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Workshop

Non-Abelian Hodge Theory March 28, 2002 - April 05, 2002
Registration Deadline: April 05, 2002 over 16 years ago
To apply for Funding you must register by: December 28, 2001 almost 17 years ago
Parent Program:
Organizers S. Bradlow, O. Garcia-Prada, M. Kapranov, L. Katzarkov, M. Kontsevich, D. Orlov, T. Pantev, C. Simpson, and B. Toen
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Description
The non-abelian Hodge theory originates in the groundbreaking works of A. Grothendieck, P. Deligne and A. Beilinson. A major development in the field was made by C. Simpson. Together with his students and collaborators, Simpson developed the theory of geometric n-stacks in a form which is particularly well adapted to understanding non-abelian cohomology of complex algebraic varieties as well as their Hodge structures. An indication of the power of the theory is that even though the notion of a geometric n-stack was invented for the task of understanding non-abelian Hodge structures it immediately acquired a life of its own and keeps popping up in different areas of modern mathematics. Part I (March 28-30): Higgs bundles and applications Introduced by Hitchin in 1987, and with pioneering work by Corlette, Donaldson and Simpson, Higgs bundles and their moduli spaces have remarkably rich geometric properties, and unexpectedly far-reaching applications. The moduli spaces are simultaneously symplectic quotients, algebraic quotients and gauge theoretic moduli spaces. They have a natural hyperkahler structure, they form an algebraic completely integrable system (the Hitchin system), they have an interpretation as representation varieties for fundamental groups, and are central in the development of non-abelian Hodge theory. Stable Higgs bundles play a key role in the theory of variations of Hodge structures, while natural gauge theoretic equations on Higgs bundles are related to harmonic maps into homogeneous spaces. Constructions based on Higgs bundles have found application in topological quantum field theory. This workshop will be the first ever to focus explicitly on the theory and applications of Higgs bundles. The participants will include experts on Higgs bundles as well as leading specialists in areas where Higgs bundles have found useful application. The workshop will be a timely opportunity to review the gains of the last 15 years and to assess future needs and opportunities. Part II (April 1 -5): Theory of high categories and applications Among the geometric applications of the stack part of the non-abelian Hodge theory one may mention: Simpson's fundamental restriction of the class of lattices that can be fundamental groups of smooth projective manifolds; Simpson's version of secondary Kodaira-Spencer classes taking values in non-abelian cohomology; the non-abelian analogues of Griffiths (p,p)-cycle theorem and Open Orbit theorem recently proven by L. Katzarkov, T. Pantev and C. Simpson; and the notion and theory of non-abelian MHS recently introduced by L. Katzarkov, T. Pantev and C. Simpson. All these indicate that until non-abelian Hodge theory has reached is full maturity it will provide us with new restrictions on the homotopy types and the monodromy of the projective varieties. These topics as well as new application for theory of n categories to homotopy theory will be discussed. Group photo of participants
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

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A block of rooms has been reserved at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is $129 per night for a Deluxe Queen/King, based on availability.

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Schedule, Notes/Handouts & Videos
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Mar 28, 2002
Thursday
08:00 AM - 05:00 PM
  An Introduction to Higher Non-Abelian Hodge Theory
Bertrand Toen
10:00 AM - 11:00 AM
  Twisted Higgs bundles and the fundamental group of compact Kahler manifolds
S. Ramanan
11:30 AM - 12:30 PM
  Representations of surface groups in unitary groups with indefinite signature
Oscar Garcia-Prada (Consejo Superior de Investigaciones Científicas (CSIC))
02:00 PM - 03:00 PM
  Branched hyperbolic structures and Rank 2 Higgs bundles
William Goldman (University of Maryland)
03:30 PM - 04:30 PM
  Higgs bundles and representation varieties
Eugene Xia
Mar 29, 2002
Friday
10:00 AM - 11:00 AM
  The gerbe of Higgs bundles
Ron Donagi (University of Pennsylvania)
11:30 AM - 12:30 PM
  Group valued Higgs bundles over a punctured elliptic curve
Eyal Markman
02:00 PM - 03:00 PM
  Intersection theory on the moduli space of Higgs bundles
Tamas Hausel
03:30 PM - 04:30 PM
  Higgs fields and variations of mixed Hodge structure
Gregory Pearlstein
Mar 30, 2002
Saturday
10:00 AM - 11:00 AM
  Higgs bundle spaces as mirror partners
Michael Thaddeus
11:30 AM - 12:30 PM
  Hitchin's connection, Higgs bundles and representations of the mapping class group
Jorgen Andersen
02:00 PM - 03:00 PM
  An integer valued SU(3) Casson invariant
Hans Boden (McMaster University)
03:30 PM - 04:30 PM
  Fundamental group of the moduli space of real and complex cubic surfaces
Domingo Toledo (University of Utah)
Apr 01, 2002
Monday
10:00 AM - 11:00 AM
  Iterated integrals and motives
Richard Hain (Duke University)
11:30 AM - 12:30 PM
  The Hodge conjecture for some moduli spaces of bundles
Donu Arapura
02:00 PM - 03:00 PM
  A Riemann-Roch theorem for (higher) determinantal gerbes and differentiable manifolds
Mikhail Kapranov (Kavli Institute for the Physics and Mathematics of the Universe )
Apr 02, 2002
Tuesday
10:00 AM - 11:00 AM
  The theory of quasi-categories
Andre Joyal (Université du Québec à Montréal)
11:30 AM - 12:30 PM
  A counterexample to a 1961 theorem in homological algebra
Amnon Neeman (Australian National University)
02:00 PM - 03:00 PM
  A language for higher categorical structures
Thomas Leinster
04:00 PM - 05:00 PM
  Formal deformations of sheaves of algebras
Vladimir Hinich (The University of Haifa)
Apr 03, 2002
Wednesday
10:00 AM - 11:00 AM
  p-Adic Hodge theory, part I
Gerd Faltings (Max-Planck-Institut für Mathematik)
11:30 AM - 12:30 PM
  Mixed Hodge structures and fiber bundles over the projective plane
Olivier Penacchio
02:00 PM - 03:00 PM
  The differential geometry of gerbes
Lawrence Breen
03:30 PM - 04:30 PM
  Sklyanin algebras and Hilbert schemes of points
Thomas Nevins (University of Illinois at Urbana-Champaign)
Apr 04, 2002
Thursday
10:00 AM - 11:00 AM
  p-Adic Hodge theory, part II
Gerd Faltings (Max-Planck-Institut für Mathematik)
11:30 AM - 12:30 PM
  A superficial overview of n-category theory
J. May
02:00 PM - 03:00 PM
  Non-abelian Hodge theory and schematization of homotopy types
Ludmil Katzarkov (University of Miami)
03:30 PM - 04:30 PM
  Higher Tannaka duality
Bertrand Toen (Centre National de la Recherche Scientifique (CNRS))
Apr 05, 2002
Friday
10:00 AM - 11:00 AM
  Topological aspects of stacks
Constantin Teleman (University of California, Berkeley)
11:30 AM - 12:30 PM
  Homotopy algebraic geometry
Gabriele Vezzosi (Università di Firenze)
02:00 PM - 03:00 PM
  Equivalences of derived categories of coherent sheaves
Dmitry Orlov (V. A. Steklov Institute of Mathematics)
03:30 PM - 04:30 PM
  n-Categories in Homology Theory
John Baez