I will review recent works on geometries underlying scattering amplitudes of (certain generalizations of) particles and strings  Tree amplitudes of a cubic scalar theory are given by "canonical forms" of the so-called ABHY associahedra defined in kinematic space. The latter can be naturally extended to generalized associahedra for finite-type cluster algebra, and for classical types their canonical forms give scalar amplitudes through one-loop order. We then consider vast generalizations of string amplitudes dubbed “stringy canonical forms”, and in particular "cluster string integrals" for any Dynkin diagram, which for type A reduces to usual string amplitudes. These integrals enjoy remarkable factorization properties at finite $\alpha'$, obtained simply by removing nodes of the Dynkin diagram; as $\alpha'\rightarrow 0$ they reduce to canonical forms of generalized associahedra, or the aforementioned tree and one-loop scalar amplitudes.

I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.