Description
MAIN TOPICS OF THE SCHOOL:
- Review of Modular forms
- Quantum Field Theory for Mathematicians
- Moduli Spaces of Sheaves
- Mock Modular forms
- Siegel Modular forms and Black Hole State counting
- Invariants of Moduli Spaces of Sheaves and Donaldson invariants

The third-week will be an international conference, where leading experts will report on the newest developments in the field of modular forms, mock modular forms and their applications.
Go to day
  • Monday, 28 February 2011
    • 08:30 - 10:30 REGISTRATION & ADMINISTRATIVE FORMALITIES
      Location: Leonardo da Vinci Building, Lobby
      • 08:30 REGISTRATION & ADMINISTRATIVE FORMALITIES 2h0'
    • 10:30 - 10:45 OPENING
      • 10:30 OPENING 15'
        Speaker: Ramadas Ramakrishnan (ICTP, Trieste, Italy)
    • 10:45 - 12:20 Introduction to modular forms 1
      • 10:45 Introduction to modular forms 1 1h35'
        Speaker: M. Vlasenko/A. Mellit (Max-Planck Institute for Mathematics, Bonn, Germany)
    • 12:20 - 14:00 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 1h40'
    • 14:00 - 15:00 Quantum field theory for mathematicians 1
    • 15:00 - 15:15 Break
      • 15:00 Break 15'
    • 15:15 - 16:50 Moduli spaces of vector bundles and sheaves 1
      • 15:15 Moduli spaces of vector bundles and sheaves 1 1h35'
        Speaker: V.B. Mehta (Tata Institute of Fundamental Research, Mumbai, India)
        Material: lecture notes
  • Tuesday, 1 March 2011
    • 09:15 - 10:45 Introduction to modular forms 2
      • 09:15 Introduction to modular forms 2 1h30'
        Speaker: M. Vlasenko/A. Mellit (Max-Planck Institute for Mathematics, Bonn, Germany)
    • 10:45 - 11:15 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:45 Coffee break 30'
    • 11:15 - 12:15 Quantum field theory for mathematicians 2
      • 11:15 Quantum field theory for mathematicians 2 1h0'
        Speaker: I. Runkel (Universität Hamburg, Hamburg, Germany)
        Material: lecture notes
    • 12:15 - 14:25 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:15 Lunch break 2h10'
    • 14:25 - 16:00 Moduli spaces of vector bundles and sheaves 2
      • 14:25 Moduli spaces of vector bundles and sheaves 2 1h35'
        Speaker: V.B. Mehta (Tata Institute of Fundamental Research, Mumbai, India)
        Material: lecture notes
  • Wednesday, 2 March 2011
    • 09:15 - 10:50 Introduction to modular forms 3
      • 09:15 Introduction to modular forms 3 1h35'
        Speaker: M. Vlasenko/A. Mellit (Max-Planck Institute for Mathematics, Bonn, Germany)
    • 10:50 - 11:20 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:50 Coffee break 30'
    • 11:20 - 12:20 Quantum field theory for mathematicians 3
      • 11:20 Quantum field theory for mathematicians 3 1h0'
        Speaker: I. Runkel (Universität Hamburg, Hamburg, Germany)
        Material: lecture notes
    • 12:20 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 2h10'
    • 14:30 - 16:05 Moduli spaces of vector bundles and sheaves 3
      • 14:30 Moduli spaces of vector bundles and sheaves 3 1h35'
        Speaker: V.B. Mehta (Tata Institute of Fundamental Research, Mumbai, India)
        Material: lecture notes
    • 18:30 - 18:30 RECEPTION
      Location: Leonardo da Vinci Building Cafeteria
      • 18:30 RECEPTION
  • Thursday, 3 March 2011
    • 09:15 - 10:50 Introduction to modular forms 4
      • 09:15 Introduction to modular forms 4 1h35'
        Speaker: M. Vlasenko/A. Mellit (Max-Planck Institute for Mathematics, Bonn, Germany)
    • 10:50 - 11:20 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:50 Coffee break 30'
    • 11:20 - 12:20 Quantum field theory for mathematicians 4
      • 11:20 Quantum field theory for mathematicians 4 1h0'
        Speaker: I. Runkel (Universität Hamburg, Hamburg, Germany)
        Material: lecture notes
    • 12:20 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 2h10'
    • 14:30 - 16:05 Moduli spaces of vector bundles and sheaves 4
      • 14:30 Moduli spaces of vector bundles and sheaves 4 1h35'
