TL;DR
This paper offers a concise survey of derived symplectic geometry, focusing on symplectic structures on derived stacks, which are crucial for understanding singular spaces in modern geometry and physics.
Contribution
It provides an overview of derived symplectic geometry and includes a case study on Casson's invariant, highlighting recent developments in the field.
Findings
Derived symplectic geometry extends classical symplectic concepts to singular spaces.
The framework is essential for applications in mathematical physics and geometry.
A case study on Casson's invariant illustrates practical applications.
Abstract
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks. Derived stacks are the main players in derived geometry, the purpose of which is to deal with singular spaces, while symplectic structures are an essential ingredient of the geometric formalism of classical mechanics and classical field theory. In addition to providing an overview of a relatively young field of research, we provide a case study on Casson's invariant.
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Taxonomy
TopicsMicrotubule and mitosis dynamics · Homotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology