Geometry of Moduli Spaces and Representation Theory
Published on 30. December 2017
Book
Hardback
436 pages
978-1-4704-3574-5 (ISBN)
Description
This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program ``Geometry of moduli spaces and representation theory'', and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory.
Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan-Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry.
Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections.
The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.
Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan-Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry.
Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections.
The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
College/higher education
Dimensions
Height: 254 mm
Width: 178 mm
Weight
915 gr
ISBN-13
978-1-4704-3574-5 (9781470435745)
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Schweitzer Classification
Persons
Roman Bezrukavnikov, Massachusetts Institute of Technology, Cambridge, MA.
Alexander Braverman, University of Toronto, ON, Canada, Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada, and Skolkovo Institute for Science and Technology, Moscow, Russia.
Zhiwei Yun, Yale University, New Haven, CT.
Alexander Braverman, University of Toronto, ON, Canada, Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada, and Skolkovo Institute for Science and Technology, Moscow, Russia.
Zhiwei Yun, Yale University, New Haven, CT.
Content
M. A. de Cataldo, Perverse sheaves and the topology of algebraic varieties
X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalaence
Z. Yun, Lectures on Springer theories and orbital intgegrals
N. G. Chau, Perverse sheaves and fundamental lemmas
A. Okounkov, Lectures on $K$-theoretic computations in enumerative geometry
H. Nakajima, Lectures on perverse sheaves on instanton moduli spaces.
X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalaence
Z. Yun, Lectures on Springer theories and orbital intgegrals
N. G. Chau, Perverse sheaves and fundamental lemmas
A. Okounkov, Lectures on $K$-theoretic computations in enumerative geometry
H. Nakajima, Lectures on perverse sheaves on instanton moduli spaces.