CONTEMPORARY MATHEMATICS

150

Mapping Class Groups and Moduli Spaces

of Riemann Surfaces Proceedings of Workshops held June 24-28, 1991 , in Gottingen, Germany, and August 6-10, 1991,

in Seattle, Washington

Carl-Friedrich B6digheimer Richard M. Hain

Editors

Recent Titles in This Series

1 50 Carl-Friedrich Bikligheimer and Richard M. Hain, Editors, Mapping class groups and moduli spaces of Riemann surfaces, 1993

149 Harry Cohn, Editor, Doeblin and modem probability, 1993 148 Jeffrey Fox and Peter Haskell, Editors, Index theory and operator algebras, 1993 147 NeD Robertson and Paul Seymour, Editors, Graph structure theory, 1993 146 Martin C. Tangon, Editor, Algebraic topology, 1993 145 Jeffrey Adams, Rebecca Herb, Stephen Kudla, Jian-Shu Li, Ron Lipsman, Jonathan

Rosenberg, Editors, Representation theory of groups and algebras, 1993 144 Bor-Luh Un and William B. Johnson, Editors, Banach spaces, 1993 143 Marvin Knopp and Mark Sheingorn, Editors, A tribute to Emil Grosswald: Number

theory and related analysis, 1993 142 Chung-Chun Yang and Sheng Gong, Editors, Several complex variables in China, 1993 141 A. Y. Cheer and C. P. van Dam, Editors, Fluid dynamics in biology, 1993 140 Eric L. Grinberg, Editor, Geometric analysis, 1992 139 Vinay Deodhar, Editor, Kazhdan-Lusztig theory and related topics, 1992 138 Donald St. P. Richards, Editor, Hypergeometric functions on domains of positivity, Jack

polynomials, and applications, 1992 137 Alexander Nagel and Edgar Lee Stout, Editors, The Madison symposium on complex

analysis, 1992 136 Ron Donagi, Editor, Curves, Jacobians, and Abelian varieties, 1992 135 Peter Walters, Editor, Symbolic dynamics and its applications, 1992 134 Murray Gerstenhaber and Jim Stashefl', Editors, Deformation theory and quantum

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noncommutative rings, 1992 129 John R. Graef and Jack K. Hale, Editors, Oscillation and dynamics in delay equations,

1992 128 Ridgley Lange and Shengwang Wang, New approaches in spectral decomposition, 1992 127 Vladimir Oliker and Andrejs Treibergs, Editors, Geometry and nonlinear partial

differential equations, 1992. 126 R. Keith Dennis, Claudio Pedrini, and Michael R. Stein, Editors, Algebraic K-theory,

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http://dx.doi.org/10.1090/conm/150

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150

Mapping Class Groups and Moduli Spaces of Riemann Surfaces

Proceedings of Workshops held June 24-28, 1991, in Gottingen, Germany, and August 6-10, 1991,

in Seattle, Washington with support from the Sonderforschungsbereich 170 "Geometrie und Analysis" and

the National Science Foundation

Carl-Friedrich Bodigheimer Richard M. Hain

Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL BOARD

Craig Huneke, managing editor Clark Robinson J. T. Stafford Linda Preiss Rothschild Peter M. Winkler

The Workshops on Mapping Class Groups and Moduli Spaces of Riemann Surfaces were held at the Mathematical Institute of the University of Gottingen, Gottingen, Germany on June 24-28, 1991 with support from the Sonderforschungsbereich 170 "Geometrie und Analysis" and at the University of Washington, Seattle on August 6-10, 1991 with support from the University of Washington, Seattle and the National Science Foundation, Grant DMS-9108213.

1991 Mathematics Subject Classification. Primary 14H10, 30F60; Secondary 14-06, 20-06, 55-06.

Library of Congress Cataloging-in-Publication Data

Mapping class groups and moduli spaces of Riemann surfaces: proceedings of workshops held June 24-28, 1991 and August 6-10, 1991 in Gottingen, Germany and Seattle, Washington ... /Carl-Friedrich Booigheimer, Richard M. Hain, editors.

p. cm.-(Contemporary mathematics; v. 150) ISBN 0-8218-5167-5 1. Riemann surfaces-Congresses. 2. Class groups (Mathematics)-Congresses.

