Derived geometry provides a way to deal solve many problems that arrise in geometry:
- infinite dimensional mapping space becomes somehow finite dimensional.
- all fiber products and quotients become smooth.
The goal of these lectures is show that it provides a suitable framework for the so-called AKSZ construction (after Alexandrov-Kontsevich-Schwartz-Zaboronski).
Here is the plan we will try to follow:
Lecture 1: Motivation + a short introduction to the theory of derived/homotopical geometry, after Toen-Vezzosi and Lurie.
Lecture 2: shifted symplectic structures in the context of derived geometry, after Pantev-Toen-Vaquie-Vezzosi.
Lecture 3: isotropic and Lagrangian structures in derived symplectic geometry.
Lecture 4: extended topological field theories from Lagrangian structures.
In this minicourse we will introduce categorical actions of quantum groups and braid groups. We will study examples of such actions in algebraic geometry and finite-dimensional algebra, and discuss how these examples are used to give constructions of homological invariants of knots.
Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. In the course we will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.
A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.
The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Soibelman and Kontsevich in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.
TBA