Totally Positive Spaces and Cluster Algebras

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Combinatorics is the study of discrete structures such as permutations and graphs, while topology deals with properties of geometric shapes that stay invariant under continuous deformations. The goal of the project is to apply a mix of topological and combinatorial techniques to questions arising in various areas of mathematics and physics. For example, recent connections between combinatorics and scattering amplitudes give rise to new methods to guide high-energy particle physics experiments. Other applications arise in determining interior properties of materials from boundary measurements, as well as in electrical impedance tomography and other types of medical imaging. The surprisingly natural combinatorial structures associated with the underlying topological spaces allow one to find unexpected connections between seemingly unrelated areas. The award provides research training of graduate studetns.This work comprises several projects that directly involve the positive Grassmannian. The topology of this space has been recently determined; however, the topology of several related spaces remains not fully understood. For example, the physics of scattering amplitudes is intimately related to the amplituhedron, which is a linear projection of the positive Grassmannian. One of the projects involves understanding its topological structure and the combinatorics of its triangulations. Another example is the space of planar Ising networks, which was shown to coincide with the positive orthogonal Grassmannian. This approach has promising applications to the questions of universality and conformal invariance of the Ising model. The resulting cell decomposition of the space of planar Ising networks suggests a surprising direct connection with the space of planar electrical networks. The underlying algebraic structure of such spaces is described by cluster algebras; one of the projects involves studying integrability properties of dynamical systems arising from cluster algebras.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.