### Overview

Spherical objects were introduced by Seidel and Thomas to construct autoequivalences of triangulated categories.
By definition, the Serre functor shifts such an object (Calabi-Yau property) and its graded endomorphism algebra is two-dimensional.
Under Kontsevichâ€™s Homological Mirror Symmetry conjecture, these autoequivalences are mirror dual to generalised Dehn twists about Lagrangian spheres.

Among the possible generalisations, there is the notion of a spherical functor, which can be considered as a relative version of a spherical object
(turned into a functor by taking the triangulated category generated by it and the inclusion functor).

Another generalisation is to drop the Calabi-Yau property, such an object is called spherelike. There the associated twist needs not to be autoequivalence of the ambient triangulated category.
Still there is a unique maximal subcategory (the so-called spherical subcategory) where it becomes spherical.

The article Spherical subcategories in algebraic geometry suggest that there is some connection between those subcategories and birational geometry. For example, given a smooth projective Calabi-Yau variety *X* and let *Y* be a blowup of *X* in an arbitrary number of points.
Then *O _{Y}* is spherelike and its spherical subcategory is equivalent to the derived category of

*X*.

This workshop aims to give some background to find further examples of this kind.

### Schedule

The workshop will take place in the **lecture room (2.03)** of the Mathematical Institute of the University of Cologne on **Monday, February 15th, 2016**.

- 10:15 – 11:30

Daniel Huybrechts (Bonn)*The K3 category of cubic fourfolds* - 11:45 – 13:00

Martin Kalck (Edinburgh)*Spherical subcategories and new invariants for triangulated categories* - 14:15 – 15:30

Timothy Logvinenko (Cardiff)*P-functors* - 16:00 – 17:15

Andreas Krug (Marburg)*Spherical functors and hyperkähler fourfolds**

* Topic of this talk might change.

### Registration

A registration is not necessary, but if you consider to come, please send a short email to one of the organisers:

Andreas Hochenegger and David Ploog.

Sponsored by the SPP 1388 Representation Theory.