Derived algebraic geometry is an extension of classical algebraic geometry which provides systematic treatment of "non-transversal situations" in a very satisfactory and workable way. Instead of using various moving lemmas, in DAG one keeps track of some additional information which roughly remembers in what way the intersection was non-transversal. Technically this is achieved by changing basic building blocks of algebraic geometry from ordinary commutative rings to dg-commutative rings or simplicial rings or, more generally, to commutative ring spectra. Consequently, the formalism of DAG turns out to be natural home for deformation theory, intersection theory and various moduli problems. More than that, this theory appears to be useful in various other parts of mathematics such as geometric representation theory, homotopy theory and mathematical physics.
In the first part of our seminar we will give a thorough introduction to the subject aimed at younger students assuming only some background in scheme algebraic geometry, basic homotopy and category theory. In the second part we will proceed to some of the major achievements in the subject such as
Formal deformation theory. In general it is not reasonable to expect that global deformations of some object are controlled by some structure of algebraic nature (instead one expect to have very non-linear moduli space of some sort). The story becomes much more tractable if one restricts to infinitesimal deformations, i.e. considers only formal moduli problems. In our course, we will try to cover formal deformations of stacks and linear categories. We will see that in the first case deformations are controlled by the tangent space (shifted by -1) with the canonical structure of Lie algebra and in the second case by the Hochschild cochain complex (with the canonical structure of E_2-algebra). In both cases the key roll is played by the notion of Koszul duality, which we will also review in an appropriate generality.
Shifted symplectic structures. Recall that one of the key features of (ordinary) symplectic manifolds is that one has a canonical equivalence of the tangent and co-tangent bundles induced by symplectic form. There is an analogous theory of symplectic structure in DAG, where roughly one defines 2-form on derived stack to be n-shifted symplectic, if it induces a non-degenerate pairing of tangent and cotangent bundle shifted by n. It turns out that a lot of familiar stacks such as cotangent bundle, BG or Perf admit shifted symplectic structure. In our course we will also show how one can build new symplectic stacks starting from given ones, use these to define deformation quantization and discuss some interesting applications of this machinery.
Application to elliptic cohomology. One can use methods of DAG to study stable homotopy theory in a geometric manner. One such example is the modern construction of the spectrum of topological modular forms which can be defined as the global sections of the structure sheaf on the moduli space of oriented derived elliptic curves. During our seminar we will study relevant theory of abelian varieties over commuative ring spectra and will sketch this construction.
Plan
Introduction to higher category theory and higher algebra.
Prestacks and stacks, quasi-coherent sheaves.
Derived schemes and Deligne-Mumford stacks, geometric stacks, examples (de Rham stacks, BG, Perf).
Ind-coherent sheaves for schemes and general stacks.
Inf-schemes and cotangent complexes.
Koszul duality. The special case of Lie-coComm duality.
Formal deformation theory in commutative world over arbitrary base.
General formal deformation theory and deformations of associative algebras over a field.
Ringed topoi approach to algebraic geometry, finite presentation and proper morphisms.