Integrable systems, gravity and moduli spaces

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The modern theory of integrable systems can be traced back to the XIXth century, when British engineer John Scott Russell observed a soliton (which he called the "wave of translation") in the Union Canal at Hermiston, Edinbourg. This event, which happened in August, 1834, found its explanation only in the middle of 20th century, when the modern theory of integrable systems (and in particular the theory of Korteveg-de-Vries equation which describes non-linear waves on shallow water observed by Russell) started its explosive development. Other roots of the theory lie in the classical geometry of surfaces: another classical integrable system - the sine-Gordon equation - appeared in works of Bianchi in 1920's; about the same time the theory of isomonodromy deformations by Schlesinger and Painlevé equations were discovered. By now the theory of integrable systems has developed into a very powerful tool used in various branches of mathematics and physics. A fundamental development in the evolution of integrable systems occurred in the '80s when the Japanese school of Jimbo connected two distant areas of science; the theory of integrable waves and statistical physics on one side and the pure mathematical endeavor of P. Painlevé, who was interested in the classification of ordinary differential equations. The result of this merge was the invention of the theory of a certain special class of functions, named "tau-functions" which now appear in ever more numerous areas of mathematics, physics and even combinatorics. The goal of the project is to develop the theory tau functions and apply to the theory of moduli spaces, certain dynamical systems introduced by N. Hitchin in the '90s and the theory of the Schrödinger equation on Riemann surfaces and its "semiclassical" analysis. We are going to define the tau-function for isomonodromic deformations on Riemann surfaces of non-trivial topology (the "higher genus surfaces") and relate it to the Hamiltonian formalism of monodromy map. Furthermore, we are going to apply the theory of tau-functions to Hitchin systems. We shall also study the Schrödiger equation on a Riemann surface, together with its Wentzel-Kramers-Brillouin (WKB) approximation and monodromy map. This object turns out to have a rich and beautiful geometric structure, and also plays an important role in the theory of Yang-Mills equations. Natural quantization of classical integrable systems produces many physically important quantum models which can be treated explicitly. One of such models appears in the theory of gravity - it is the model of Einstein-Rosen waves with two polarizations. Such model can be quantized on algebraic level, and it turns out to arise in one of natural approaches to quantization of the full four-dimensional gravity. The completion of the development of the quantum model of non-linear Einstein-Rosen waves is one of the main goals of the proposal.