Tilings and cluster algebras for the amplituhedron

Cluster algebras (Fomin-Zelevinsky 2000) are a type of commutative algebras with various applications in representation theory and geometry, originating in the theory of total positivity (Lusztig 1994). In recent years, connections emerged between cluster algebras and scattering amplitudes in certain quantum field theories (starting with Golden-Goncharov-Spradlin-Vergu-Volovich 2013). In this talk I will discuss the origin of this connection in the setting of the amplituhedron (Arkani Hamed-Trnka 2013), a generalization of the nonnegative Grassmannian which admits a related algebraic and combinatorial structure whose "volume" is conjectured to express certain scattering amplitudes. I will introduce these topics and discuss how they are connected and clarified using the recursion relations for scattering amplitudes introduced by Britto, Cachazo, Feng and Witten (BCFW 2005).

Based on joint works with Even-Zohar, Parisi, Sherman-Bennett, Tessler and Williams.