LECTURER:Ralph Kaufmann TIME:Monday 16:30-18:00 Starting date:Monday, April 28 LOCATION:Seminarraum B, Beringstr. 4 EMAIL:ralphk@mpim-bonn.mpg.de HOMEPAGEhttp://guests.mpim-bonn.mpg.de/ralphk

Moduli spaces of curves or Riemann surfaces have been studied for a

long time and quite extensively in conjunction with problems ranging

from topology to arithmetic algebraic geometry.These spaces parameterize in a certain precise sense all possible structures

of a Riemann surface of a fixed underlying topological surface.

There are several formulations of this theory coming from conformal,

hyperbolic and algebraic geometry, each beautiful and rich in its own right.Recently, these spaces have entered into the spotlight due to their

relation to string theory. This has lead to the definition of quantum

cohomology, which was used to solve many enumerative problems.

In a variation this has also lead for instance to new results on Hurwitz numbers.We will start by studying moduli spaces of curves from the different

points of view as stated above. This together with a study of

the geometry of these spaces and their compactifications

(e.g. their cohomology, intersection theory, tautological bundles)

will be the bulk of the course.Later, we intend to specialize to a selection of topics of current

research such as quantum cohomology. Other possible subjects

could be the recent solution of the Mumford Conjecture

or the relation to integrable systems.

**CAVEAT:**

**This list is preliminary and only for general
information.**
**It is neither exhaustive nor the basis for
the course!**

Some Monographs:

Weyl, Hermann Die Idee der Riemannschen Fläche. (German) [The concept of a Riemann surface] Reprint of the 1913 German original. With essays by Reinhold Remmert, Michael Schneider, Stefan Hildebrandt, Klaus Hulek and Samuel Patterson. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997.

E. Arbarello, M. Cornalba, P. Griffiths and J. Harris: Geometry of Algebraic Curves I. Grundlehren Math. Wiss. 267, Springer Verlag (1985).

Manin, Yuri I. Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society. Colloquium Publications, 47. American Mathematical Society, Providence, RI, 1999..

Harris, Joe; Morrison, Ian. Moduli of curves. Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998.

O.Lehto, K.I. Virtanen .Quasikonforme Abbildungen. Springer 1965 .

S.Nag .The Complex Analytic Theory of Teichmueller Theory. Wiley & Sons 1988.

F.P.Gardiner, N.Lakic . Quasiconformal Teichmueller Theory. American Mathematical Society, Providence, RI, 2000.

Travaux de Thurston sur les surfaces. (French) [The works of Thurston on surfaces]

Séminaire Orsay. With an English summary. Astérisque, 66-67.

Société Mathématique de France, Paris, 1979.F.P. Gardiner .Teichmueller theory and Quadratic differentials. John Wiley & Sons, Inc., New York (1987).

W.Abikoff . The Real Analytic Theory of Teichmueler Space. LNM 820. (Springer 1980).

Penner, R. C.; Harer, J. L. Combinatorics of train tracks. Annals of Mathematics Studies, 125.

Princeton University Press, Princeton, NJ, 1992.

A survey articleHain, Richard; Looijenga, Eduard Mapping class groups and moduli spaces of curves.

Algebraic geometry-Santa Cruz 1995, 97-142, Proc. Sympos. Pure Math., 62, Part 2,

Amer. Math. Soc., Providence, RI, 1997. http://front.math.ucdavis.edu/alg-geom/9607004

Some original articles:M. Dehn: Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-206.

P. Deligne and D. Mumford: The irreducibility of the space of curves of given genus, Inst. Hautes Etudes Sci. Publ. Math. 36 (1969), 75-109.

C.J.Earle, J.Eells On the differential geometry of Teichmueller theory. J.Analyse.Math 1967.

Hubbard, John; Masur, Howard Quadratic differentials and foliations. Acta Math. 142 (1979), no. 3-4, 221--274

M. Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23.

E. Witten: Two dimensional gravity and intersection theory on moduli space, Surveys in Di . Geom. 1 (1991), 243-310.

Links to Max-Planck Institute , the Math Department of the University of Bonn and my homepage