the accessory parameter and the classical conformal block for the first several orders, illustrated with several examples which may have irregular singularities.   

This talk is based on ongoing joint work with Kohei Iwaki (University of Tokyo) and Hajime Nagoya (Kanazawa University). 

 We introduce a moduli space of decorated twisted G-local systems over a marked surface with prescribed boundary condition by the Weyl group (or braid group) elements, and investigate its cluster structure. The corresponding quivers are related by a kind of “degeneration” when the Weyl group elements are related in the Bruhat order.  This talk is based on a joint work with Linhui Shen.

There are four real forms of the Tzitzeica (or Bullough-Dodd) equation. They are distinguished from each other by their geometrical interpretations, as well as by their analytic properties. We focus on the special case of radial solutions, which can be regarded as real solutions of a version of the Painleve III equation, and hence correspond to monodromy data of a linear isomonodromic system. We discuss the interplay between the geometrical properties of this monodromy data and the analytic properties of solutions .

It is widely believed that a period integral is connected to special values of L-functions and there are a number of conjectures by Beilinson, Deligne and others. Motivated by this philosophy, we study an arithmetic nature of the asymptotic expansion of a period integral. It is an active playground where arithmetics goes along well with special functions. The theorem we propose is the following: for a generic one-parameter degeneration of projective hypersurfaces defined over a subfield K of the field of complex numbers, the periods of the limiting mixed Hodge structure are generated by certain special values of log, Gamma and Dirichlet L-functions over a cyclic extension of K. The genericity condition of a one-parameter degeneration is defined in terms of the secondary fan. Our proof strategy is in two steps. First, we show that ANY generic one-parameter degeneration is reduced to a SINGLE one-parameter degeneration, which we call the Fermat deformation. Second, we perform an analytic continuation of solutions of a period integral by expanding it as a linear combination of solutions to a GKZ system. This is a joint project with Masanori Asakura (Hokkaido University).