One can associate a `heart fan’ $\Delta(H)$ to the data of an abelian category $H$ and a finite rank quotient lattice of its Grothendieck group $K(H)$. The fan is generated by the `heart cones’ of $H$ and its (forward) Happel—Reiten—Smal\o tilts. (These are defined inside a choice of ambient triangulated category, but the heart fan is independent of this choice.) 

 

I will explain the construction and illustrate it with a number of low-dimensional examples. If there is time I will also explain how its local structure is governed by heart fans of Serre subcategories of $H$, how the heart fan relates to the g-fan when $H$ is the module category of a finite-dimensional algebra, and how the heart fan relates to Bridgeland stability conditions. 

 

This is joint work with Nathan Broomhead, David Pauksztello and David Ploog (arXiv:2310.02844).

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