Monday (July 18):
Chair:C.S. Seshadri
9:3010:30: V. Balaji
11:0012:00: Y. Hu
Lunce break
1:302:30: V. Mehta
Introductory lecture: (only students attend)
2:454:15: Yi Hu
4:306:00: Xiaowei Wang
Tuesday (July 19):
Chair: R. Thomas
9:3010:30: A. Bayer
11:0012:00: XW. Wang
Lunch breqak
Chair: KF. Liu
1:302:30: ST. Yau
2:453:45 K. Yoshioka
4:155:15: XT. Sun
Banquet
Wednesday (July 20):
Chair: J. Li
9:0010:00: YH. Kiem
10:3011:30: ZB. Qin
Vikraman Balaji: Parahoric bundles and parabolic bundles
Description: In this talk I will discuss some recent work of mine with Seshadri. Let $X$ be an irreducible smooth projective algebraic curve of genus $g \geq 2$ over the ground field $\bc$ and let $G$ be a semisimple simply connected algebraic group. We introduce the notion of a {\em semistable and stable parahoric} torsor under a certain BruhatTits group scheme $\mathcal G$, construct the moduli space of semistable parahoric $\mathcal G$torsors and identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of $G$. The results give a complete generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.
Arend Bayer: Bridgeland stability conditions on threefolds and a conjectural
BogomolovGieseker inequality
Description: I will explain a conjectural construction of Bridgeland stability conditions for the derived category any smooth projective threefold. It is based on a conjectured BogomolovGieseker type inequality for the third Chern character of "tiltstable" twoterm complexes.
The conjectured BogomolovGieseker inequality turns out to be interesting by itself, as it implies a Reidertype theorem on freeness and very ampleness of adjoint line bundles, and a version of Fujita's conjecture. This is based on joint work with Aaron Bertram, Emanuele Macri and Yukinobu Toda.
Yi Hu, Modular desingularization of moduli of genustwo stable maps
Description: For the moduli space of genusg stable maps into a projective space, we describe the local structures of certain associated canonical derived objects and hence that of the moduli space itself, using some moduli parameters. For genustwo (as well as genusone), this enables us to provide an explicit modular desingularization of the moduli of stable maps.
YoungHoon Kiem: Moduli spaces of quasimaps with perfect obstruction theories
Description: A quasimap to projective space P^{n1} refers to a line bundle over a curve together with n ordered sections. I will introduce the notion of deltastable quasimaps and show that deltastable quasimaps form a proper separated DeligneMumford stack for each value of delta>0 except for a finite set of walls. I will also talk about the GuffinSharpeWitten models and the wall crossing behaviors.
Vikram Mehta: Stratified Bundles on Simply Connected Varieties in char p
Description: Gieseker had conjectured that there are no stratified bundles
on simply connected varieties in char p. We prove this conjecture, using
the boundedness theorems of Langer and the work of Hrushovski on the first
order theory of the Frobenius automorphism. This is a joint work with Esnault.
Ian Morrison: GIT and birational geometry of moduli spaces of curves
Description: In the mid1970s, Gieseker explained how to construct the moduli space $\overline{M}_g$ of stable curves as a GIT quotient of the locus of pluricanonical Hilbert points. After a long hiatus, variations on his ideas have recently reemerged as tools for studying questions about the birational geometry of $\overline{M}_g$ and related moduli spaces of pointed curves and maps. In the talk, I will review the applications to the log minimal model program and the $F$conjectrue.
Zhenbo Qin: GiesekerUhlenbeck morphisms and extremal GromovWitten invariants
Description: Using techniques of KiemJun Li regarding localized virtual fundamental cycles in GromovWitten theory, we determine 1point extremal GromovWitten invariants of the Gieseker moduli spaces of sheaves over the projective plane.
Xiaotao Sun: Stratified bundles and ètale fundamental group
Description: This is a joint work with H. Esnault. Let X be a smooth projective variety
over an algebraically closed field of characteristic p>0, we show the following:
(1) the commutator of etale fundamental group is a propgroup if and only if all of irreducible straitified bundles on X has have rank one.
(2) the category of stratified bundles on X is semisimple with irreducible objects of rank one if and only if the etale fundamental group of X is abelian without ppower quotient.
Xiaowei Wang: Hilbert Mumford Criterion of nodal curve
Description: We prove the GIT stability of connected nodal curve by directly apply Hilbert Mumford criterion. Also several applications will be discussed.
Shingtung Yau: NonKahler CalabiYau Manifolds
Description: We will discuss the existence of special metrics on bundles over nonKahler CalabiYau threefolds.
Kota Yoshioka: Bridgeland stability conditions and FourierMukai transforms
Description: Bridgeland stability condition is preserved under the FourierMukai transform by its definition. I will explain the relation with Gieseker stability. In particular, I will explain known results on the birational maps of moduli spaces by using Bridgeland stability condition.
Introductory Lectures
Yi Hu: An introduction to Geometric Invariant Theory
Description: In the first part of the talk, I will provide motivations, basic definitions and some general theorems (including the variation of Geometric Invariant Theory quotients), using elementary examples. In the second part of the talk, I will focus on the applications, mainly on factorization Theorem in birational geometry.
Xiaowei Wang: Numerical criterion of stability
Description: I show how numerical criterion of stability can lead to the proof that SLOPE stable bundle are GIT stable, and SMOOTH curves are (GIT) Hilbert stable.
