Japan-Taiwan workshop on

Moduli spaces of flat connections on surfaces

and related topics

Waseda University

Nishi-Waseda Campus, Building 55N (near Tully's Coffee)

1st floor (ground floor), 1st conference room

15-16 November 2014

  

The 1982 paper of Atiyah and Bott on "The Yang-Mills equations over Riemann surfaces" started a vigorous period of research into the symplectic and Poisson geometry of moduli spaces of flat connections on vector bundles over surfaces. By generalizing to bundles with complex structural groups, Hitchin introduced Higgs bundles, which have deep connections to the theory of integrable systems. Donaldson, Jeffrey-Weitsmann and Alekseev-Malkin-Meinrenken-Woodward studied bundles over surfaces with boundary (or connections with regular singularities), revealing new connections with Poisson geometry. Boalch developed the theory for connections with irregular singularities, which arise naturally in the theory of Painleve equations and also in the theory of Frobenius manifolds. At the same time, the early relations with differential geometry and variations of Hodge structures remain important. The talks in this workshop will address several of these topics.

Nishi-Waseda Campus access map

Campus map

Photographs from the workshop: Group 1 Group 2

Speakers:

Chi-Kwong Fok (National Tsing-Hua University)

TITLE: Twisted Poisson manifolds and their almost symplectically complete isotropic realizations
ABSTRACT: Let B be a twisted Poisson manifold with a fixed tropical affine structure. In this talk, we will discuss the classification of almost symplectically complete isotropic realizations (ASCIRs) over B in the spirit of Dazord-Delzant. We will construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of B. We will give descriptions of the Picard group and the corresponding 'Neron-Severi group' using certain sheaf cohomology groups. This is joint work with Reyer Sjamaar.

Martin Guest (Waseda University)

TITLE: Moduli spaces of real solutions of the third Painleve equation (joint work with Claus Hertling)

Nan-Kuo Ho (National Tsing-Hua University)

TITLE: Hitchin's equations over a nonorientable manifold
ABSTRACT: We study Hitchin's equations over a non-orientable manifold whose oriented cover is compact Kahler. Using the involution induced by the deck transformation, we show that the Hitchin's moduli space over a nonorientable manifold is Langrangian/Kahler with respect to the hyper-Kahler structure on Hitchin's moduli space over its orientable cover. We then establish a Donaldson-Corlette type correspondence which implies that the moduli space of flat connections over a nonorientable manifold is Kahler. Finally, we study Hitchin's moduli space via the use of representation varieties. This is a joint work with G. Wilkin and S. Wu.

Kohei Iwaki (RIMS, Kyoto University)

TITLE: Part 1: Introduction to exact WKB analysis, Part 2: Exact WKB analysis and cluster algebras
ABSTRACT: Exact WKB analysis is an effective method for the global study of differential equations (containing a large parameter) defined on a complex domain. On the other hand, a cluster algebras is a particular class of commutative subalgebra of a field of rational functions with distinguished generators. In this talk I'll explain about a hidden cluster algebraic structure in exact WKB analysis. I'll give an introduction to the theory of exact WKB analysis in the first talk. The second talk will be devoted to an explanation of our main results. My talk is based on a joint work with T. Nakanishi (Nagoya).

Masa-Hiko Saito (Kobe University)

TITLE: Canonical coordinates of moduli spaces of parabolic connections and parabolic Higgs bundles (joint work with Szilard Szabo)
ABSTRACT: In the first part of the talk, we will review on the algebraic construction of the moduli spaces of parabolic connections and parabolic Higgs bundles on curves. Then we review new results on canonical coordinates of their moduli spaces by so called apparent singularities.

Yoshihiro Ohnita (Osaka City University and OCAMI)

TITLE: Gauge-theoretic approach to harmonic maps and subspaces in moduli spaces (joint work with Mariko Mukai-Hidano)

Eugene Xia (National Cheng-Kung University)

TITLE: Higgs bundle and representation variety
ABSTRACT: This presentation introduces the subject of non-abelian cohomology, more specifically, the correspondences of the representation varieties, the moduli spaces of smooth, holomorphic and algebraic flat connections and the moduli spaces of Higgs bundles on a smooth projective variety. Striking geometric and topological results are obtained through these correspondences.

Daisuke Yamakawa (Tokyo Institute of Technology)

TITLE: Isomonodromic deformations from Hamiltonian point of view

SCHEDULE OF TALKS:

Saturday 15 November 2014 10:00-

10:15-11:15 Ohnita

11:30-12:30 Xia

lunch break

14:00-15:00 Ho

15:15-16:15 Iwaki (1)

coffee/tea break

16:45-17:45 Saito

Sunday 16 November 2014 10:00-

10:15-11:15 Fok

11:30-12:30 Guest

lunch break

14:00-15:00 Iwaki (2)

15:15-16:15 Yamakawa

Organizing committee: Martin Guest (Waseda University), Nan-Kuo Ho (National Tsing-Hua University, Taiwan), Yoshihiro Ohnita (Osaka City University)

Sponsors:

JSPS Grant-in-Aid for Scientific Research (A)25247005 (PI: Martin Guest) "Systematic development and application of methods in differential geometry and integrable systems motivated by quantum cohomology"

JSPS Grant-in-Aid for Scientific Research (S)24224001 (PI: Masa-Hiko Saito) "Developments in Interactions between Algebraic Geometry and Integrable
Systems"

JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (Oct. 2014-Mar. 2017) "Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI" (Osaka City University - Kobe University - Waseda University, PI: Yoshihiro Ohnita)