Integrable systems and applications

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In its modern formulation (since the discovery of the "inverse spectral" method), the theory of integrable systems has had enormous impact in a variety of domains, both in mathematical physics and pure mathematics, including: 1) nonlinear integrable dynamics (solitons and nonlinear quasi-periodic flows, with applications to: optics, fluid dynamics, superconductivity phenomena); 2) the quantum inverse scattering method (with applications to quantum spin chains, vertex models and other integrable lattice models in statistical mechanics); 3) the spectral theory of random matrices (with applications to the statistical theory of spectra of large nuclei; graphical enumeration problems relating to moduli spaces of Riemann surfaces, and "universality" phenomena regarding the eigenvalues of random operators, and discrete random processes; 4) enumerative geometry, moduli spaces and combinatorics (Gromov-Witten and Donaldson-Thomas invariants; Hurwitz numbers, Hodge invariants; Topological Recursion); ***5) random growth processes and integrable probabiity (crystal growth, exclusion processes, Schur processes) and 6) Isomonodromic deformations of meromorphic covariant derivatives on Riemann surfaces.*** A key element is the notion of "Tau functions", as introduced by Sato et al. These may be seen, variously, as: 1) generating functions, in the sense of canonical transformation theory, of a complete set of commuting flows; 2) generating functions for isomonodromic deformations dynamics; 3) partition functions and multipoint correlators for families of random matrix models, with respect to parametric families of measures; 4) generating functions for transition probabilities underlying random dynamics of "integrable" random processes; 5) generating functions, in the sense of combinatorics, of the various geometric, enumerative, geometrical and topological invariant mentioned above. *** This proposal aims at the further development of some key concepts and methods introduced by the author in the theory of integrable systems, as discussed above. Namely, we propose to:***1. Develop further the notion of "weighted Hurwitz numbers" and their generating functions within the framework of tau functions and integrable systems and embed this within the framework of Topological Recursions of Eynard and Orantin. (In collaboration with Eynard, and others).***2. Study the semiclassical asymptotics and small parameter limits of "Quantum Hurwitz numbers" (recently introduced by the author).***3. Analyze the discrete integrable dynamics generated by "cluster mutations" as isospectral flows of Lax matrices and to analyze the discrete integrable dynamics generated by polytope recursion relations in the framework of isotropic Grassmannians and tau functions.*****