Research

My current research revolves around geometric Galois actions, operads and questions related to finite type invariants of knots. The questions I am working on are motivated by the study of embedding spaces and Drinfeld’s Grothendieck-Teichmueller group which, in turn, has links to the absolute Galois group of rational numbers and the theory of motives.

I am a US based participant of the International Research Network Arithmetic & Homotopic Galois Theory.

Exploration of Grothendieck-Teichmueller(GT)-shadows and their action on Grothendieck’s child’s drawings

  • In this paper (joint with Ivan Bortnovskyi, Borys Holikov, Vadym Pashkovskyi), we gave the first examples of non-abelian quotients of the Grothendieck-Teichmueller group and these quotients receive surjective homomorphisms from the absolute Galois group of rational numbers.
  • The paper (joint with Jacob Guynee) is devoted to GT-shadows for the gentle version of the Grothendieck-Teichmueller group.
  • My talk at the AHGT Seminar in December 2023.
  • The final research paper by Ivan Bortnovskyi and Vadym Pashkovskyi “Exploration of the Grothendieck-Teichmueller (GT) shadows for the dihedral poset”.
  • My seminar talk at Topology Seminar at the University of Minnesota. The virtual talk was given on Sept 26, 2022. The slides are available here.
  • My seminar talk at Algebra/Galois Theory Seminar at Penn. The virtual talk was given on Sept 12, 2022. The slides are available here.
  • My seminar talk at the University of Angers, France. The virtual talk was given on May 27, 2022. The slides (without pauses) are available here.
  • My virtual 50 minute colloquium talk at the University of Nevada, Reno about GT-shadows and their action on child’s drawings. The virtual talk was given on March 3, 2022.
  • Here is Jingfeng Xia’s master thesis. It is devoted to the groupoid of GT-shadows for the gentle version of the Grothendieck-Teichmueller group. It also contains partial results about the connected components of this groupoid related to finite quotients of the full modular group.
  • My 46 minute video presentation about GT-shadows and their action on child’s drawings. This video presentation is very similar to my virtual talk at the conference “Koszul Duality & Operads” that took place in October 2020 and it was organized by CIRM (Marseille, France)and MPI MiS (Leipzig, Germany).
  • The paper is devoted to the action of GT-shadows on child’s drawings. Note that, in this paper, I work with GT-shadows for the original version of the Grothendieck-Teichmueller group.
  • The final version (03/16/2022) of the software package (as a zip-file) for working with GT-shadows and their action on child’s drawings can be found here. The detailed documentation for this package includes many examples.
    • Here is Session 1 for working with the package GT (recorded on 03/28/2022).
    • Here is Session 2 for working with the package GT (recorded on 03/30/2022).
    • Here is Session 3 for working with the package GT (recorded on 03/31/2022).
  • The joint paper with K.Q. Le and A.A. Lorenz What are GT-shadows? will appear in Algebraic and Geometric Topology.

The term “GT-shadow” could have been introduced in paper Approximating Galois orbits of dessins by David Harbater and Leila Schneps from 1997. The authors used a different (but equivalent) definition of the Grothendieck-Teichmueller group but all the original ideas for “approximations” can be found in their paper.

Other useful things

Documentation for the package that allows one to compute Tamarkin’s Ger-infinity structure on Hochschild cochains recursively. This package accompanies the joint paper with Geoffrey Schneider.

My PhD thesis in mathematics. And… here is the erratum.

My PhD thesis in theoretical physics.