2019 Thematic Program (IV)
Derived loop spaces
Time: 2:00-5:00pm, October 17-18, 2019
Venue: W303 School of Mathematics, Sichuan University
The purpose of the present series of lectures is to introduce notions from derived algebraic geometry in order to define the derived loop space of an algebraic variety (or a scheme or a stack). I will explain the relations between the derived loop spaces and classical notions such as de Rham cohomology and characteristic classes. One of the major result of this course is a highly structured version of the Hochschild-Kostant-Rosenberg theorem, and its applications to the construction of symplectic structures. In the final lecture, I will also present some new results concerning positive or mixed characteristic situations.
• Lecture 1
Derived schemes I: simplicial commutative rings and their homotopy theory, examples.
• Lecture 2
Derived schemes II: derived schemes as ringed spaces, derived stacks as functors, cotangent complexes and formal comple- tion. Example : the derived moduli space of representations.
• Lecture 3
The derived loop space: definition and examples, the HKR theorem in characteristic zero, shifted symplectic structures.
• Lecture 4
The filtered loop space: the filtered circle and the HKR- filtration, applications to symplectic structures and the theory of sin- gular supports in non-zero characteristic.
Bertrand Toën, Derived Algebraic Geometry. EMS Surv. Math. Sci. 1 (2014), no. 2, 153-245.