# An algebraic variety related to tau-tilting theory

Let A be a finite-dimensional algebra of finite representation type. I will describe an affine algebraic variety whose totally non-negative part reflects the combinatorics of the tau-tilting fan of A. Starting from a Dynkin quiver, one obtains something closely related to the corresponding Fock–Goncharov cluster X variety, while in general, points on (one component of) the variety can be given in terms of ratios of F-polynomials; the upshot is that this construction can be viewed as an extension of some of the beautiful features of cluster algebras to a more general setting. Nonetheless, familiarity with cluster algebras will not be needed to understand the talk. A conjecture related to functoriality properties of the construction will be discussed. This is part of a joint project with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.