Misha Kornev

Misha Kornev

PhD student in mathematics

Posts by Collection

musica

publications

talks

Dynamics and Multivalued Groups

Published:

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 Read more

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

Published:

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 Read more

Principal ∞-Bundles

Published:

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

Homology Spheres, Acyclic Groups and Kan-Thurston Theorem

Published:

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 site of the conference. Read more

teaching

Little Mech-Math

Olympliad School Course, Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, 2022

During the period 2016-2022, I taught students not only olympiad mathematics but some interesting topics in Topology, Abstract Algebra, Differential Geometry, Probability in Combinatorics etc. In each case, I tried to simplify a language and gave some intuition of main ideas without loss of strictness as much as possible. Read more