Quantized Lagrangian submanifolds of moduli spaces and representation theory

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Despite its relatively recent development, the theory of cluster algebras has proven to be a powerful and versatile tool across a broad range of areas of modern mathematics and physics, and has been instrumental in building bridges between these disciplines. This project focuses on exploring new structures in representation theory, quantum topology and enumerative geometry from the cluster-algebraic perspective. In many problems in these areas, identifying an underlying cluster structure reveals hidden combinatorial structures and symmetries, thereby leading to explicit, constructive proofs of deep results. This research program will closely involve early career researchers, with plans to disseminate both the necessary background ideas and cutting edge results from the project through the organization of mini-schools aimed at graduate students and postdocs in adjacent areas of research.More specifically, this project focuses on the quantum geometry of moduli spaces of local systems on surfaces, and the problem of quantizing Lagrangian submanifolds of these symplectic moduli spaces. Constructing such a quantization amounts to producing a canonical vector in the Hilbert space associated to the surface, and in accordance with the philosophy of topological quantum field theory, these quantized Lagrangians are closely related to the geometry of three-manifolds. The PI will systematically study this quantization problem, developing along the way new structures on the underlying moduli spaces of local systems based on their connection with representation theory. New directions to be explored include the construction of integrable systems providing higher Teichmueller-theoretic analogs of the classical Fenchel-Nielsen Hamiltonians on Teichmueller spaces, as well understanding the behavior of the cluster structure for moduli spaces of local systems with non-generic monodromy data at punctures, which is intimately connected with the theory of double affine Hecke algebras and their higher genus analogs.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.