|Title and Abstract of Talks ：
| Priya Kshirsagr (UC Davis, USA) |
|Title: Counting cell graphs and topological recursion |
This talk is aimed at surveying surprising appearances of cell graphs in algebraic geometry.
We first review the theorem of Belyi that gives a topological description of algebraic curves defined over the field of algebraic numbers.
Then we introduce Grothendieck's dessins d'enfants, which are often called ribbon graphs or cell graphs,
and appear as Feynman diagrams in matrix models. The enumeration problem of these graphs are solved by "topological recursion."
The Laplace transform of the number of cell graphs leads to global topology of the moduli space of pointed algebraic curves.
| Chang-Shou Lin (National Taiwan University, Taiwan) |
|Title: Toda system and Hypergeometric equation |
The Toda system is a nonlinear elliptic PDE of second order, but it is also an integrable
system, which could be discussed from two aspects at least: geometric aspect or the monodramic aspect.
The monodramy is related to an n-th order ODE in complex variable. For the case with three singular points,
the n-th order ODE , n is greater than 2, is very interesting case to study. There is the Beukers and Heckman theory
on hypergeometric equation, or more general, the theory of local rigidity. I will review these theory and see how to connect
the Toda system and those ODE theory.
The talk will try to be a survey without too many technical details.
| Motohico Mulase (UC Davis, USA) |
|Title: Quantization of Hitchin spectral curves, opers, and their WKB analysis via topological recursion
|Abstract: A new idea of quantization of Hitchin spectral curves for Higgs bundles is introduced.
The quantization is motivated by my recent joint paper with Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo
and Andrew Neitzke. The quantum spectral curve is realized as an "oper."
We then give all-order WKB analysis of opers through a PDE version of the topological recursion for the group SL(2).
| Akane Nakamura (Josai University, Japan) |
|Title: Isospectral limit of the Painlev\'e-type equations and degeneration of curves |
My work is motivated by a list of 40 types of 4-dimensional Painlev\'e-type equations derived
from isomodromic deformation and degeneration process (Sakai, Kawakami-Nakamura-Sakai, Kawakami).
My goal is to characterize these systems in a geometrical way.
I deal with the integrable systems derived as the isospectral limit of these Painlev\'e-type equations
and consider the degeneration of the theta divisors of their Liouville tori in two different ways.
| Masa-Hiko Saito (Kobe University) |
Moduli spaces of Vector bundles, Higgs bundles and Connections on curves. |
This is a joint work with Szilard Szabo in Budapest.
We have been interested in isomonodromic differential equations (ISD)
associated to singular connections on curves from the view point of
algebro-geometric constructions of moduli spaces of parabolic
connections on curves. The main advantage of this approach is the fact
that one can understand the Painleve property of ISDs in clear geometric
pictures of Riemann-Hilbert correspondences between the
moduli space of connections and the moduli space of monodromy data.
Moreover this clear geometric picture of RHC leads to a very clear proof
of Painleve property of ISD under a deep fact that Riemann-Hilbert
correspondence is a proper analytic bimeromorphic map.
However most people would like to look at the ISD in the coordinates,
so we need nice analytic coordinates of the phase space, that is,
the moduli space of parabolic connections. I will talk about our theory
of canonical coordinates associated to apparent singularities
of connections and Higgs bundles. We will also explain about a difference
of Higgs bundles case and connections case. Some problems on moduli
space of vector bundles related to this problem will be also explained.
| Siye Wu (National Tsing Hua University, Hsinchu, Taiwan) |
|Title: Non-orientable surfaces and electric-magnetic duality |
We consider a twisted N=4 gauge theory on a 4-dimensional spacetime
that incorporates a non-orientable surface and show that it reduces to a sigma-model on a worldsheet with boundary,
where branes appear naturally. Special attention is paid to the topology of the bundle over the 4-manifold
and its interpretation from the 2-dimensional point of view in terms of Hitchin's moduli spaces.