The main focus of my research is differential geometry, in particular Nijenhuis geometry and its applications. I study Nijenhuis operators and related tensorial structures in neighborhoods of singular points. In particular, I classified almost non-degenerate Nijenhuis operators with simple singularities. My work lies in local differential geometry and relies on methods of tensorial and geometric analysis.

Currently, I work on integrable Hamiltonian systems, with a particular emphasis on the integrability of geodesic flows, the construction of first integrals and Killing tensors, and their relationship with Nijenhuis operators. I also study geodesically equivalent metrics in the pseudo-Riemannian setting and investigate the integrability of geodesic flows both locally and globally. In addition, I am interested in topological and differential-geometric obstructions to integrability, such as the existence or non-existence of specific geometric structures on manifolds. In particular, I classified all two-dimensional local coordinate systems admitting separation of variables and completely solved the Beltrami problem in dimension two.

In my undergraduate thesis, I proved a series of general theorems for Nijenhuis operators with differential singularities and constructed new examples. In my master’s thesis, I generalized these results to three-dimensional operators and subsequently to operators in arbitrary dimensions. I formulated a conjecture concerning the admissible singularities of almost non-degenerate Nijenhuis operators in two dimensions. The thesis also addressed several related problems, including the construction of regular conservation laws for certain gl-regular Nijenhuis operators and the classification of all geodesically compatible metrics in these cases. I derived a formula relating the determinant of a metric to the characteristic map of a geodesically compatible operator; in particular, this formula allows one to recover the classical Levi-Civita theorem.