3. Moduli theory-Congresses. I. Bodigheimer, Carl-Friedrich, 1956- . II. Hain, Richard M. (Richard Martin), 1953- . III. Workshop on Mapping Class Groups and Moduli Spaces of Riemann Surfaces (1991: Gottingen, Germany, and Seattle, Wash.) IV. Series: Contemporary mathematics (American Mathematical Society); v. 150. QA333.M37 1993 5151 .9223--dc20

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CONTENTs·

Preface vii

Introduction ix

Participants xiii

L~tof~ ~

A combinatorial approach to reducibility of mapping classes D. BENARDETE, M. GUTIERREZ, Z. NITECKI 1

Interval exchange spaces and moduli spaces C.-F. BODIGHEIMER 33

Cohomology of the group of motions of n strings in 3-space A. BROWNSTEIN, R. LEE 51

Mapping class groups and classical homotopy theory F. COHEN 63

Completions of mapping class groups and the cycle C - c-R. HAIN 75

The rational Picard group of the moduli space of Riemann surfaces with spin structure

J. HARER 107

On certain families of compact Riemann surfaces W. HARVEY 137

On the homology stability for Teichmiiller modular groups: closed surfaces and twisted coefficients

N. IVANOV 149

Higher Franz-Reideme~ter torsion: low dimensional applications J. KLEIN 195

Cohomology of Ms and Ml E. LOOIJENGA 205

Logarithmic law for geodesics in moduli space H. MASUR 229

v

vi CONTENTS

Riemann's moduli space and the symmetric groups J. MILGRAM, R. PENNER 247

Primitive Mumford classes J. MORAVA 291

The structure of the mapping class group and characteristic classes of surface bundles

S. MORITA 303

Subvarieties of moduli spaces of curves: open problems from an algebra-geometric point of view

I. MORRISON 317

£ 2-cohomology of the Weil-Petersson metric L. SAPER 345

On the moduli space of principally polarized abelian varieties W. WANG 361

The Weil-Petersson volume of the moduli space of punctured spheres P. ZOGRAF 367

PREFACE

During the summer of 1991, there were two workshops on mapping class groups and moduli spaces of Riemann surfaces. They were held independently in Gottingen (June 24-28) and Seattle (August 6-10). This volume contains the joint proceedings of those two meetings, as well as a previously unpublished manuscript of John Harer.

One can approach mapping class groups and moduli spaces from radically differ-ent perspectives (e.g. geometric topology, dynamical systems, algebraic geometry, group theory, physics ... ) and with correspondingly dissimilar tools. Our goal in organizing the two conferences was to attempt to bring these points of view closer together. We hope this volume will further contribute to achieving this goal.

The workshop in Gottingen was financed by the Sonderforschungsbereich 170 "GEOMETRIE UND ANALYSIS" at the Mathematical Institute of the University of Gottingen. The workshop in Seattle was supported by the National Science Foun-dation and the University of Washington. Without the support of these institutions, these conferences would not have been possible, and we would like to take this op-portunity to thank them for their generous support. The editors would also like to thank those who devoted their time and energy to refereeing the papers and other-wise helping with the preparation of this volume. Finally, we would like to thank Donna Harmon and Christine Thivierge of the American Mathematical Society who ably helped us assemble this volume.

All papers in this volume are in final form and no version of any will be submitted for publication elsewhere.*

Carl-Friedrich Bodigheimer Richard Hain

February 1993

*Except the papers by D. Benardete, M. Gutierrez, and Z. Nitecki, and S. Morita.

vii

INTRODUCTION

We now give a terse overview of mapping class groups and moduli spaces of Riemann surfaces, and their connections with various fields. Further information can be found in the book by Joan Birman [3], and the survey papers of Eisenbud and Harris [6], Harer [11], Harris [13], Ivanov [15], McMullen [19], Thurston [23] and Wolpert [26].

