Conference: Amplituhedra, Cluster Algebras and Positive Geometry

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The conference "Amplituhedra, Cluster Algebras and Positive Geometry" will be held on May 29-31 2024 at the Harvard Center of Mathematical Sciences and Applications (CMSA). In recent years, a remarkable paradigm shift has occurred in our understanding of quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron, the first and major example of a positive geometry, which describes particle interactions in a certain quantum field theory (QFT). We aim to explore connections between the amplituhedron and cluster algebras, a mathematical theory with broad applications across various areas of mathematics and mathematical physics. The conference will also actively engage and empower junior researchers and women, ensuring their integral presence and impactful contributions to the conference.More precisely, building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that "explains" the so-called BCFW recurrence for computing scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras – originally introduced by Fomin and Zelevinsky to study total positivity – have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. Our program will bring together a wide range of mathematicians and physicists working on adjacent areas both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT. The conference website is https://cmsa.fas.harvard.edu/event/amplituhedra2024/This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.