## Abstract

Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps C → X. This space admits certain compactification whose points are called quasi-maps. In the last decade it has been discovered that in the case when X is a (partial) flag variety of a semi-simple algebraic group G (or, more generally, of any symmetrizable Kac- Moody Lie algebra) these compactifications play an important role in such fields as geometric representation theory, geometric Langlands correspondence, geometry and topology of moduli spaces of G-bundles on algebraic surfaces, 4-dimensional super-symmetric gauge theory (and probably many others). This paper is a survey of the recent results about quasi-maps as well as their applications in different branches of representation theory and algebraic geometry.

Original language | English |
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Pages | 1145-1170 |

Number of pages | 26 |

Publication status | Published - 2006 |

Externally published | Yes |

Event | 25th International Congress of Mathematicians, ICM 2006 - Madrid, Spain Duration: 22 Aug 2006 → 30 Aug 2006 |

### Conference

Conference | 25th International Congress of Mathematicians, ICM 2006 |
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Country/Territory | Spain |

City | Madrid |

Period | 22/08/06 → 30/08/06 |

## Keywords

- Geometric Langlands duality
- Quasi-maps
- Schubert varieties
- Supersymmetric gauge theory