Fix a compact oriented surface S of genus g. The mapping class group asso-ciated to s, denoted r 9• is the group of isotopy classes of orientation preserving diffeomorphisms of S. The study of this group began with the work of Dehn [4] and Nielsen [20]. Mapping class groups are, in some sense, generalizations of braid groups.

One original impetus for studying mapping class groups comes from the study of 3-manifolds. One can show that every 3-manifold can be constructed from the 3-sphere by splitting the sphere into two handlebodies along an imbedded surface S and then reglueing the pieces by composing the original identification map with an element of the mapping class group of S. This point of view has undergone a renaissance with the advent of topological quantum field theories (e.g., (21]).

A marked Riemann surface is a compact Riemann surface (i.e., a compact !-dimensional complex manifold) X and an orientation preserving diffeomorphism f : S-+ X. Two marked Riemann surfaces II : S-+ X1 and h : S-+ X2 are equivalent if there is a conformal homeomorphism 1/J: x1 -+ x2 (i.e., a biholomor-phism) such that 1/J o II is isotopic to f2. The set of equivalence classes of marked Riemann surfaces naturally forms a metric space Tg, which, by a fundamental result of Teichmiiller (22], has a real analytic structure and is real analytically equivalent to R.69-6 when g ~ 2 and R.2 when g = 1. Alhfors [2]subsequently showed that Tg has a natural complex analytic structure whose underlying real analytic structure is the one discovered by Teichmiiller.

There are several equivalent ways to define the topology on Tg. The Teichmiiller distance between two marked Riemann surfaces [II: S-+ X1] and [h: S-+ X2] is defined to be (logK['I/J]) /2, where 1/J : X1 -+ X2 is a quasiconformal homeo-morphism whose dilatation K['I/J] is minimal among all quasiconformal homeomor-phisms X 1 -+ X2 which are homotopic to h o /}1 • Alternatively, when g ~ 2, the universal covering of each Riemann surface of genus g is the upper half plane. Consequently, each marked Riemann surface f : S -+ X determines a homomor-phism 1r1(S) -+ PSL2(R.) which is well defined up to conjugation by elements of PSL2(R.). In this way, one obtains a one-to-one function

Tg <-+ Hom(1r1(S),PSL2(R.))jPSL2(R).

If we give the right hand side the quotient topology, then this injection takes Tg homeomorphically onto one component of the right hand side. Another important method of describing the topology on Tg when g ~ 2 is to use Fenchel-Nielsen coordinates. These are described, for example, in [1] and [7, expose 7].

The group r 9 acts on Tg on the left as a group of biholomorphisms: the mapping class [4>] e r 9 of the diffeomorphism 4> of S takes the class of f : S -+ X to the class of f o q,-1. The quotient space r 9 \ Tg is the set of isomorphism classes of complex structures on S. It is called the moduli space of compact Riemann surfaces

ix

X INTRODUCTION

of genus g and is usually denoted by Mg. It follows from standard Riemann surface theory that this is also the moduli space of smooth projective curves over C. The isotropy group of a point is the group of automorphisms of the corresponding Riemann surface. One can deduce from this that the mapping class group has a subgroup of finite index which acts freely on Tg, so that Mg is an analytic space with only finite quotient singularities. Deligne and Mumford [5) proved that Mg is, in fact, an irreducible quasi-projective algebraic variety.1 They also showed that the moduli space Mg of stable curves of genus g is projective, and that it is a natural compactification of Mg where Mg - Mg is locally the quotient of a divisor with normal crossings by a finite group.

When g = 1, the mapping class group is SL2(Z), 1i. is biholomorphic to the upper half plane, the Teichmiiller distance between two points is the distance between them in the Poincare metric, and the action of r 1 on 1i. is the standard action of SL2(Z) on it. The quotient M 1 is the j-line which is isomorphic to C, and its Deligne-Mumford compactification M 1 is isomorphic to the Riemann sphere JPI1.

