Moduli Spaces in Geometry

CIRM, Luminy, Marseille

Date(s) : 26/10/2015 - 30/10/2015   iCal
0 h 00 min

Moduli Spaces in Geometry 
October 26 – 30, 2015

Classification problems in Complex Geometry   often lead to  so-called moduli spaces. These are complex analytic spaces, algebraic varieties, or stacks whose points parameterize the isomorphy classes of the objects in question. Famous examples include the moduli space of smooth curves of fixed genus and moduli spaces of stable vector bundles on a fixed variety.

A well-known   construction method for such moduli spaces comes from GIT. This technique works   for vector and principal bundles with or without extra structures on projective algebraic varieties. The notion of  stability from GIT has to be translated into an intrinsic notion of  stability for the objects one would like to study. One of the striking and interesting discoveries, known under the name of Kobayashi-Hitchin correspondence, is the fact that moduli problems for stable objects are often related to  gauge theoretical moduli problems.

It is a remarkable fact that the moduli stack of Higgs bundles features prominently in many aspects of the Langlands program.  Ngô Bào Châu used the topology of the moduli stack of Higgs bundles and the  Hitchin map to prove the fundamental lemma in the Langlands program over function fields over finite fields.  Drinfeld and Laumon proposed a geometric version of the Langlands program which works over arbitrary fields, in particular, over C. It postulates an equivalence between the derived category of D-modules on the moduli stack of principal G-bundles and the derived category of O-modules on the stack of local systems for the Langlands dual group on an algebraic curve. Donagi and Pantev showed that the Hitchin integrable system for a simple algebraic group   is dual to the Hitchin system for the Langlands dual group. This can be interpreted as a « classical limit » of the Geometric Langlands Conjecture. The workshop will discuss recent results related to these spectacular developments.


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