        Speaker: V.B. Mehta (Tata Institute of Fundamental Research, Mumbai, India)
        Material: lecture notes
  • Friday, 4 March 2011
    • 09:15 - 10:50 Introduction to modular forms 5
      • 09:15 Introduction to modular forms 5 1h35'
        Speaker: M. Vlasenko/A. Mellit (Max-Planck Institute for Mathematics, Bonn, Germany)
    • 10:50 - 11:20 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:50 Coffee break 30'
    • 11:20 - 12:20 Quantum field theory for mathematicians 5
      • 11:20 Quantum field theory for mathematicians 5 1h0'
        Speaker: I. Runkel (Universität Hamburg, Hamburg, Germany)
        Material: lecture notes
    • 12:20 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 2h10'
    • 14:30 - 16:05 Quantum field theory and modular forms
      • 14:30 Quantum field theory and modular forms 1h35'
        Speaker: A. Dabholkar (LPTHE, Université Paris VI, Paris, France)
  • Monday, 7 March 2011
    • 09:15 - 10:15 Mock theta functions, mock modular forms and indefinite theta series 1
      • 09:15 Mock theta functions, mock modular forms and indefinite theta series 1 1h0'
        Speaker: S.P. Zwegers (Universität zu Köln, Köln, Germany)
        Material: lecture notes
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 12:20 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 1
      • 10:45 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 1 1h35'
        Speaker: L. Goettsche (ICTP, Trieste, Italy)
        Material: lecture notes
    • 12:20 - 14:00 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 1h40'
    • 14:00 - 15:00 Mock modular forms and Jacobi forms 1
      • 14:00 Mock modular forms and Jacobi forms 1 1h0'
        Speaker: Don B. Zagier (Max-Planck Institute for Mathematics, Bonn, Germany)
        Material: lecture notes
    • 15:00 - 15:10 Break
      • 15:00 Break 10'
    • 15:10 - 16:10 Mock modular forms and Jacobi forms 1 (continuation)
      • 15:10 Mock modular forms and Jacobi forms 1 (continuation) 1h0'
        Speaker: Don B. Zagier (Max-Planck Institute for Mathematics, Bonn, Germany)
        Material: lecture notes
  • Tuesday, 8 March 2011
    • 09:15 - 10:15 Mock theta functions, mock modular forms and indefinite theta series 2
      • 09:15 Mock theta functions, mock modular forms and indefinite theta series 2 1h0'
        Speaker: S.P. Zwegers (Universität zu Köln, Köln, Germany)
        Material: lecture notes
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 12:20 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 2
      • 10:45 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 2 1h35'
        Speaker: L. Goettsche (ICTP, Trieste, Italy)
        Material: lecture notes
    • 12:20 - 14:00 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 1h40'
    • 14:00 - 15:35 Mock modular forms and Jacobi forms 2
      • 14:00 Mock modular forms and Jacobi forms 2 1h35'
        Speaker: Don B. Zagier (Max-Planck Institute for Mathematics, Bonn, Germany)
        Material: lecture notes
    • 15:35 - 15:50 Break
      • 15:35 Break 15'
    • 15:50 - 17:25 Quantum black holes and modular forms 1
      • 15:50 Quantum black holes and modular forms 1 1h35'
        Speaker: A. Sen/A. Dabholkar (Harish-Chandra Research Institute, Allahabad, India/LPTHE, Paris VI, France)
  • Wednesday, 9 March 2011
    • 09:15 - 10:15 Mock theta functions, mock modular forms and indefinite theta series 3
      • 09:15 Mock theta functions, mock modular forms and indefinite theta series 3 1h0'
        Speaker: S.P. Zwegers (Universität zu Köln, Köln, Germany)
        Material: lecture notes
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 12:20 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 3
      • 10:45 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 3 1h35'
        Speaker: L. Goettsche (ICTP, Trieste, Italy)
        Material: lecture notes
    • 12:20 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 2h10'
    • 14:30 - 16:05 Quantum black holes and modular forms 2
      • 14:30 Quantum black holes and modular forms 2 1h35'
