Jonathan Woolf  - University of Liverpool
Title: Heart fans

Abstract: 

The aim of this talk is to describe some new convex-geometric invariants of abelian and triangulated categories, illustrated by some low-dimensional examples. This is joint work with Nathan Broomhead, David Pauksztello and David Ploog.

In slightly more detail, let $D$ be a triangulated category and $K(D)\to \Lambda$ a homomorphism from its Grothendieck group to a finite rank free abelian group. Each bounded heart $H \subset D$ determines a `heart fan’ $\Sigma(H)$ in the dual of $\Lambda\otimes \mathbb{R}$. This fan depends only on the heart $H$, not the ambient triangulated category $D$, and is complete when $H$ is a length category.

When $H$ is the module category of a finite-dimensional algebra the heart fan is a completion of the g-fan, which is the sub-fan of simplicial cones. The heart fan is also closely related to the wall-and-chamber structure of the algebra, and the associated scattering diagram.

The local structure of the heart fan at a point in its support is encoded in a tangent fan; points of the tangent fan can be interpreted as (lax) stability functions on certain bounded hearts in $D$.

If there is time I will discuss how the heart fans, and their tangent fans, of all bounded hearts in $D$ can be glued together to form a global invariants of $D$.