DAG learning seminar

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.

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.