        Speaker: A. Sen/A. Dabholkar (Harish-Chandra Research Institute, Allahabad, India/LPTHE, Paris VI, France)
  • Thursday, 10 March 2011
    • 09:15 - 10:15 Mock theta functions, mock modular forms and indefinite theta series 4
      • 09:15 Mock theta functions, mock modular forms and indefinite theta series 4 1h0'
        Speaker: S.P. Zwegers (Universität zu Köln, Köln, Germany)
        Material: lecture notes
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 12:20 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 4
      • 10:45 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 4 1h35'
        Speaker: L. Goettsche (ICTP, Trieste, Italy)
        Material: lecture notes
    • 12:20 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 2h10'
    • 14:30 - 16:05 Quantum black holes and modular forms 3
      • 14:30 Quantum black holes and modular forms 3 1h35'
        Speaker: A. Sen/A. Dabholkar (Harish-Chandra Research Institute, Allahabad, India/LPTHE, Paris VI, France)
  • Friday, 11 March 2011
    • 09:15 - 10:15 Mock theta functions, mock modular forms and indefinite theta series 5
      • 09:15 Mock theta functions, mock modular forms and indefinite theta series 5 1h0'
        Speaker: S.P. Zwegers (Universität zu Köln, Köln, Germany)
        Material: lecture notes
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 12:20 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 5
      • 10:45 Wallcrossing for invariants of Moduli spaces of sheaves and Donaldson invariants 5 1h35'
        Speaker: L. Goettsche (ICTP, Trieste, Italy)
        Material: lecture notes
    • 12:20 - 14:00 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:20 Lunch break 1h40'
    • 14:00 - 15:35 Mock modular forms and Jacobi forms 3
      • 14:00 Mock modular forms and Jacobi forms 3 1h35'
        Speaker: Don B. Zagier (Max-Planck Institute for Mathematics, Bonn, Germany)
        Material: lecture notes
  • Monday, 14 March 2011
    • 08:45 - 09:45 Theta liftings, cycles, and harmonic weak Maass forms
      • 08:45 Theta liftings, cycles, and harmonic weak Maass forms 1h0'
        Speaker: J. Funke (University of Durham, Durham, UK)
    • 09:45 - 10:00 Break
      • 09:45 Break 15'
    • 10:00 - 11:00 Weak Harmonic Maass forms of weight zero: algebraicity and congruences
      • 10:00 Weak Harmonic Maass forms of weight zero: algebraicity and congruences 1h0'
        Speaker: P. Guerzhoy (University of Hawaii, Honolulu, USA)
    • 11:00 - 11:30 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 11:00 Coffee break 30'
    • 11:30 - 12:30 Harmonic Siegel-Maass forms and their Fourier-Jacobi expansions
      • 11:30 Harmonic Siegel-Maass forms and their Fourier-Jacobi expansions 1h0'
        Speaker: M. Raum (Max-Planck Institute for Mathematics, Bonn, Germany)
    • 12:30 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:30 Lunch break 2h0'
    • 14:30 - 15:30 Mock Jacobi forms and moment asymptotics
      • 14:30 Mock Jacobi forms and moment asymptotics 1h0'
        Speaker: K. Mahlburg (Princeton University, Princeton NJ, USA)
    • 15:30 - 15:45 Break
      • 15:30 Break 15'
    • 15:45 - 16:45 Faber polynomials and Poincaré series
      • 15:45 Faber polynomials and Poincaré series 1h0'
        Speaker: Ben Kane (University of Cologne, Cologne, Germany)
  • Tuesday, 15 March 2011
    • 08:45 - 09:45 The simplest reflective modular forms
      • 08:45 The simplest reflective modular forms 1h0'
        Speaker: V. Gritsenko (Université Lille 1, Villeneuve d'Ascq, France)
    • 09:45 - 10:00 Break
      • 09:45 Break 15'
    • 10:00 - 11:00 Generalised Kac-Moody Algebra, Automorphic Forms, and the sporadic group M24
      • 10:00 Generalised Kac-Moody Algebra, Automorphic Forms, and the sporadic group M24 1h0'
        Speaker: M. Cheng (Harvard University, Cambridge MA, USA)
    • 11:00 - 11:30 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 11:00 Coffee break 30'
    • 11:30 - 12:30 Symmetries of K3 in M_{24}: I Geometry
