Research

My current research is concerned with algebraic operads, their generalizations and related structures. The questions I am working on are motivated by the study of embedding spaces, deformation quantization, and Drinfeld's Grothendieck-Teichmueller group which, in turn, has links to the absolute Galois group of rational numbers and the theory of motives.

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

For the introduction to GT-shadows, please see the paper What are GT-shadows? (joint with Khanh Q. Le, Aidan A. Lorenz).
Please also see the paper The Action of GT-Shadows on Child's Drawings.
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).
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).
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. Note that, in this talk, I tried to avoid the language of operads.
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.
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 seminar talk at Algebra/Galois Theory Seminar at Penn. The virtual talk was given on Sept 12, 2022. The slides (without pauses) are available here.
My seminar talk at Topology Seminar at the University of Minnesota. The virtual talk was given on Sept 26, 2022. The slides (without pauses) are available here.
The final research paper by Ivan Bortnovskyi and Vadym Pashkovskyi "Exploration of the Grothendieck-Teichmueller (GT) shadows for the dihedral poset".

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.

Algebra Seminar at Penn.

Galois Seminar at Penn.

Slides of some selected talks

The slides of my talk "A manifestation of the Grothendieck-Teichmueller group in geometry".
The slides of my talk "The Goldman-Millson Theorem revisited".
The slides of my talk "The Intricate Maze of Graph Complexes". Variations of this talk were presented
- at the Deformation Theory Seminar at Penn,
- at the Closing Conference for the Program on Higher Structures in Geometry and Physics at the Max Planck Institute for Mathematics,
- at several places in the Midwest in March of 2016, and
- at a couple of places in Boston area in April of 2016.
The slides of my talk "Kontsevich's formality quasi-isomorphism is "demystified"" presented at the conference Cyclic Homology. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland, October 2016.

Handwritten lectures on deformation quantization and the corresponding homework sets.

The software related to my joint paper with Geoffrey Schneider. This software allows one to compute Tamarkin's Ger-infinity structure on Hochschild cochains recursively. The documentation for this software can be found here.

The algebraic index theorem can be applied to the study of energy bands of molecular systems.

I serve on the editorial board of Tbilisi Mathematical Journal.