Title: Summer school on "Geometry of Teichmuller spaces and moduli spaces of curves"
Time: July 14 -- July 20, 2008.
The venue: Center of Mathematical Sciences, Zhejiang University,Hangzhou
The organizers:
Lizhen Ji (Michigan University)
Kefeng Liu (UCLA & Zhejiang University)
Shing-Tung Yau (Harvard & Zhejiang University)
The local organizer:
Hongwei Xu (Zhejiang University)
Hao Xu (Zhejiang University)
The speakers:
Richard Hain, (Duke University)
Yi Hu, (The University of Arizona)
Jun Li, (Standford University)
Chiu-Chu Liu,(Columbia University)
Feng Luo, (Rutgers University)
Yair Minksy, (Yale University)
Pan Peng,(Harvard)
Robert Penner, (University of Southern California and Aarhus University)
Xiaofeng Sun, (Lehigh University)
Andrey Todorov,(UCSC)
Richard Wentworth, (Johns Hopkins University)
Hao Xu, CMS,(Zhejiang University)
Introduction:
Riemann surfaces are fundamental objects in complex analysis, differential and complex geometry, low dimensional topology, algebraic geometry, number theory, mathematical physics etc. There are two moduli spaces associated with Riemann surfaces. If the Riemann surfaces are marked, the corresponding moduli space is the Teichmuller space. On the other hand, the moduli space of unmarked Riemann surfaces is the moduli space of algebraic curves, which is arguably the most important and basic space in algebraic geometry. These two moduli spaces are closely related by forgetting the markings, i.e., the quotient of the Teichmuller space by the mapping class group is the moduli space of curves. These moduli spaces and their variants have played a undamental role in algebraic geometry and string theory.
In this summer school, experts from around the world will give comprehensive introductions to many different aspects of these important moduli spaces.
Speaker: Richard Hain
Abstract:
I was thinking of doing something more elementary. Namely, discussing the moduli of elliptic curves in detail; using it to motivate stacks/orbifolds; constructing the Deligne-Mumford compactification of the moduli space of elliptic curves (as a stack/orbifold) and computing the Picard groups of the moduli space and its compactification; discussing modular forms and their relation to topology. Prerequisites: complex analysis, some algebraic topology. I would review some basic facts about elliptic curves.
Speaker: Yi Hu
Title: Curves and their moduli in tropical geometry.
I will give some very elementary introduction to tropical curves (definitions, examples, and the tropical versions of classical theorems), their moduli (the genus-0 case), and their applications to enumerative geometry (such as the Kontsevich formula)
Speaker: Feng Luo
Title: Lectures on Feynman diagrams and matrix models
Abstract:
We give a brief introduction to the Feynman diagram technique used in the study of moduli space of Riemann surfaces.
In the first lecture, we introduce the notion of arc-complex and ribbon graphs. In the second lecture, we relate the ribbon graph with the matrix models and the moduli space of curves. In the last lecture, we introduce Kontsevich's proof of Harer-Zagier-Penner's theorem on the Euler characteristic of the moduli space of curves.
There will be some overlaps with Professor Penner's lectures which uses hyperbolic geometry point of view.
Speaker: Robert Penner
Abstract:
Penner
Lecture 1 Punctured surfaces
Abstract: We shall review the coordinates on and cell decomposition of the decorated Teichm\"uller space of a punctured surface emphasizing the transformation properties of the coordinates and their applications.
Lecture 2 Profinite and pro-nilpotent versions
Abstract: The basic structure of decorated Teichm\"uller space of a punctured surface extends to the punctured solenoid, a variant of the solenoid introduced by Sullivan for closed surfaces which is a natural universal object and corresponds to profinite completion of the underlying surface group; this is joint work with \v Sari\'c and Bonnot-\v Sari\'c.
There is likewise a pro-nilpotent version of the theory in the setting of Torelli-Johnson-Morita theory, which will be discussed and indeed emphasized; this is joint work with Morita and Bene-Kawazumi.
