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
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.
- 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
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 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
- 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
final research paper
by Ivan Bortnovskyi and Vadym Pashkovskyi "Exploration of the Grothendieck-Teichmueller (GT) shadows
for the dihedral poset".
Algebra Seminar at Penn.
Galois Seminar at Penn.
Slides of some selected talks
Handwritten lectures on deformation quantization and
the corresponding homework sets.
- 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
- 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.
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.