# Convex geometry for fans of triangulated categories

Fans and other convex-geometric objects have recently appeared in homological algebra in several related contexts. For example, as g-fans in the silting theory of finite-dimensional algebras and as scattering diagrams in Bridgeland stability theory. I will discuss joint work with David Pauksztello, David Ploog and Jon Woolf on a general construction which we hope will provide a natural and unifying framework.
Starting with a triangulated category D and a finite rank quotient lattice L of its Grothendieck group, we show that each heart H in D determines a closed convex `heart cone' in the dual vector space V=Hom(L,R). The heart cones of H and all its forward tilts form a `

heart fan’ in V. If H is `algebraic’, i.e. is a length category with finitely many simple objects, then the heart cone is simplicial and the heart fan is complete.