Geometrical structures in mathematical physics

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The proposed research aims at establishing and strengthening connections between mathematics and quantum physics. We believe this to be an important direction as such connections can bring (as they have done in the past) new insights in both fields. In general, using techniques of one field in another might lead to new ways of looking at problems and thus to discoveries that would be less easy to make if one stays within a closed area of research. On one hand, we are planning to use our insight on one differential equation important in mathematics (Painlevé-6) to reveal similarities between its structure and the structure of some other equations arising in mathematical and quantum physics (Korteweg-de Vries and Schrödinger equations). On the other hand, we propose to study the topological recursion approach in application to quantum mechanics. Topological recursion method, having recently appeared in mathematics, suggests that information about quantum mechanical systems might be encoded in an associated mathematical object called Riemann surface. We plan to discover consequences for physics of this new approach to quantum mechanics. Another part of the proposal suggests, on the contrary, to use methods of quantum field theories to enumerate mathematical objects, namely graphs on surfaces with certain properties. This is possible due to the correspondence established in our recent work between such graphs and Feynman diagrams of one particular quantum field theory. The fourth part of the proposal concerns the relationship between mathematical structures called cluster algebras and the very special so-called BPS (Bogomolnyi-Prasad-Sommerfield) states in quantum field theories. It turned out that certain numbers in cluster algebra theory count the number of such BPS states. This fact has been discovered by both physicists and mathematicians studying the same questions but using quite different approaches. We are planning to translate results of one area into the language of the other, and discover new implications of results of each field for the other. Students working on these projects, in addition to the mathematics involved, will also learn some physics, which will give them an excellent start in research on mathematical physics.