Higher representation theory: a view towards applications

The aim of this project is to further develop general tools and techniques to study 2-categories of 2-representations of finitary 2-categories with the emphasis on the structure theory and on classification of various classes of 2-representations.

In the abstract half of the project,we plan to understand the role of Drinfeld center and double centralizer phenomena in finitary 2-representation theory and to classify simple transitive  2-representations of of Soergel bimodules in positive characteristic.

We also aim at developing a Koszul duality theory for finitary 2-representations.In the applied half of the project, we plan to use results from finitary 2-representation theory to study representations of Lie algebras and superalgebras and to understand McKay correspondence in higher dimensions.

In particular, we plan to attempt classification of  parabolic projective functors in all types, prove a general theorem reducing classification of simplesupermodules for Lie superalgebras to classification ofsimple modules over Lie algebras and study a Lie theoreticanalogue for the desingularization map of the singular braidmonoid and its application to Vassiliev knot invariants.Our methods  range over several  mathematical areas, from homological algebra and representationtheory, to category theory and advanced combinatorics.

The general results of this project should have potentialapplications in lower dimensional topology,algebra, geometry, category theory and theoretical physics.