Recent years have seen the discovery of many intriguing connections between algebraic combinatorics and theoretical particle physics, specifically in scattering amplitudes. One such connection is the amplituhedron (Arkani-Hamed--Trnka 2013) a generalization of the nonnegative Grassmannian which admits a related combinatorial and geometric structure. Its "volume'' expresses scattering amplitudes in certain quantum field theories. Another such connection is the appearance of combinatorial cluster-algebraic structures in scattering amplitudes (starting with Golden-Goncharov-Spradlin-Vergu-Volovich 2013). In this talk I will introduce these topics and discuss how they are illuminated by 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.