The 2022 distinguished lecture series will be given by Professor Mikahail Kapranov , U. of Tokyo, 9th, 10th and 11th of May 2022

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
comultiplication) 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 cohomology. 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.

All lectures will take place in room 570 of the Science & Education Building. For further information call 04-8249294