Combinatorics of Total Positivity: Amplituhedra and Braid Varieties

The answers to real world problems, such as determining the behavior of particles in particle accelerators, are often quite complicated. Mathematics abstracts these complicated behaviors, and often reveals hidden structures; abstraction allows one to see the forest rather than the trees. For example, physicists Arkhani-Hamed and Trnka uncovered a high-dimensional mathematical object called the "amplituhedron" whose geometry should govern particle scattering. However, as abstraction increases, intuition decreases; it is easy to lose sight of the trees among the clouds. Algebraic combinatorics, as a mathematical discipline, is a tool to represent abstract mathematics in a more concrete way--similar to how a bar graph or scatter plot is a tool to represent a long list of numbers in a more intuitive way. In the case of the amplituhedron, combinatorics provides a way to break the amplituhedron up into smaller, simpler pieces. It also provides a way to visualize each piece, even though the pieces do not fit in three dimensions. It is through this combinatorics that the conjectural relationship between the amplituhedron and particle scattering is most apparent. In one project the PI will work to prove this conjectural relationship with collaborators Even-Zohar, Lakrec, Parisi, Tessler, and Williams. In general, the PI will seek to better understand the combinatorics of amplituhedra and related mathematical objects called cluster varieties. The PI will involve both undergraduate and graduate students in thisd research. The broader mathematical context for the proposed projects is the theory of total positivity. Classically, a matrix is totally positive if all minors are positive. Lusztig extended the notion of total positivity to partial flag varieties, while Postnikov independently defined the positive Grassmannian. The combinatorics of total positivity is incredibly rich, leading to the definition of cluster algebras by Fomin and Zelevinsky. The PI proposes to study two generalizations of total positivity through a combinatorial lens. The first project concerns amplituhedra, which generalize the positive Grassmannian and arise in particle physics. The PI will work to resolve conjectures on the relationship between tilings of m=4 amplituhedra and the computation of scattering amplitudes, as well as various conjectures on tilings of m=2 amplituhedra. The second project concerns cluster structures on braid varieties, which generalize positive partial flag varieties. The PI will further develop the combinatorics of this cluster structure, investigating 3D plabic graphs and their relationship to weaves, and explore applications. 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.