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PhD Student
I am primarily interested in the application of modern mathematical (especially geometric) methods to describe gauge field theories, in particular the general theory of relativity.
Previously, I was engaged in embedding theory, some modification of the general theory of relativity, in which physical space-time is considered embedded in a flat space of a larger dimension. For the purposes of this theory, we constructed a global embedding of a BTZ black hole with an angular momentum in a seven-dimensional flat space, despite the fact that previously only a local embedding in a ten-dimensional was known. We have also developed a new method for constructing explicit isometrically bendable embeddings of sufficiently symmetric spaces with not necessarily a diagonal metric.
I am currently studying gauge theories on spaces with boundaries. The Gauge PDE formalism, a generalization of the Batalin-Vilkovyssky formalism on the jet bundle, is extremely convenient in this context. It allows us to give a completely invariant and geometric definition of the boundary theory. Currently we have constructed Gauge PDE formulations of boundary theories for gravity and for tractor geometry. In the future, we plan to apply the Gauge PDE formalism to theories about whose boundary behavior is much less known than about gravity, for example, to higher spin theories.
Selected Publications:
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A. A. Sheykin; M. V. Markov; S. A. Paston, Global embedding of BTZ spacetime using generalized method of symmetric embeddings construction, J.Math.Phys. 62 (2021) 10, 102502, https://aip.scitation.org/doi/10.1063/5.0062060
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A. A. Sheykin, M. V. Markov, Ya. A. Fedulov and S. A. Paston, Explicit isometric embeddings of pseudo-Riemannian manifolds: ideas and applications, J.Phys.Conf.Ser. 1697 (2020) 1, 012077, iopscience.iop.org/article/10.1088/1742- 6596/1697/1/012
Publications affiliated with ITMP:
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Grigoriev, M.; Markov, M. (2023). "Asymptotic symmetries of gravity in the gauge PDE approach". arXiv: 2310.09637 [math-ph].
