Talks

An Introduction to Cartan Geometries

December 02, 2024

Talk, Texas Tech Quantum Homotopy Seminar, Zoom

Consider any geometry $(G,H)$ in the sense of F. Klein, consisting of a Lie group $G$ and its subgroup $H$. This setup provides the following data: the smooth manifold $G/H$, the principal $H$-bundle $H\to G\to G/H$, and the Maurer–Cartan form $\omega_G: TG→\mathfrak{g}$. Moreover, the form $\omega_G$ establishes a linear isomorphism on each fiber, respects the fundamental vector fields on $G$, and converts the tangent right action into the Ad-action. These properties, together with the Cartan development technique in Darboux contexts, play a crucial role in introducing the so-called Cartan geometries. In the talk, we will discuss this idea in greater detail. The page of the seminar.

Homology Spheres, Acyclic Groups and Kan-Thurston Theorem

May 14, 2024

Talk, Conference 'From Analysis to Homotopy Theory' , Greifswald, Germany

The theory of non-simply connected manifolds, the well-known area of algebraic topology, is closely related to combinatorial group theory and group cohomology. It is well known that for any finitely presented group $G$ and $n \geqslant 4$, there exists a smooth closed n-manifold whose fundamental group is isomorphic to $G$. This observation motivates the following question. Let us fix the class $\mathcal{C}$ of manifolds. What finitely presented groups can be fundamental groups of manifolds from that given class $\mathcal{C}$? The question is of interest in the case of $\mathcal{C}$ being the class of homology $n$-spheres, i.e., smooth closed $n$-manifolds with the homology of $n$-sphere. We review remarkable connections appearing in the study of homology spheres: we discuss applications of acyclic groups, the Kan-Thurston theorem, and smooth structures on spheres. We also present related author’s results. The page of the conference.

Euler-Lagrange dynamics via the infinite jet bundle

March 06, 2024

Talk, Texas Tech Quantum Homotopy Seminar, Zoom

We will discuss Chapter 5 from Field Theory via Higher Geometry I: Smooth Sets of Fields by G. Giotopoulos, H. Sati (two talks on March 6 and March 20). The page of the seminar.

Principal ∞-Bundles

December 09, 2023

Talk, Study Seminar on Abstract Homotopy Theory and Applications, Independent University of Moscow, Zoom

We consider a generalization of principal bundles and gerbes following the paper by Thomas Nikolaus, Urs Schreiber and Danny Stevenson.

An Introduction To The ∞-Category Of Motivic Coarse Spaces. A Series Of Talks

May 22, 2023

Talk, Study Seminar on Abstract Homotopy Theory and Applications, Higher School of Economics, Independent University of Moscow, Zoom

We introduce the category $\mathsf{BornCoarse}$ of bornological coarse spaces, define the category of motivic coarse spaces $\mathrm{Spc}\, \chi$, consider some examples of coarse homology theories, including coarse ordinary homology, equivariant ordinary homology and equivariant coarse topological $K$-theory. We discuss also connections with index theory. This series of talks is based on the book by Ulrich Bunke and Alexander Engel “Homotopy Theory with Bornological Coarse Spaces”. Read more

A First Glimpse On ∞-Cosmoi. A Series Of Talks

April 03, 2023

Talk, Study Seminar on Abstract Homotopy Theory and Applications, Higher School of Economics, Independent University of Moscow, Zoom

In this series of talks, we will learn about $\infty$-cosmoi in the sense of Emily Riehl and Dominic Verity. Read more

Dynamics and Multivalued Groups

March 13, 2023

Talk, Conference 'SIMC Youth Race', Steklov Mathematical Institute of RAS

In 1971, S. P. Novikov and V. M. Buchstaber gave the construction, predicted by characteristic classes. This construction describes a multiplication, with a product of any pair of elements being a non-ordered multiset of $n$ points. It led to the notion of $n$-valued groups. Soon after that, V. M. Buchstaber gave the axiomatic definition of $n$-valued groups, obtained the first results on their algebraic structure, and began to develop the theory. At present, a number of authors are developing $n$-valued (finite, discrete, topological or algebra geometric) group theory together with applications in various areas of Mathematics and Mathematical Physics. In this talk, we will give some key notions of $n$-valued group theory and discuss the author’s recently obtained results, which are related to multivalued discrete dynamical systems. We will describe some connections with famous results on symbolic dynamics, combinatorics on words and constructions of quasicrystals

The preprint and slides