Program [pdf]
9月15日（金）
13:00--15:00 齋藤政彦(神戸大)
Geometry of moduli spaces of
connections and Higgs bundles over curves and
Integrable Systems
15:30--17:30 稲場道明（京都大）
Moduli space of
parabolic connections, isomonodromic deformation and
compactification problem
18:00 -- Dinner (discussion)
9月16日(土)
9:30 -- 10:30
三田史彦(名古屋大)
An analogue of Dubrovin's conjecture
10:45--12:15 宮地秀樹(大阪大)
Deformation of Riemann surfaces
via affine deformations
13:30 -- 15:00 光明 新(神戸大) （南４号館205号室へ移動)
"Chekhov-Mazzocco-Rubtsov: Algebras of quantum monodromy data and decorated character
varieties" の紹介
15:00 -- 17:00 自由討論
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齋藤政彦
Title: Geometry of moduli spaces of
connections and Higgs bundles over curves and
Integrable Systems
Abstract: This is an introductory
talk of geometry of moduli spaces of connections
and Higgs bundles and their relation to Integrable Systems. I will start with a summary
of constructions of moduli spaces of parabolic
connections and singular parabolic Higgs bundles on a smooth
projective curve by
geometric invariant theory. Next, we will explain about
Riemann-Hilbert correspondence from a family of moduli spaces of
singular connections to
the corresponding moduli spaces of (generalized) monodromy
data. An analysis of RH correspondence shows the geometric
Painlevé property of
isomonodromic differential equations associated to each type
of singular connections. Next, I will investigate explicit
geometric structures of
moduli spaces of parabolic connections and Higgs bundles. On
a Zariski dense open set of each moduli space one can define a
canonical coordinate
system associated to apparent singularities and their duals.
The spectral curves for Higgs bundles play essential roles for
this explicit geometry. If
time permits, we will explain more geometric
structures of moduli spaces.
稲場道明
Title: Moduli space of parabolic connections, isomonodromic
deformation and compactification problem
Abstract: This talk is a survey on the
series of joint work with Katsunori Iwasaki
and Masa-Hiko Saito. First
I will explain the notion of regular singular parabolic connection on a smooth projective curve and the moduli space of
stable parabolic connections. This
moduli space is in fact constructed by its embedding to
the projective moduli space of parabolic $\Lambda^1_D$-triples. This embedding gives a
compactification of the moduli space
of parabolic connections in the case of rank two.
On the other hand, there is Simpson's procedure of compactifying
the moduli space of
connections by adding the projectivized moduli space
of Higgs bundles as a boundary locus.
In the case of rank two connections on $\mathbb{P}^1$
with 4 distinct singular
points, our
compactification gives a resolution of singularities of
the Higgs compactification and the compactification
is nothing but the Okamoto Painleve pair.
I will also explain the isomonodoromic deformation on the moduli space of parabolic
connections.
三田史彦
Title: An analogue of Dubrovin's
conjecture
Abstract:
Dubrovinはミラー対称性予想に基づいて、Fano多様体の連接層の導来圏がfull exceptional
collectionを持つことと量子コホモロジー環が半単純であることが同値であると予想した。 更に量子D加群のStokes行列がfull
exceptional collectionのオイラーpairingによって与えられることを予想した。
これらの予想はGalkin-Golyshev-Iritaniによってガンマ予想と いう形で精密化された。 本講演ではfull
exceptional collectionを持つとは限らない場合のガンマ予想の類似について述べる。
時間が許せばCalabi-Yau超曲面の量子D加群との関係についても言及す る。
これは京都大学数理解析研究所の社本陽太氏との共同研究に基づく。
宮地秀樹
Title: Deformation of Riemann surfaces via affine deformations
Abstract: An orientable closed surface of negative Euler characteristic
admits a singular Euclidean structure, and any singular Euclidean
Euclidean structure defines a complex structure on the surface. In this
talk, we consider singular Euclidean structures defined from ``generic”
holomorphic quadratic differentials on Riemann surfaces, and discuss the
deformation of the underlying conformal structures via the deformation
of the singular Euclidean structures in the Teichmueller theoretical
formulation.
光明 新
Title: "Chekhov--Mazzocco--Rubtsov: Algebras of quantum monodromy data
and
decorated character
varieties"の紹介
Abstract: Riemann--Hilbert対応は, 接続のモジュライ空間からcharacter varietyと呼ばれる
モノドロミー表現のモジュライ空間への同型写像を与える. Hitchinは,
リーマン球面上の高々１位の極をもつ接続の場合にこの写像がシンプレクティック同相写像であることを示した. この理論を,
一般のリーマン面上の接続に２位以上の極を許す場合へ拡張する問題を考える. Chekhov--Mazzocco--Rubtsovは,
この論文の中でこの問題のdecorated character varietyを用いた定式化を行った. この論文について概略を紹介する.
参考文献
・Chekhov--Mazzocco--Rubtsov: Algebras of quantum monodromy data and decorated character varieties(arXiv:1705.01447)
・Chekhov--Mazzocco--Rubtsov: Painlev'e; monodromy manifolds, decorated character varieties and cluster algebras (arXiv:1511.03851)