Description
Moduli spaces of stable pointed curves play an important role in algebraic geometry. This School will have one course on vector bundles of coinvariants and on conformal blocks and another one on their cohomology classes in relation with those of moduli of abelian varieties. The cohomology of moduli spaces of curves and abelian varieties carries several natural classes. We focus on the tautological classes and the cohomology classes related to spaces of modular forms. The problem of determining relationships between the tautological classes turns out to be particularly interesting. Moduli spaces of curves carry vector bundles of coinvariants and conformal blocks; they are invariants of a curve C attached to a Lie group G that are canonically isomorphic to global sections of an ample line bundle on the moduli stack of certain G-bundles on C. These are generalized theta functions in case C is smooth. In case g=0, the bundles of co-invariants are globally generated, and their first Chern classes are semi-ample line bundles on the moduli of curves, and shed light on its birational geometry. We can also use the moduli space of curves to learn about generalized theta functions.




TOPICS:

Cohomology classes on moduli of curves and abelian varieties
  • moduli spaces of curves and abelian varieties
  • properties of the moduli spaces and their compactifications
  • natural cohomology classes and their relations
  • tautological classes and cohomology classes related to spaces of modular forms
 
Vector bundles of coinvariants and conformal blocks
  • introduction to moduli spaces of curves
  • open problems and F-conjecture
  • vector bundles of coinvariants and conformal blocks
  • case g=0: global generation and semi-ample divisors
  • any genus: nef divisors
  • Chern classes of bundles of coinvariants
  • Global sections of ample line bundles on Bun_G(C): smooth and nodal case

 
PARTICIPATION:
Women are particularly encouraged to apply.
Should you come to Trieste with your child(ren), please send an e-mail to smr3215@ictp.it to describe your family needs and we will do our best to meet them.
A limited number of grants are available to support the attendance of selected participants, with priority given to participants from developing countries. There is no registration fee.
To apply, please use the link on the left side of this web page.


The deadline for submitting applications expired on 15 March 2018.
Go to day
  • Monday, 18 June 2018
    • 09:00 - 16:00
      Location: Adriatico Guest House - Kastler Lecture Hall
      • 09:00 Registration 30' ( Adriatico Guest House - Kastler Lecture Hall Area (Lower Level 1) )
        REGISTRATION
        Participants NOT STAYING in ICTP guesthouses and all faculty should register for the Workshop. Registration of participants staying in one of the guesthouses is done upon arrival, at Reception desks. 
        
      • 09:30 Opening Remarks 10'
        Speaker: Fernando Rodriguez Villegas (ICTP)
        Material: Video
      • 09:40 Conformal blocks and the moduli space of curves - Lecture 1: The moduli space of curves 1h30'
        Speaker: A. Gibney (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 11:10 Coffee Break 20'
      • 11:30 Lecture 1 1h30'
        Speaker: O.Tommasi (Università degli Studi di Padova, Italy)
        Material: Video
      • 13:00 Lunch break and free afternoon for administrative formalities 3h0'
        After registration, visitors funded by the ICTP must go to the Operations Office (E. Fermi building, ground floor), with their badge and passport or I.D. Card, to collect their dues. It is strongly recommended that you visit the Operations Office during this slot, as their opening hours are limited to Monday, Tuesday and Friday (8:30-12:00/13:30-14:30) and the Bank is only open on Mondays (8.20-13.20 and 14.30-16.00).
  • Tuesday, 19 June 2018
    • 09:00 - 16:55
      • 09:00 Lecture 2 1h30'
        Speaker: O.Tommasi (Università degli Studi di Padova, Italy)
        Material: Video
      • 10:30 Coffee break 30' ( Adriatico Guest House - Terrace )
      • 11:00 Conformal blocks and the moduli space of curves - Lecture 2: Conformal blocks, Co-invariants, Factorization, and Propagation of Vacua 1h30'
        Speaker: A. Gibney (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 12:30 Lunch Break 1h30'
      • 14:00 Exercise session - Part 1 1h30'
        Speaker: C. Damiolini (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 15:30 Coffee Break 30' ( Adriatico Guest House - Terrace )
      • 16:00 Contributed Talk by participant - Gromov-Witten invariants of the complex Lagrangian Grassmannian 25'
        Speaker: H. Dang (Da Lat University, Vietnam)
        Material: Video
      • 16:30 Contributed talk by participant - On the dimension of the moduli space of monomials curves: an version of Deligne formule 25'
        Speaker: J. Quispe Vargas (Universidade Federal de Minas Gerais, Brazil)
        Material: Video
  • Wednesday, 20 June 2018
    • 09:00 - 16:55
      • 09:00 Conformal blocks and the moduli space of curves - Lecture 3: Beauville’s quotient construction and global generation in case g = 0 1h30'
        Speaker: A. Gibney (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 10:30 Coffee Break 30' ( Adriatico Guest House - Terrace )
      • 11:00 Lecture 3 1h30'
        Speaker: O.Tommasi (Università degli Studi di Padova, Italy)
        Material: Video
      • 12:30 Lunch Break 1h30'
      • 14:00 Contributed Talk by participant - Cohomology of the moduli space of non-hyperelliptic genus four curves 25'
        Speaker: M. Fortuna (Leibniz Universität Hannover, Germany)
      • 14:30 Contributed Talk by participant - A Torelli-type theorem for moduli space of stable sheaves over a singular curve 25'
        Speaker: I. Kaur (IMPA, Brazil)
      • 15:00 Contributed Talk by participants - The average size of 2-Selmer groups over function fields 25'
        Speaker: A. Landesman (Stanford University, USA)
        Material: Video
      • 15:30 Coffee Break 15'
      • 15:45 Contributed Talk by participants - The d-elliptic Locus in Genus 2 25'
        Speaker: C. Lian (Columbia University, New York, USA)
        Material: Video
      • 16:15 Contributed Talk by participant -On the birational geometry of moduli spaces of points on the line 25'
        Speaker: A. Massarenti (Universidade Federal Fluminense, Brazil)
        Material: Video
  • Thursday, 21 June 2018
    • 09:00 - 21:00
      • 09:00 Lecture 4 1h30'
        Speaker: O.Tommasi (Università degli Studi di Padova, Italy)
        Material: Video
      • 10:30 Coffee Break 30' ( Adriatico Guest House - Terrace )
      • 11:00 Conformal blocks and the moduli space of curves - Lecture 4: Chern classes: vanishing, identities and open questions 1h30'
        Speaker: A. Gibney (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 12:30 Lunch Break 1h30'
      • 14:00 Exercise session - Part 2 1h30'
        Speaker: C. Damiolini (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 15:30 Coffee Break 15'
      • 15:45 Contributed Talk by participant - Counting the number of trigonal curves of genus five over finite fields. 25'
        Speaker: T. Wennink (University of Liverpool, UK)
        Material: Video
      • 16:15 Contributed Talk by participant - Computing the Gopakumar-Vafa Invariant 25'
        Speaker: L. Zhao (University of Illinois at Urbana-Champaign, USA)
        Material: Video
      • 19:30 Social Dinner 1h30' ( Adriatico Guest House - Terrace )
  • Friday, 22 June 2018
    • 09:30 - 13:00
      • 09:30 Conformal blocks and the moduli space of curves - Lecture 5: Geometric interpretations 1h30'
        Speaker: A. Gibney (Rutgers University Hill Center for the Mathematical Sciences, USA)
      • 11:00 Coffee Break 30' ( Adriatico Guest House - Terrace )
      • 11:30 Lecture 5 1h30'
        Speaker: O.Tommasi (Università degli Studi di Padova, Italy)
        Material: Video