Derived Algebraic and Differential Geometry

Explore advanced mathematical concepts through this comprehensive lecture series covering derived algebraic and differential geometry. Delve into foundational topics including model and ∞-categories, Grothendieck topologies, and homotopy descent before progressing to more specialized areas such as derived Artin stacks and De Rham complexes. Examine S¹-equivariant schemes and loop spaces, study Chern character theory, and investigate the local structure of closed differential forms in the derived sense across multiple detailed sessions. Master cyclic homology concepts and explore definition and existence results for various geometric structures. Focus extensively on Lagrangian geometry through multiple lectures covering Lagrangians, Lagrangian fibrations, and their intersections. Apply theoretical knowledge through concrete examples and applications, culminating in an exploration of the Uhlenbeck-Yau construction and correspondence with practical examples to solidify understanding of these sophisticated mathematical frameworks.

Syllabus

Special Lecture Series on Derived Algebraic/Differential Geometry
Lecture 1: Model and с-categories
Lecture 2: Grothendieck topologies and homotopy descent
Lecture 3: Derived Artin stacks
Lecture 5: De Rham complexes and S1-equivariant schemes (loop spaces)
Lecture 6: Chern character
Lecture 7: Local structure of closed differential forms in the derived sense Part I
Lecture 8: Local structure of closed differential forms in the derived sense Part II
Lecture 9: Cyclic homology
Lecture 10: Definition and existence results
Lecture 11: Lagrangians and Lagrangian fibrations
Lecture 12: Lagrangians and Lagrangian fibrations
Lecture 13: Intersections of Lagrangians
Lecture 14: Examples and applications 2 (Part I)
Lecture 15: Examples and applications 2 (Part II)
Lecture 16: Examples and applications 2 (Part III)
Lecture 17: Uhlenbeck–Yau construction and correspondence Examples (Part I)