Since every Riemann surface of genus 2 is hyperelliptic, M 2 is isomorphic to the moduli space of unordered 6-tuples of distinct points in JPI1, modulo projec-tive equivalence. Since we can always find a unique automorphism of JP11 which takes any three of the points onto {0, 1, oo }, it follows that M 2 is the quotient of (C- {0, 1} )3 - a by the natural action of the symmetric group on 6 letters. Here a is the subset of (C- {0, 1})3 consisting of those 3-tuples where at least 2 of the coordinates are equal. As g increases, it becomes more and more difficult to give such explicit descriptions of Mg.

The action of r g on Tg is analogous to the action of an arithmetic group on a hermitian symmetric space - the analogue of the natural symmetric space metric being the Weil-Petersson metric on Teichmiiller space. The Thurston boundary [7), to which the action of r g on Tg extends, is the analogue of the spherical boundary of a symmetric space. The analogy between mapping class groups and arithmetic groups has proved to be very fruitful. Thurston has used his compactification of Teichmiiller space to classify diffeomorphisms of a surface [7) and to prove that cer-tain 3-manifolds have hyperbolic structures (cf. [19]). Harer [10) has constructed an analogue for mapping class groups of the building associated to an arithmetic group and used it to prove that mapping class groups have three important prop-erties enjoyed by arithmetic groups: that their cohomology stabilizes as g --+ oo [9); that each rg is a virtual duality group; and, with Zagier [12), that the orbifold Euler characteristic of Mg is the value ((1- 2g) of the Riemann zeta function. These results may lead one to suspect that r g is arithmetic, or at least has a finite index subgroup which is. However, when g ~ 2, rg is not arithmetic [14, 11), and has no finite index subgroup which is (see [16]).

Since the mapping class group r g acts virtually freely on Teichmiiller space (i.e., it has a subgroup of finite index which acts freely), and since Teichmiiller space is

1 It seems difficult to give the correct attributions here. The variety M 9 is defined over all fields. The irreducibility of M 9 over the complex numbers follows from Teichmiiller theory; the difficult part of the theorem of Deligne and Mumford is to prove irreducibility in finite characteristic. Also, the quasi-projectivity of Mg over the complex numbers seems to follow from the Torelli theorem and the fact that the moduli spaces of principally polarized abelian varieties are quasi-projective.

INTRODUCTION xi

contractible, there is a natural isomorphism

That is, the computation of the rational group cohomology of r 9 is the same aB

the computation of the rational cohomology of the moduli space of curves of genus g. A knowledge of the topology of this space is important in algebraic geometry. Examples of applications of topology to algebraic geometry can be found among the results of John Harer. Perhaps the most striking is his computation of H 2(r 9 ; Q); it is one dimensional when g ~ 5 [8). This implies that the Picard group of M 9 ,

tensored with Q, is isomorphic to Q when g ~ 5. The mapping class group r 9 haB a natural subgroup, the Torelli group T9 • It con-

sists of those mapping classes of diffeomorphisms of the surface S which act trivially on its homology. That is, T9 is the kernel of the natural surjective homomorphism

r 9 -+ SP29 (Z).

Since T9 measures the difference between r 9 and Sp29 (Z), it should contain a lot of deep geometric information about surfaces. Much of what is known about this mysterious group was discovered by Dennis Johnson. His results are surveyed in [17).

In the last few years an extraordinary connection between physics and the topol-ogy and geometry of moduli spaces haB developed through the ideas of Witten in conformal field theory (see, e.g., [24]). The most remarkable example of this con-nection is a conjecture of Witten [25) and its solution by Kontsevich [18). The conjecture states that a suitable generating function constructed from the intersec-tion numbers of certain natural divisors on the moduli spaces of n-pointed curves is a solution of the KP hierarchy. This allows the computation of these intersection numbers which are of interest in the theory of algebraic curves.

