Positive geometries are semialgebraic sets with certain special properties that play an important role in particle physics and cosmology. The notion was introduced by Arkani-Hamed, Bai and Lam in 2017. In recent work, Brown and Dupont recast this notion in the language of mixed Hodge theory. The motivation for the development of the machinery of positive geometries was the discovery of the amplituhedron, a geometric object dramatically simplifying the computation of scattering amplitudes. It is however still a conjecture that ampliutehdra are positive geometries. In this talk I will compate the two notions of positive geometries and discuss recent progress on the amplituhedron conjecture. Based on joint work with Joris Koefler and Rainer Sinn.

In 1975, Wachspress introduced the notion of polycons and their adjoint curves. He conjectured that the adjoint of a regular real polycon would never vanish anywhere in the polycon's interior. In this talk we present a counterexample to this conjecture.

In establishing this counterexample, some beautiful geometry of the space of (adjoints of) polycons is revealed. it will be shown that adjoint curves satisfy a recursive property expressed in terms of contact curves. We will see how this implies a natural correspondence between equivalence classes of linear determinantal representations of cubic curves, families of polycons bounded by three conics, and their (cubic) adjoint curves.