This book is a significant contribution within and across High Energy Physics and Algebraic Combinatorics. It is at the forefront of the recent paradigm shift according to which physical observables emerge from geometry and combinatorics. It is the first book on the amplituhedron, which encodes the scattering amplitudes of N=4 Yang-Mills theory, a cousin of the theory of strong interactions of quarks and gluons. Amplituhedra are generalizations of polytopes inside the Grassmannian, and they build on the theory of total positivity and oriented matroids. This book unveils many new combinatorial structures of the amplituhedron and introduces a new important related object, the momentum amplituhedron. Moreover, the work pioneers the connection between amplituhedra, cluster algebras and tropical geometry. Combining extensive introductions with proofs and examples, it is a valuable resource for researchers investigating geometrical structures emerging from physics for some time to come.
1. Introduction.-
2. The Amplituhedron.-
3. The Hypersimplex.-
4.
T-duality: the Hypersimplex VS the Amplituhedron.-
5. Positroid
Triangulations.-
6. The Momentum Amplituhedron.-
7. Cluster Algebras and
Amplituhedra.-
8. Conclusions
Matteo Parisi is a joint postdoctoral fellow in physics at IAS and in math at Harvard CMSA. He is also a Lecturer in Physics at Princeton University. He obtained his DPhil (PhD) in Mathematics at the University of Oxford. He has been recipient of several honors and awards, among which the Daniel Sachs and Della Riccia scholarships during his DPhil at Oxford, the DAAD scholarship during his Master in Germany, and two Certificates of Merit by the Rector of his alma mater University of Bologna. His research lies at the intersection of high energy physics and algebraic combinatorics. He has been working on a new program in which quantum mechanical observables in particle physics and cosmology arise from underlying novel mathematical objects. Physical properties purely emerge from their combinatorics and geometry. In particular, his research focuses on scattering amplitudes in quantum field theories, in relationto the(positive) Grassmannian, Amplituhedra, tropical geometry and cluster algebras. Being one of the leading contributors in this program, he produced over ten publications in major peerreviewed journals. These ranged from theoretical physics (Journal of High Energy Physics) to mathematical physics (Communications of Mathematical Physics) and pure mathematics (Communications of American Mathematical Society), with more than 250 total citations. Teaching, outreach and advocating for inclusivity and diversity, within and beyond academia, are a key part of his identity. His teaching experiences ranged from holding Lectureships at two Oxford colleges (St. Peters and Keble), to volunteering to organize and teach teenagers at the first ever Math Camp in Cameroon. He has worked with equality and diversity units of the university; he is a trained LGBT+ Role Model and organized conferences and interacted with charities on LGBTQ+ issues. In his free time, he loves playing the piano, playing volleyball, experiencing the beauty of art (concerts,
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