REFERENCES

1. W. Abikoff: The Reo.l Analytic ThflOf'JI of Teichmuller Space, LNM 820, Springer-Verlag, 1980. 2. L. V. Ahlfors: The complex analytic structure of the space of closed Riemann surfaces, in:

Analytic Functions, Proceedings of a Conference, Princeton 1957, R. Nevanlinna et al. (eds.), Princeton University Press (1960).

3. J. Birman: Braids, Links and Mapping Class Groups, Annals of Math. Studies 82, Princeton University Press, 1975.

4. M. Dehn: Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-206. 5. P. Deligne, D. Mumford: The irreducibility of the space of curoes of a given genus, Publ.

Math. IHES 36 (1969), 75-109. 6. D. Eisenbud, J. Harris: Progress in the theory of complex algebraic curoes, Bull. Amer. Math.

Soc. 21 (1989), 205-232. 7. A. Fathi, F. Laudenbach, V. Poenaru: 7tuvaw: de Thurston sur les Surfaces, Asterisque 66&

67 (1979). 8. J. Harer: The second homolog11 group of the mapping class group of an orientable surface,

Invent. Math. 72 (1983), 221-239. 9. J. Harer: Stabilit11 of the homology of the mapping class groups of orientable surfaces, Ann.

Math. 121 (1985), 215-249. 10. J. Harer: The virtual cohomological dimension of the mapping class group of an orientable

surface, Invent. Math. 84 (1986), 157-176. 11. J. Harer: The cohomology of the moduli space of curoes, in: Theory of Moduli, E. Sernesi

(ed.), LNM 1337, Springer-Verlag, Berlin, Heidelberg, New York, 1988, 139--221.

xii INTRODUCTION

12. J. Harer, D. Za.gier: The Euler chamcteristic of the moduli space of curves, Invent. Math. 85 (1986), 457-485.

13. J. Harris: Curves and their moduli, Algebraic Geometry Bowdin 1985, Proc. Symp. Pure Math. 46 (1987), 99-143.

14. N. Ivanov: Algebmic properties of mapping class groups of surfaces, in Geometric and Al-gebraic Topology, Banach Center Publications vol. 18, Polish Scientific Publishers, Warsaw 1986, 15-35.

15. N. Ivanov: Complexes of curves and the Teichmiiller modular group, Uspekhi Mat. Nauk 42 (1987), 49-91; English translation: Russian Math. Surveys 42 (1987),55-107.

16. N. Ivanov: Teichmiiller modular groups and arithmetic groups, Research in Topology 6, Notes of LOMI scientific seminars, V.167 (1988}, 95-110. English translation: J. Soviet Math., 52, (1990), 2809-2818.

17. D. Johnson: A survey of the Torelli group, Contemp. Math. 20 (1983), 165-179. 18. M. Kontsevich: Intersection theory on the moduli space of curves, Funk. Anal. Prilozh. 25

(1991), 5Q-57. 19. C. McMullen: Riemann surfaces and geometrization of9-manifolds, Bull. Amer. Math. Soc.

27 (1992), 207-216. 20. J. Nielsen: Untersuchungen zur Topologie der zweiseitigen Fliichen, I-III, Acta Math. 50

(1927), 189-358; 53 (1929), 1-76; 58 (1932), 87-167. 21. N. Reshetikhin, V. Turaev: Invariants of 9-manifolds via link polynomials and quantum

groups, Invent. Math. 103 (1991), 547-597. 22. 0. Teichmiiller: Extremale quasikonforme Abbildungen und quadmtische Differentiate, Abh.

Preuss. Akad. Wiss. 22 (1939), 3-197. 23. W. Thurston: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull.

Amer. Math. Soc. 6 (1982), 357-381. 24. E. Witten: Quantum field theory, Gmssmannians and algebmic curves, Commun. Math.

Phys. 113 (1988), 529-600. 25. E. Witten: Two-dimensional gmvity and intersection theory on moduli space, Surveys in Diff.

