Organizer: Su, Tao (苏桃)
Time: Spring 2021, 1:30-3:30pm, Thursday.
Place: Room S11, Ning Zhai (宁斋), YMSC.
The ultimate goal of this seminar is to learn something about shifted symplectic structures and their applications. To get there, we'll start with the basics in derived algebraic geometry.
I. Background
Speaker: Su, Tao Notes01
Abstract: Derived intersection: a derived perspective on Bezout's theorem for degenerate intersections, Serre's intersection formula. Deformation theory: a derived interpretation of the full cotangent complex. Derived/homotopy fiber product and the base change theorem. A preview for derived symplectic geometry: shifted symplectic and Lagrangian structures, derived Lagrangian intersections, derived mapping stacks, etc.
Speaker: Su, Tao Notes02
Abstract: Simplicial sets/objects. Relation with topological spaces: geometric realization and the singular set functor. Relation with categories: the first truncation functor and nerve. Relation with chain complexes: the Dold-Kan correspondence.
Speaker: Su, Tao (expanded) Notes03
Abstract: Motivation and axioms for model categories. Examples: topological spaces, simplicial sets, cochain complexes in nonpositive degrees. Localization and homotopy category. Derived functors, Quillen adjunction, and homotopy limits/colimits. Construction of model structures: combinatorial model categories, projective/injective model structures on diagram categories, right transferred model structures via adjunction.
Speaker: Su, Xiao-Yu Notes04
Abstract: In this talk, we will introduce two models for (infinity, 1)-category namely the simplicial enriched category and the quasi-category. We will talk about their model structures, i.e. the Dywer-Kan model structure on the category of simplicial enriched categories and the Joyal model structure on the category of quasi-categories. The localizations and limits/colimits. At the end, we will introduce the adjoint pair of path category functor and homotopy coherent nerve , which induces a Quillen equivalences between the two models.
II. Basics in derived algebraic geometry
Speaker: Su, Tao (expanded) Notes05
Abstract: The (right transferred) model structure on derived algebras, semi-free resolutions. Proper model categories, computation of homotopy pushout/pullback. The (infinity,1)-category of derived affine schemes: mapping spaces and polynomial differential forms on simplices. From Quillen adjunction to (infinity,1)-adjunction. From homotopy limits/colimits to (infinity,1)-limits/colimits.
Speaker: Su, Tao (expanded) Notes06
Abstract: Cotangent complexes for derived affine schemes and their applications:
Derived derivations and cotangent complexes. Applications: square-zero extensions, deformation, and obstruction theory. Truncations for derived algebras, Postnikov towers, and connectivity results.
Speaker: Su, Tao (expanded) Notes07 Complementary notes on EGA IV 17.9.1
Abstract: Various notions for morphisms of derived affine schemes and their basic properties:
Finiteness conditions: finitely presented morphisms, finite cell objects. Flatness, and Zariski open embeddings. Smoothness and infinitesimal lifting properties: (formal) unramified, etale and smooth morphisms. Strongness and alternative characterizations.
Speaker: Su, Xiao-Yu
Abstract: In this talk, we will introduce the motivation and basic concepts of topoi, i.e. sites, topoi, and morphisms between them. Then we will give several examples.
Speaker: Su, Xiao-Yu
Abstract: In this talk, we will introduce some basic concepts of infinity-topoi. We will first recall some concepts about stacks and then give a sketch of different versions of higher topos theory.
Speaker: Su, Tao
Abstract: In this talk, we will introduce the basic concepts of derived stacks:
Left Bousfield localization of model categories. HA contexts and prestacks, examples: higher algebraic geometry, and derived algebraic geometry. Model sites and homotopy sheaves of prestacks, examples: etale site in AG, and derived etale site in DAG. Stacks on a model site, examples: higher stacks in AG, and derived stacks in DAG.
Speaker: Su, Tao
Abstract: Homotopy hypercovers and hyperdescent. HAG contexts and representable stacks, examples: Higher AG, DAG. Derived schemes. Derived algebraic stacks. Properties of morphisms of derived stacks. Truncations and extensions.
Speaker: Su, Tao
Abstract: In this talk, we will introduce a few examples of derived schemes and derived algebraic stacks:
Injective model structures and (derived) mapping stacks. Derived loop spaces. Moduli space of local systems. Moduli space of vector bundles on a projective variety. Moduli space of stable maps.
Speaker: Su, Xiao-Yu
Abstract: In this talk, we will introduce the basic concepts of quasi-coherent sheaves on geometric stacks in an ad-hoc way. We will talk about the direct and inverse image functors, projection formula and the base change theorem.
Speaker: Su, Tao
Abstract: In this talk, we will focus on cotangent complexes and representability of derived stacks, plus an application to the geometricity of certain derived mapping stacks:
Cotangent complexes and obstruction theory for derived stacks. Lurie's representability theoreom. Examples of applications: derived schemes of maps between schemes, moduli of algebraic G-bundles, de Rham and Dolbeault moduli space, etc.