Masterclass and PhD course on

The tautological ring

Lectures by

Carel Faber, KTH

with additional material by Oscar Randal-Williams, KU

December 12-16, 2011.


This masterclass will introduce the participants to the topic of the tautological ring of the moduli space of Riemann surfaces. It is primarily aimed at graduate students and postdocs wishing to approach this research subject.

Topics of Faber's course:

  1. Definition of the tautological classes \kappa_i in both the Chow ring and the cohomology of the moduli space of Riemann surfaces.
  2. Definition of the Hodge bundle over the moduli space of Riemann surfaces, and Mumford's calculation of its Chern classes in terms of the \kappa_i.
  3. Mumford's proof that \kappa_1, ..., \kappa_{g-2} generate the tautological ring.
  4. Faber's method of producing relations.
  5. Faber's conjectures on the structure of the tautological ring, and discussion of the present status of each part of the conjecture.

Topics of Randal-Williams' course:

  1. Definition of the universal degree zero Picard variety, and Morita's class \Omega; proof that \Omega^{g+1}=0.
  2. Construction of the space parametrising degree zero complex line bundles, and description of \Omega on this space in terms of generalised Mumford--Morita--Miller classes.
  3. Simplified version of Morita's proof that \kappa_1, ..., \kappa_{g-2} generate the tautological ring.
  4. The generalised Morita relations; reduction to counting graphs.

    Venue: Centre for Symmetry and Deformation, KU.

    Format of the course: Principal lecture series by Carel Faber, with additional talks by Oscar Randal-Williams.

    Monday Tuesday Wednesday Thursday Friday

    Coffee and



    Faber I

    Aud 1

    Faber III

    Aud 9

    Faber V

    Aud 10

    Faber VI

    Aud 9

    Faber VII

    Aud 9

    Coffee Coffee Coffee Coffee Coffee

    Faber II

    Aud 1

    Faber IV

    Aud 9

    Randal-Williams I

    Aud 10

    Randal-Williams II

    Aud 9

    Randal-Williams III

    Aud 9


    Practical Information:  Registration on Monday will also be in the common room on the fourth floor, and starts at 13:00. For Information how to get to the Department of Mathematical Sciences click here.

    Organiser: Oscar Randal-Williams (KU)


    • Carel Faber (KTH)
    • Oscar Randal-Williams (KU)
    • Emanuele Dotto (KU)
    • Nathalie Wahl (KU)
    • Federico Cantero Moran (Universitat de Barcelona)
    • Hiroaki Tanaka (Northwestern)
    • Paolo Masulli (AU)
    • Angela Klamt (KU)
    • Anssi Lahtinen (KU)
    • Daniela Egas Santander (KU)
    • Kristian Moi (KU)
    • Mauricio Esteban Gomez Lopez (KU)
    • Olof Bergvall (KTH)

        Miscellaneous links and documents: