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This review is a primer on recently established geometric methods for observables in quantum field theories. The main emphasis is on amplituhedra, i.e. geometries encoding scattering amplitudes for a variety of theories. These pertain to a broader family of geometries called positive geometries, whose basics we review. We also describe other members of this family that are associated with different physical quantities and briefly consider the most recent developments related to positive geometries. Finally, we discuss the main open problems in the field. This is a Topical Review invited by Journal of Physics A: Mathematical and Theoretical.
Original languageEnglish
Article number033001
Number of pages43
JournalJournal of Physics A: Mathematical and Theoretical
Issue number3
Early online date29 Dec 2020
Publication statusE-pub ahead of print - 29 Dec 2020


  • Positive geometries for amplitudes
  • Scattering amplitudes in maximally supersymmetric Yang-Mills theory
  • Scattering amplitudes in scalar cubic theory


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