Geom. 1 (1991), 243-310. 26. S. Wolpert: The topology and geometry of the moduli space of Riemann surfaces, Arbeitsta-

gung Bonn 1984, F. Hirzebruch, J. Schwermer, S. Suter (eds.), LNM 1111, Springer-Verlag, 431-451.

Carl-Friedrich Bodigheimer Richard Hain

PARTICIPANTS

GOTTINGEN PARTICIPANTS:

Diego Benardete Department of Mathematics Trinity College Hartford, CT 06106, USA

Carl-Friedrich Bodigheimer Mathematisches Institut Universitat Gottingen 3400 Gottingen, Germany

Alan Brownstein

Ralf Ehrenfried

Thomas Fiedler

Richard Hain

Hans-Werner Henn

Michael Keane

John Klein

Maxim Kontsevich

Department of Mathematics Rutgers University Newark, NJ 07102, USA

Mathematisches Institut Universitat Gottingen 3400 Gottingen, Germany

Akademie der Wissenschaften Berlin Current address: SFB 170 "Geometrie und Analysis" Mathematisches Institut Universitat Gottingen 3400 Gottingen, Germany

Department of Mathematics Duke University Durham, NC 27706, USA

Mathematisches Institut Universitat Heidelberg 6900 Heidelberg, Germany

Department of Mathematics Delft University of Technology 2628 BL Delft, The Netherlands

Fachbereich Mathematik Universitii.t Siegen 5900 Siegen, Germany

Institute for Problems of Information Transmission Academy of Sciences of the USSR Current address: Max-Planck-Institut fiir Mathematik 5300 Bonn, Germany

xiii

xiv

Ronnie Lee

Gregor Masbaum

James Milgram

Jack Morava

Shigeyuki Morita

Zbigniew Nitecki

Scott Wolpert

Yining Xia

Peter Zograf

PARTICIPANTS

Department of Mathematics Yale University New Haven, CT 06520, USA

URA CNRS Mathematiques Universite de Nantes 44072 Nantes, France

Department of Mathematics Stanford University Stanford, CA 94305, USA

Department of Mathematics Johns-Hopkins-University Baltimore, MD 21218, USA

Department of Mathematics Tokyo Institute of Technology Tokyo 152, Japan

Department of Mathematics Tufts University Medford, MA 02155, USA

Department of Mathematics University of Maryland College Park, MD 207 42, USA

Department of Mathematics Ohio State University Columbus, OH 43210, USA

Wissenschaftskolleg Berlin Wallot Str. 19 1000 Berlin 33, Germany

PARTICIPANTS

SEATTLE PARTICIPANTS:

Carl-Friedrich BOdigheimer Mathematisches Institut Universitii.t Gottingen 3400 Gottingen, Germany

Alan Brownstein

Jim Carlson

Fred Cohen

Igor Dolgachev

Carel Faber

Henry Glover

Richard Hain

William Harvey

Ou-Henry Ho

Nicolai Ivanov

Ludmil Katzarkov

Department of Mathematics Rutgers University Newark, NJ 07102, USA

Department of Mathematics University of Utah Salt Lake City, UT 84112, USA

Department of Mathematics U Diversity of Rochester Rochester, NY 14627, USA

Department of Mathematics University of Michigan Ann Arbor, MI 48109, USA

Fakulteit Wiskunde en Informatica Universiteit Amsterdam 1018 TV Amsterdam, The Netherlands

Department of Mathematics Ohio State University Columbus, OH 43210, USA

Department of Mathematics Duke University Durham, NC 27706, USA

Department of Mathematics King's College London, WC2R 2LS, United Kingdom

P.O. Box 495 New Brunswick, NJ 08903, USA

Department of Mathematics Michigan State University East Lansing, MI 48824, USA

Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA

XV

xvi

Steven Kerckhoff

Alex Kouvidakis

Eduard Looijenga

Howard Masur

John McCarthy

Ian Morrison

Michael Nyenhuis

Tony Pantev

Chris Poor

Les Saper

Lev Slutskin

PARTICIPANTS

Department of Mathematics Stanford University Stanford, CA 94303, USA

Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA

Fakulteit Wiskunde en Informatica Universiteit Utrecht 3508 TA Utrecht, The Netherlands

