Lecture Series - Mathematics

By Prof. Mikhail Kapranov


Distinguished Lecture Series – Perverse sheaves and their categorification

Date &Time:

May 11, 2022


Room 570, Science & Education Building
Prof. Mikhail Kapranov
( Kavli Inst. for Physics & Mathematics of the Universe – University of Tokyo)

Perverse sheaves and their categorification
Perverse sheaves are by now a classical tool in algebraic geometry, having many applications in other areas such as representation theory. The lectures are based on joint works in progress with T. Dyckerhoff, V. Schechtman, Y. Soibelman and L. Soukhanov. They will present an approach to understanding perverse sheaves using real skeleta of complex manifolds and a program of “categorification” of the very concept of a perverse sheaf in which vector spaces are replaced by (enhanced) triangulated categories.
* On Monday May 9, 2022 at 4:00 PM,
The first lecture will give an elementary introduction to perverse sheaves and present several instances of their description in terms of data of mixed functoriality (Janus sheaves). Among natural examples of such data one finds Hochschild “bicomplexes” of Hopf algebras (involving both multiplication and comultipli-cation) and various induction/restriction diagrams in representation theory.
* On Tuesday, May 10, 2022 at 2:00 PM
The second lecture will outline the program of perverse schobers (categorical lifts of perverse sheaves), a concept which is conjectural in general but can be defined directly in simple situations. I will explain the role of perverse schobers as coefficient data for Fukaya categories (which can be themselves considered as categorifcations of middle (co)homology) similar to the role of ordinary sheaves as coefficient data for co- homology. I will also explain the connection with the Waldhausen S-construction known from algebraic K-theory.
* On Wednesday, May 11, 2022 at 11:00 AM
The third lecture will discuss the analog of the (geometric) Fourier transform for perverse schobers on the complex line and its relation to the Algebra of the Infrared of Gaiotto-Moore-Witten. In this way schobers can be seen to encode some of the features of 2d quantum field theories in the infrared limit.