Mathematics

Description
Derived algebraic geometry is the natural setting in which mathematicians can study
deformation theory of schemes, moduli spaces, and highly singular objects employing the
ideas and the technical machinery of homological and categorical algebra. In a broader
sense, it also provides a unifying theoretical and conceptual framework encompassing
simultaneously the worlds of algebraic geometry, algebraic topology, homotopy theory and
higher category theory.
The goal of this course is to get the students acquainted with the main formalisms and
concepts of homotopical algebra, which provides the "affine charts" of the objects of derived
algebraic geometry (i.e., derived schemes and derived stacks). We will be mainly interested
in removing the "esoteric" aura surrounding the subject via motivating examples, clarifying
exercises and the development of the theory. As such, we will focus on the applications to
intersection theory, deformation theory in characteristic 0, derived categories and Morita
theory, rather than on the technical and cumbersome details of the proofs.
Prerequisites
General knowledge of algebraic geometry and commutative algebra. Basic notions of cat-
egory theory (adjunctions, equivalence of categories, Yoneda lemma, sheaves and presheaves
over Grothendieck sites, abelian categories, monoidal categories) will be necessary, but
since they will be developed in the framework of higher category theory we will always
recall the classical definitions and main features. Some knowledge of algebraic topology
(CW-complexes, homotopy groups, singular and cellular homology and cohomology) is
helpful but not necessary by any means.
Essential list of references
• Lurie, J.: "Higher Topos Theory", "Higher Algebra", the "Derived Algebraic Geometry"
series, "Spectral Algebraic Geometry".
• Toën, B. and Vezzosi, G.: "Homotopical Algebraic Geometry" I and II.
• Gaitsgory, D. and Rozenblyum, N. "A Study in Derived Algebraic Geometry", vol. 1
and 2.