Marcolli works from her invited lectures delivered at several universities to address questions and reinterpret results and constructions from number theory and arithmetric algebraic geometry, generally is they are applied to the geometry and arithmetic of modular curves and to the fibers of archimedean places of arithmetic surfaces and varieties. One of the results is to refine the boundary structure of certain classes of spaces, such as moduli spaces (like modular curves) or arithmetric varieties completed by suitable fibers at infinity by adding boundaries that are not visible within algebraic geometry. Marcolli defines the noncommutative spaces and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois theory, and noncommutative geometry at arithmetric infinity.