Research

Quantum groups

  • "Categorical approach to dynamical quantum groups" (with Pavel Safronov), arXiv:2008.09081, published
    We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups F(G) and F_q(G) are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group    .

  • "Geometric and categorical approaches to dynamical quantum groups", PhD thesis, link
    Essentially, based on the paper above. Additionally, we show that the parabolic restriction functor is naturally isomorphic to the Kostant-Whittaker reduction introduced by Bezrukavnikov-Finkelberg,     arXiv:0707.3799. In the case of gl_n, we show there exists a "trivialization" of the latter functor by the Gelfand-Tsetlin subalgebra of Ugl_n such that the monoidal structure is constant and quantizes the so-called Cremmer-Gervais r-matrix. In particular, this provides a categorical interpretation of the so-called vertex-IRF transformation from the theory of integrable systems (for instance, see arXiv:math/0512500).

Mirror symmetry

The works below are to be modified and eventually published on arxiv.

  • "Mirror symmetry for Jacobians of hyperelliptic curves", bachelor project, link
    We suggest a mirror family for the Jacobian varieties of hyperelliptic curves based on computation of the (surprisingly, nontrivial) J-function via the abelian/nonabelian correspondence of Ciocan-Fontanine--Kim--Sabbah, arXiv:math/0610265, exploring in detail genus two case.

  • "Gauged linear sigma model for Jacobians of hyperelliptic curves", master project (HSE), link
    We study the GLSM describing in one limit a geometric theory on the Jacobians of hyperelliptic curves. In particular, we show how the other limit is related to the "symmetric product" of geometric theories on the corresponding hyperelliptic curve.

Other projects