Goal of research
To conduct a research in various areas of topology and to develop the connections between the topology and applications. In particular, to develop methods and approaches in the brain studies.
Equivariant topology and symplectic geometry (cohomology, equivariant cohomology), homological algebra (exact and spectral sequences), low-dimensional topology, crystallography, discrete mathematics (graphs, hypergraphs, multigraphs, simplicial complexes, colorings, partially ordered sets), random graph theory, applied topology, data analysis (methods of linear and nonlinear data dimension reduction, supervised learning, data visualization).
Empirical base of research
Open dataset OASIS-3, with fMRI data of healthy people and people with Alzheimer disease. Brain parcellation atlases: Automated Anatomical Labeling (AAL), Harvard, Bootstrap Analysis of Stable Clusters (BASC). EEG and MEG data, obtained in Center for neurocognitive research (MEG-center), longitudal MRI data from Burdenko National Medical Research Center. Data of invasive neurobiological experiments on rodents, obtained in Institute for advanced brain studies MSU and National Research Center «Kurchatov Institute».
Results of research
A series of theoretical results in the areas of topology, geometry and discrete mathematics. Description of the orbit spaces of equivariantly formal torus actions of complexity one, description of homeomorphism types of misorientation spaces, description of the topology of the space of isospectral Hermitian arrow-matrices, new estimates for the asymptotics of signal distribution on metric graphs, new results in the area of probabilistic combinatorics. A series of results in the area of neurobiological applications. New methods are proposed to distinguish resting state networks in the brain by fMRI data based on topological invariants, new approach to solve the inverse task of encephalography.A new set of algorithms automatizing on the stages of glioblastoma therapy, in particular, the algorithm for relapse prediction by control images.
Level of implementation, recommendations on implementation or outcomes of the implementation of the results
Theoretical results on misorientation spaces have potential applications in material science and crystallography. Theoretical results on multigraphs have potential applications in construction of numeric characteristics of resting state networks in the brain. Results on fMRI data analysis can potentially be applied for automated recognition of various neurodegenerative diseases. Results in the analysis of invasive experiments can be used to construct more efficient and demonstrative algorithms for visualizing the activity of separate neurons.