Vol 22 No. 2

On the equivarant De-Rham cohomology for non-compact Lie groups
Camilo Arias Abad, Bernardo Uribe
Let G be a connected and non-necessarily compact Lie group acting on the connected manifold M. In this short note we announce the following result: for a G-invariant closed differential form on M, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient.
Symplectic Lefschetz fibrations from a Lie theoretical viewpoint
B. Callander, E. Gasparim, L. Grama, L. A. B. San Martin
This is an announcement of results proved in [15], [16], [10], and [11] where methods from Lie theory were used as new tools for the study of symplectic Lefschetz fibrations.
Applications of Gauged Gromov-Witten Theory: A Survey
Eduardo González
This is a short survey article on applications of gauged Gromov-Witten theory into the understanding of Gromov-Witten theory of GIT quotients of a smooth projective variety by a reductive group. In particular we will explain how several classical results in equivariant cohomology extend to quantum cohomology. These include wall crossing results, Witten localisation and abelianisation. We also describe a GIT version of the so called crepant conjecture.
Derived Mackey functors and profunctors: an overview of results
D. Kaledin
In this paper we overview the theory of derived Mackey functors and profunctors.
Perverse Schobers
Mikhail Kapranov, Vadim Schechtman
In this paper we introduce the notion of perverse schober and talk about how it is related to the understanding of Fukaya categories.
Sheaf of categories and categorical Donaldson theory
Ludmil Katzarkov, Yijia Liu
In this paper we take a new look at categorical linear systems applying the technique of sheaves of categories. We combine this technique with the theory of categorical K ̈ahler metrics in order to build two parallels:
1) A parallel with Donaldson theory of K ̈ahler-Einstein metrics.
2) A parallel with Donaldson theory of polynomial invariants.
As an outcome we introduce sheaves of categories which cannot be connected to potentials and obstructions to that are the moduli spaces of stable objects. Connections of sheaves of categories with Homological Mirror Symmetry for non-complete intersections and the procedure of arborealization are discussed as well.