Department of Mathematics University of Illinois Chicago, IL 60680, USA

Department of Mathematics Michigan State University East Lansing, MI 48824, USA

Department of Mathematics Fordham University Bronx, NY 10458, USA

Department of Mathematics University of British Columbia Vancouver, B.C. V6T 1Y4, Canada

Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA

Department of Mathematics Fordham University Bronx, NY 10458, USA

Department of Mathematics Duke University Durham, NC 27706, USA

School of Business Yeshiva University 500 West 185th Street New York, NY 10033, USA

PARTICIPANTS xvii

Neal Stoltzfus Department of Mathematics Louisiana State University Baton Rouge, LA 70803, USA

Michel Vaquie Centre de Mathematiques Ecole Normale Superieure 75230 Paris, France

Wen-Xiang Wang Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794, USA

Michael Wolf Department of Mathematics Rice University Houston, TX 77251, USA

Yining Xia Department of Mathematics Ohio State University Columbus, OH 43210, USA

Jun Yang Department of Mathematics Duke U niverisity Durham, NC 27706, USA

Jietein Yu Department of Mathematics University of Notre Dame Notre Dame, IN 46556, USA

David Yuen Department of Mathematics Colgate University Hamilton, NY 13346, USA

LIST OF TALKS

GOTTINGEN TALKS:

C.-F. BOdigheimer Moduli of surfaces and interval exchange transformations.

A. Brownstein

Th. Fiedler

R. Hain

H.-W. Henn

J. Klein

M. Kontsevich

R. Lee

G. Masbaum

J. Milgram

J. Morava

S. Morita

Z. Nitecki

S. Wolpert

Y. Xia

P. Zograf

The cohomology of the symplectic group Sp4(Z).

A simple conjugacy invariant for braids.

Rational cohomology of mapping class groups and Torelli groups.

Refinements of Quillen's description of equivariant coho-mology and applications.

Higher Franz-Reidemeister torsion with aspiring connec-tions to mapping class groups.

Witten's conjecture, matrix models and A00-structures.

Invariants of 3-manifolds and the geometry of moduli spaces.

Topological quantum field theories derived from the Kauff-man bracket.

Instantons and holomorphic 2-plane bundles on CIP'2 : the Atiyah-Jones conjecture.

Hopf algebras of operations in the cohomology of moduli spaes.

On the structure of the mapping class group of surfaces and invariants of 3-manifolds.

Braids and the Thurston-Nielsen classification.

The spectral conundrum for hyperbolic surfaces.

Farrell-Tate cohomology of the mapping class groups.

The Weil-Petersson volume of the moduli space of punc-tered spheres.

xix

XX LIST OF TALKS

SEATTLE TALKS:

C.-F. Bodigheimer Interval exchange maps, moduli of Riemann surfaces and conformal field theories.

A. Brownstein

James Carlson

F. Cohen

I. Dolgachev

C. Faber

R. Rain

W. Harvey

N. Ivanov

St. Kerckhoff

E. Looijenga

H. Masur

N. Stoltzfus

Y. Xia

J. Yang

Cohomology of the group Sp( 4, Z).

Introduction to mixed Hodge theory.

K(1r, I)'s and the cohomology of the hyperelliptic mapping class group.

New compactifications of configuration spaces.

Tautological classes.

Hodge theory and the cohomology of the mapping class groups and Torelli groups.

Fuchsian groups and Teichmiiller deformations.

Mapping class groups and Teichmiiller spaces versus arith-metic groups and symmetric spaces.

Infinitesimal deformations and Hodge theory.

Mixed Hodge structures and the Torelli group.

Metrics in the moduli space.

Biderivations and applications to braid and mapping class groups.

Torsion in the cohomology of the mapping class group.

Cohomology of arithmetic groups and relations to mapping class groups.