Misha Kornev

Misha Kornev

PhD student in mathematics

Talks

$n$-Valued Groups, Kronecker Sums, and Wendt Matrices

August 15, 2025

Talk, 5th Conference of Mathematical Centers of Russia, Krasnoyarsk, Russia

In 1971, V.,M. Buchstaber and S.,P. Novikov proposed a construction motivated by the theory of characteristic classes. This construction describes a multiplication in which the product of any pair of elements is a multiset of $n$ points. The axiomatic definition of $n$-valued groups and the first results of their algebraic theory were obtained in a subsequent series of works by V.,M. Buchstaber. At present, the theory of $n$-valued (formal, finite, discrete, topological, and algebro-geometric) groups and its applications in various areas of mathematics and mathematical physics is being developed by a number of authors.

$n$-Valued Groups, Kronecker Sums, and Wendt Matrices

May 20, 2025

Talk, Seminar of the Department of Geometry and Topology: Geometry, Topology and Mathematical Physics (Novikov Seminar), Steklov Mathematical Institute of RAS

In the first part of the talk, for each $n$ the notion of classes of symmetric $n$-algebraic $n$-valued groups will be introduced. For $n = 2, 3$, a description of the universal objects of these classes will be given. The second part of the talk is devoted to the $n$-valued groups $G_n$ over the field of complex numbers $\mathbb{C}$. It will be shown that the $n$-valued addition $x * y = [z_1, \ldots, z_n]$ in $\mathbb{G}_n$ is realized in terms of the eigenvalues of the Kronecker sum of the companion matrices of the Frobenius polynomials $t^n - x$ and $t^n - y$ in the variable $t$, where $z = t^n$. $(n \times n)$-matrices $W_n(z; x,y)$ will be introduced, whose determinants are integral polynomials $p_n(z; x,y)$ defining the law $x*y$. It will be shown that $p_n(x; (-1)^n, (-1)^n)$ is the characteristic polynomial of the classical Wendt matrix, which was introduced in 1894 in connection with Fermat’s Last Theorem. As corollaries, results on the structure of the polynomials $p_n(z; x,y)$ will be presented.

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.

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

Acyclic Groups and Spaces

April 11, 2022

Talk, 29th International Scientific Conference for Undergraduate and Graduate Students and Young Scientists Lomonosov, Lomonosov Moscow State University