      • 11:30 Symmetries of K3 in M_{24}: I Geometry 1h0'
        Speaker: A. Taormina (University of Durham, Durham, UK)
    • 12:30 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:30 Lunch break 2h0'
    • 14:30 - 15:30 Symmetries of K3 in M_{24}: II Group Theory
      • 14:30 Symmetries of K3 in M_{24}: II Group Theory 1h0'
        Speaker: K. Wendland (University of Augsburg, Augsburg, Germany)
    • 15:30 - 15:45 Break
      • 15:30 Break 15'
    • 15:45 - 16:45 Rigid semistable objects and kinds of modular forms
      • 15:45 Rigid semistable objects and kinds of modular forms 1h0'
        Speaker: So Okada (Kyoto University, Kyoto, Japan)
  • Wednesday, 16 March 2011
    • 08:45 - 09:45 Harmonic Maass forms and periods
      • 08:45 Harmonic Maass forms and periods 1h0'
        Speaker: H. Bruinier (Technical University of Darmstadt, Darmstadt, Germany)
    • 09:45 - 10:00 Break
      • 09:45 Break 15'
    • 10:00 - 11:00 Harmonic Weak Maass Forms and CM values of automorphic green functions
      • 10:00 Harmonic Weak Maass Forms and CM values of automorphic green functions 1h0'
        Speaker: T. Yang (University of Wisconsin, Madison WI, USA)
    • 11:00 - 11:30 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 11:00 Coffee break 30'
    • 11:30 - 12:30 Generalized Mathieu Moonshine
      • 11:30 Generalized Mathieu Moonshine 1h0'
        Speaker: S. Govindarajan (Indian Institute of Technology Madras, Chennai, India)
    • 12:30 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:30 Lunch break 2h0'
  • Thursday, 17 March 2011
    • 09:15 - 10:15 Modularity in Chern-Simons theory
      • 09:15 Modularity in Chern-Simons theory 1h0'
        Speaker: T. Dimofte (Institute for Advanced Study, Princeton NJ, USA)
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 11:45 Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes
      • 10:45 Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes 1h0'
        Speaker: A. Klemm (Humboldt University of Berlin, Berlin, Germany)
    • 11:45 - 12:00 Break
      • 11:45 Break 15'
    • 12:00 - 12:30 Kac-Wakimoto characters, asymptotics and mock modular forms
      • 12:00 Kac-Wakimoto characters, asymptotics and mock modular forms 30'
        Speaker: A. Folsom (Yale University, New Haven CT, USA)
    • 12:30 - 14:30 Lunch break
      Location: Leonardo da Vinci Building Cafeteria
      • 12:30 Lunch break 2h0'
    • 14:30 - 15:30 The Betti numbers of the moduli space of sheaves of rank 3 on the projective plane
      • 14:30 The Betti numbers of the moduli space of sheaves of rank 3 on the projective plane 1h0'
        Speaker: J. Manschot (IPhT, Gif-sur-Yvette, France)
    • 15:30 - 15:45 Break
      • 15:30 Break 15'
    • 15:45 - 16:45 Quantum black holes, wall crossing and mock modular forms
      • 15:45 Quantum black holes, wall crossing and mock modular forms 1h0'
        Speaker: S. Murthy (TIFR, Mumbai, India)
  • Friday, 18 March 2011
    • 09:15 - 10:15 Congruences for smallest parts functions
      • 09:15 Congruences for smallest parts functions 1h0'
        Speaker: J. Lovejoy (Université Denis Diderot Paris 7, Paris, France)
    • 10:15 - 10:45 Coffee break
      Location: Leonardo da Vinci Building, Lobby
      • 10:15 Coffee break 30'
    • 10:45 - 11:45 p-adic properties of the partition function
      • 10:45 p-adic properties of the partition function 1h0'
        Speaker: Z. Kent (Emory University, Atlanta GA, USA)
    • 11:45 - 12:00 Break
      • 11:45 Break 15'
    • 12:00 - 12:30 Bounds for coefficients of cusp forms and extremal lattices
      • 12:00 Bounds for coefficients of cusp forms and extremal lattices 30'
        Speaker: P. Jenkins (Brigham Young University, Provo UT, USA)
    • 12:30 - 14:30 Lunch break
      • 12:30 Lunch break 2h0'
    • 16:00 - 18:00 ICTP Colloquium "Quantum Modular Forms"
      • 16:00 ICTP Colloquium "Quantum Modular Forms" 2h0'
        Speaker: Professor Don Zagier (Max Planck Institute for Mathematics (Bonn) and Collège de France)
        Material: lecture notes