Geometry of Moduli Spaces and Representation Theory
About this Title
Roman Bezrukavnikov, Massachusetts Institute of Technology, Cambridge, MA, Alexander Braverman, Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada and Zhiwei Yun, Yale University, New Haven, CT, Editors
Publication: IAS/Park City Mathematics Series
Publication Year: 2017; Volume 24
ISBNs: 978-1-4704-3574-5 (print); 978-1-4704-4234-7 (online)
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.
Table of Contents
- Mark de Cataldo – Perverse sheaves and the topology of algebraic varieties
- Xinwen Zhu – An introduction to affine Grassmannians and the geometric Satake equivalaence
- Zhiwei Yun – Lectures on Springer theories and orbital intgegrals
- Ngô Châu – Perverse sheaves and fundamental lemmas
- Andrei Okounkov – Lectures on $K$-theoretic computations in enumerative geometry
- Hiraku Nakajima – Lectures on perverse sheaves on instanton moduli spaces