Graduate School of Science Bldg. No.3, Room 108
(map)

For inquiries, please contact the
organizer So.Okada@gmail.com.

12/17 (Monday)

14:45-16:15 Jack
Huizenga (UI Chicago)

*Birational geometry of the Hilbert scheme of points in the plane
and Bridgeland stability*

16:30-18:00
Tom Sutherland (U. Sheffield & U. Oxford)

*Quadratic differentials as stability conditions*

12/18 (Tuesday)

09:00-10:30
Akishi Ikeda (U. Tokyo)

*The space of stability conditions for preprojective algebras of
m-Kronecker quivers*

10:45-12:15 Jack
Huizenga (UI Chicago)

*Interpolation for vector bundles and divisors on the Hilbert
scheme of points in the plane*

13:15-14:45
Tom Sutherland (U. Sheffield & U. Oxford)

*Stability conditions for Painlevé quivers*

12/19 (Wednesday)

10:30-12:00
Takahisa
Shiina (Kogakuin U.)

*The space of stability conditions for quivers*

13:00-14:30
Kotaro Kawatani (Nagoya U.)

* A hyperbolic metric on the space of stability conditions on K3
surface with $\rho=1$ and applications*

14:45-16:15
Zeng Hua (Kansas State U. & Chinese U. of HK)

*Spin structure on moduli space of sheaves on CY 3-folds*

Jack Huizenga

Title : Birational geometry of the Hilbert scheme of points in the
plane and Bridgeland stability

Abstract : The Hilbert scheme of $n$ points in the projective plane
parameterizes zero-dimensional subschemes of length $n$. An
interesting problem is to describe the birational geometry of this
space, and give modular interpretations for its various birational
models. Recent work with Daniele Arcara, Aaron Bertram, and Izzet
Coskun carries out this program, at least for small $n$. We study
the stable base locus decomposition of the effective cone of the
Hilbert scheme, and observe that this decomposition can be
understood via a similar decomposition of a space of Bridgeland
stability conditions. We formulate a precise conjectural
correspondence between the two pictures, and observe this
correspondence holds in some small examples.

Title : Interpolation for vector bundles and divisors on the Hilbert
scheme of points in the plane

Abstract : A first step in understanding the birational geometry of
the Hilbert scheme of points in the plane is to determine the cone
of effective divisors. A natural way to construct effective
divisors on the Hilbert scheme is to consider loci of schemes which
fail to impose independent conditions on sections of a vector bundle
on the plane. We construct the effective cone by showing general
collections of points impose independent conditions on sections of
suitable vector bundles. Along the way, we will be lead to consider
a natural generalization of Gaeta's theorem on the resolution of a
general ideal sheaf of n points. This resolution has a natural
interpretation in terms of Bridgeland stability, and we observe that
general ideal sheaves are always destabilized by exceptional
bundles. We also verify that our computation of the effective cone
is consistent with the conjectural correspondence between the stable
base locus decomposition and the decomposition of the space of
Bridgeland stability conditions.

Akishi Ikeda

Title : The space of stability conditions for preprojective algebras
of $m$-Kronecker quivers

Abstract : We study the space of stability conditions on a derived
category of a preprojective algebra associated with the m-Kronecker
quiver. We show that there is a connected component which is a
covering space of some open subset of the Cartan subalgebra of the
Kac-Moody Lie algebra defined by the underlying graph of this
quiver. This is a generalization of Bridgeland's result where
underlying graphs are finite or affine type.

Tom Sutherland

Title : Quadratic differentials as stability conditions

Abstract : This talk will outline how to interpret spaces of
meromorphic quadratic differentials on Riemann surfaces as spaces of
stability conditions for certain CY3 triangulated categories. We
will show how a generic quadratic differential determines via its
trajectories a quiver with potential drawn on the surface following
Gaiotto, Moore and Neitzke. There are codimension 1 walls in the
space of quadratic differentials with fixed pole orders and simple
zeroes along which the quiver with potential jumps by a mutation.
This suggests viewing this space as a space of stability conditions
of the bounded derived category of modules of the Ginzburg dg
algebra of the quiver with potential, which is the subject of
upcoming work of Bridgeland and Smith.

Title : Stability conditions for Painlevé quivers

Abstract : We consider some of the simplest examples of spaces of
stability conditions in the previous talk, namely those associated
to quivers with potential of rank two drawn on the Riemann sphere.
The mutation-equivalence classes of these quivers are in bijection
with the Painlevé equations describing isomonodromic
deformations of flat SL(2,C)-connections with certain (possibly
irregular) singularities on the Riemann sphere. We will describe a
connected component of the space of numerical stability conditions
associated to these "Painlevé Quivers" more explicitly by
considering hypergeometric equations satisfied by the periods of the
Seiberg-Witten differential on the family of spectral elliptic
curves. Finally we will interpret the spaces of initial conditions
of the Painlevé equations as cluster X-varieties of these
quivers.

Takahisa Shiina

Title : The space of stability conditions for quivers

Abstract : I will talk about the space of stability conditions on a
derived category of an abelian category of finite dimensional
representations on the quiver. The quiver considered in is the one
with two vertices
(arXiv.math: 1108.2099)
and the one with three vertices. I explain the shape of the orbit
space by C-action and the local homeomorphism map. Besides, I
introduce the work of Macrì-"Stability conditions on curves"
(arXiv.math:0705.3794).

Kotaro Kawatani

Title : A hyperbolic metric on the space of stability conditions on
K3 surface with $\rho=1$ and applications

Abstract : The main aim of this talk is to give a review of my
recent paper
arXiv.math:1204.1128
for experts of Bridgeland stability conditions. More precisely I
introduce a hyperbolic metric on the (normalized) space of stability
conditions on K3 surface with $\rho=1$. Furthermore I give some
applications of the hyperbolic metric.

Zheng Hua

Title : Spin structure on moduli space of sheaves on CY 3-folds

Abstract : Konstevich and Soibelman's orientation data is roughly
speaking a consistent choice of spin structure on moduli space of
objects on CY 3-folds. Such spin structure is necessary for
constructing any refined version of Donaldson-Thomas invariant. I
will discuss some recent progress on this subject.

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