International Laboratory of Cluster Geometry

Scientific results:

For a given spectral curve, a family of symplectic dual spectral curves is constructed, for this family an explicit formula which expresses the corresponding n-pointed functions in terms of the n-point functions of the original curve. As a corollary, topological recursion for generating functions enumerating generalized totally simple maps is proved. Dependence of topological recursion on the initial data is studied, in particular, for the case of extending the recursion to the degenerate initial data.

Weight systems for permutations taking values in the universal enveloping algebras of the classical Lie algebras are developed.

Closed explicit formulas for primary potentials in genus 0 of 3-dimensional Fan-Jarvis-Ruan-Witten theories for simple elliptic singularities with non maximal symmetry group are found.

An explicit resolution of the symplectic groupoid conditions is constructed by means of Fock-Goncharov-Shen cluster coordinates. Several examples of explicit resolution of the symplectic groupoid conditions for higher genera are constructed.

Baxter operators for the quantum hyperbolic Ruijsenaars system are constructed. It is proven that these operators form a commuting system of integral operators commuting with McDonald operators.

Education and personnel occupational retraining:

In 2021-24 four research assistants completed successfully their PhD studies and defended their theses.

In 2023, two members of the Lab defended Doctor of Science theses.

More than 100 students and PhD students took part in season schools organized by the Lab.