This course is put on by the Taught Course Centre, and will be transmitted to the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London.

I am sorry, but if you are not at one of these five universities, you can't come to the course. I have no control over this. Even if someone gave you the Teams link, you can only get into the Teams meeting if your e-mail address has been authorized by the TCC staff.

Derived Algebraic Geometry is the study of "derived" schemes and "derived"
stacks. These are enhanced versions of classical schemes and (Artin) stacks
which have a richer geometric structure. This richer structure is particularly
important in problems involving virtual classes and enumerative invariants, and
in moduli spaces of objects in Calabi-Yau categories. As an example, let
*X* be a smooth projective scheme of dimension *m*. Then we can
form the classical moduli stack *M* and the derived moduli stack
** M** of coherent sheaves on

Derived Algebraic Geometry is famously hard to learn -- the foundational papers, due to Lurie and Toën-Vezzosi, run to 1000's of pages. Part of the difficulty is that the theory must be set in the world of ∞-categories.

After a brief discussion of derived geometry in the first week, I will spend several weeks talking about other stuff which is interesting for its own sake: classical stacks, triangulated categories and derived categories (derived categories are a necessary prequel to derived stacks and derived algebraic geometry, which were developed much earlier than DAG), and ∞-categories, before trying to explain derived schemes and derived stacks at the end of the course.

I am not an expert in DAG. I've mostly picked the subject up by osmosis, without actually spending two years reading Lurie and Toën-Vezzosi. So if you ask me technical questions about the fppf topology, etc, I won't know the answer. I can probably teach you how to bluff about DAG at parties, though.

Lecture 1: Different kinds of spaces
in algebraic geometry: schemes, stacks, higher stacks, derived stacks. Basics
of category theory, categories, functors. The Yoneda Lemma. Schemes as functors
Alg_{K} → Sets, and stacks as functors Alg_{K} → Groupoids. Grothendieck’s approach to
moduli spaces as ‘representable functors’.

Lecture 2: What is derived geometry? Derived schemes and stacks. Commutative differential graded algebras, examples. Bézout’s Theorem and derived Bézout’s Theorem. Patching together local models in derived geometry. Fibre products in ordinary categories; why we need higher categories in derived geometry.

**Lecture 3:** Introduction to 2-categories. Fibre products in
2-categories. First look at ∞-categories.

**Lecture 4:** Global quotient stacks, as a way of introducing
Deligne-Mumford and Artin stacks before giving the formal definition, and
emphasizing the 2-categorical aspects. 1- and 2-morphisms of quotient stacks.
The classifying stack of principal bundles. Fibre products. Sheaves and stacks
on topological spaces.

**Lecture 5:** Classical stacks. Moduli functors, moduli
schemes and moduli stacks. Grothendieck topologies. Sheaf-type properties of
functors. Definition of stacks. Deligne-Mumford and Artin stacks.

**Lecture 6: **Abelian categories. Coherent sheaves coh(X) and
quasicoherent sheaves qcoh(X). Examples.

**Lecture 7:** Introduction to derived categories and
triangulated categories.

**Lecture 8:** More on triangulated categories and derived
categories. Derived categories of coherent sheaves D^{b}coh(X).
Relation to ∞-categories.

**Lecture 9:** Informal introduction to ∞-categories.
Model categories. Simplicial sets. Simplicial categories.

**Lecture 10:** More about simplicial sets. Simplicial objects.
Kan complexes and weak Kan complexes. Quasicategories, a model for
∞-categories.

**Lecture 11:** Stable ∞-categories as the "correct"
definition of triangulated category. Dg-categories. Segal categories.
Foundational theories of DAG in the literature. Higher stacks.

**Lecture 12:** The definitions of derived schemes and derived
stacks, following Toën-Vezzosi. Cotangent complexes of derived schemes and
stacks. Quasi-smooth derived stacks. Examples of derived schemes and stacks.

**Lecture 13:** Enumerative invariants in Algebraic Geometry
(e.g. Gromov-Witten invariants, algebraic Donaldson invariants). Obstruction
theories and virtual classes on classical schemes and stacks, following
Behrend-Fantechi. How they are used to define enumerative invariants. Relation
to Derived Algebraic Geometry: a classical stack with obstruction theory is the
semi-classical shadow of a quasi-smooth derived stack. Enumerative invariants
usually count only 'semistable' objects. Stability and semistability, in the
sense of stability conditions on abelian categories, and in Geometric Invariant
Theory, and the relation between these.

**Lecture 14:** Shifted symplectic derived stacks, following
Pantev-Toën-Vaquié-Vezzosi, and applications to Donaldson-Thomas theory of
Calabi-Yau 3- and 4-folds.

[1] B. Toën, ‘Derived Algebraic Geometry’, arXiv:1401.1044.

[2] B. Toën, *'Higher and derived stacks: a global overview'*, math.AG/0604504.

[3] J. Lurie, ‘Derived Algebric Geometry V: Structured spaces’, arXiv:0905.0459.

[4] B. Toën and G. Vezzosi, ‘Homotopical Algebraic Geometry II: Geometric
Stacks and Applications’, Mem. Amer. Math. Soc. 193 (2008), no.
902. math.AG/0404373.

[5] B. Toën and G. Vezzosi, *'From HAG to DAG: derived moduli
stacks'**, *math.AG/0210407.

[6] T. Pantev, B. Toën, M. Vaquié, and G. Vezzosi, *'Shifted Symplectic
Structures'*, arXiv:1111.3209.

[7] R. Hartshorne, *'Algebraic Geometry'*, Graduate Texts in Math.
52, Springer, New York, 1977.

[8] T.L. Gomez, *'Algebraic stacks'*, Proc. Indian Acad. Sci. Math.
Sci. 111 (2001), 1-31. math.AG/9911199.

[9] M. Olsson, *'Algebraic Spaces and Stacks'*, A.M.S. Colloquium
Publications 62, A.M.S., Providence, RI, 2016.

[10] G. Laumon and L. Moret-Bailly, *'Champs algebriques'*, Ergeb.
der Math. und ihrer Grenzgebiete 39, Springer-Verlag, Berlin, 2000.

[11] R.P. Thomas, *'Derived categories for the working
mathematician'*, math.AG/0001045.

[12] S.I. Gelfand and Y.I. Manin, *'Methods of Homological Algebra'*,
second edition, Springer, 2002.

[13] D. Huybrechts, *'Fourier-Mukai transforms in Algebraic
Geometry'*, Oxford University Press, Oxford, 2006.