Lecture 3 Compactifications
Abstract: A natural goal is to compactify Riemann's moduli space respecting or in some way extending the combinatorial decomposition of Teichm\"uller space say in the once-punctured case.Two manifest possibilities are: to naturally extend the combinatorics and understand the resulting degenerations and ideal combinatorics; or to study the combinatorial implications of the natural degenerations to stable curves.The former approach leads to a stratified space with an interesting recursive structure, and the latter approach, which is joint work with McShane and will be emphasized,leads to an enhancement of the basic combinatorics by a Fulton-MacPherson-type construction which derives from the coordinates.
Lecture 4 Recent Applications
Abstract: Recent joint work with Andersen-Bene and Andersen-Bene-Meilhan shows that just as the classical Johnson homomorphisms and Nielsen representations extend from their usual domains to an appropriate fundamental path groupoid, so too extends the Le-Murakami-Ohtsuki invariant say of homology cylinders.This will be briefly discussed as the computationally definitive theorem is still forthcoming.Another joint project with Andersen-Knudsen-Wiuf has applied the basic combinatorics of decorated Teichm\"uller space to a model for folded proteins in
bioinformatics with positive results.The latter topic will be emphasized in this first-ever public lecture on this project.
Here are three papers on which the first couple of lectures will be based:
``The decorated Teichm\"uller space of punctured surfaces",
{\it Communications in Mathematical Physics}{\bf 113}(1987),299-339.
``Perturbative series and the moduli space of Riemann surfaces",
{\it Journal of Differential Geometry}{\bf 27}(1988),35-53.
``The simplicial compactification of Riemann's moduli space'',
Proceedings of the 37th Taniguchi Symposium, World Scientific (1996), 237-252.
And here are several papers which will be treated in the latter several lectures:
(with G. McShane) ``Stable curves and screens on fatgraphs'', math.GT/0707.1468, preprint (2007).
``The structure and singularities of arc complexes'' (2004), math.GT/0410603, revision to appear Journal of Topology.
(with S. Morita)``Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves'', math.GT/0602461, to appear Mathematical Proceedings Cambridge Philosophical Society.
(with A. Bene and N. Kawazumi) ``Fatgraph Magnus expansions and groupoid Johnson maps'', preprint (2007), math.GT/0710.2651.
(with J. Andersen and A. Bene) ``Groupoid Lifts of Mapping Class Representations for Bordered Surfaces'', math.GT/0710.2651, preprint (2007).
Speaker: Richard Wentworth
Title: The Action of the Mapping Class Group on Representation Varieties
Abstract:
I was thinking about talking about surface group representations and mapping class group actions. It would involve a little analysis some harmonic maps, Higgs bundles, Atiyah-Bott style Morse theory and some things about mapping class groups and such. This would be a little removed from the main thread of the summer school, namely Weil-Petersson geometry, but it is closer to what I'm working on right now and I feel I'd be better able to make a coherent set of lectures.
Speaker:Jun Li
Title: Moduli space of curves
I will cover the basic property of the moduli of curves from the angle of
algebraic geometry. The topics include:
1. coarse moduli space of curves;
2. local property of the moduli space of curves;
3. global property of moduli spaces.
Speaker: C.C. Liu
Lecture 1 (Wed)
Title: Equivariant Cohomology and Localization
Example: CP^n.
Lecture 2 (Fri) and 3 (Sat)
Title:Using localization on moduli space of relative stable maps to CP^1 to derive some Hodge
integral identities
These two talks are after Li's 4 talks and Xu's first 2 talks, so it is likely that Hodge integrals
will be defined before I give these two talks; These two talks are after Li's 4 talks and Peng's
first talk, so it is possible that Gromov-Witten invariants will be defined before I give these two talks.
In any case, I will adjust my talks based on the exact contents of Li, Peng, and Xu's talks.
Speaker: Hao Xu
Title: Hurwitz-Hodge Integrals
Abstract: We will introduce Hurwitz-Hodge integrals. The main references are two papers
Jian Zhou, On computations of Hurwitz-Hodge integrals, arXiv:0710.1679.
T. J. Jarvis, T. Kimura, Orbifold quantum cohomology of the classifying space of a finite group, arXiv: 0112037.