Misha Kornev

Misha Kornev

PhD student in mathematics

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Pages

Posts

Representation Homology

5 minute read

Published:

We can think about any commutative associative algebra $A$ as a functor $\mathcal{L}(A):\mathsf{Fin}\to k\text{-}\mathsf{mod}$ from the category of finite sets and arbitrary functions to the category of $k$-modules ($k$ is a commutative ring, the letter $\mathcal{L}$ is devoted to Jean-Louis Loday). It can be constructed as follows. On objects it is defined by sending a set $\mathbf{n}$ to $A^{\otimes_k n}$. For a morphism $f: \mathbf{n}\to\mathbf{m}$, define the morphism $\mathcal{L}(A)(f)$. Read more

De Bruijn Construction

4 minute read

Published:

As well known, a plane can be covered aperiodically by two types of rhombi: thick ones having angles $72^\circ$ and $108^\circ$, and thin ones having angles $36^\circ$ and $144^\circ$ (see Fig. 1). Read more

Koszul Duality And Gröbner bases

3 minute read

Published:

Koszul duality is a fundamental notion of Math. Its reincarnations can be found in algebra, geometry and homotopy theory. For example, Sullivan and Quillen’s approaches to rational homotopy theory are Koszul dual to each other with respect to the Koszul duality between commutative and Lie operads. In algebra, we have the PBW-theorem (by Poincaré, Birkhoff and Witt), which states that the universal enveloping algebra of a Lie algebra looks like a polynomial algebra meaning that it has as a monomial basis of a vector space with variables being sorted by some fixed order. Further, by algebras we mean associative differential graded algebras $A = T(V)/R$ with $R$ being quadratic or quadratic-linear relations (e. g., tensor algebras $T(V),$ symmetric algebras $S(V),$ universal enveloping algebras $U(\mathfrak{g}),$ etc). Read more

Stable Homotopy Groups Via Motivic Ones

3 minute read

Published:

Briefly, there is a wonderful application of motivic homotopy theory to Agriculture classic Algebraic Topology. Classical and motivic Adams-Novikov spectral sequences coincide in the case of algebraic closed fields with zero characteristic (Marc Levine). This connection allowed to eliminate misunderstanding and mistakes in classical calculations. I will follow the Marc Levin overview. Read more

The Mandell Theorem

5 minute read

Published:

Recently, I learned about a wonderful result, Mandell’s theorem. We use algebraic invariants like (co-)homology, Massey products, Steenrod squares, etc, to prove two spaces, say, $X$ and $Y$ to be different. Anyway, each topological space can be encoded by means of some algebraic data, i. e., Postnikov’s system and $k$ invariants belonging to some cohomology groups with local coefficients — some sort of higher data. The rational part of the one, in the case of simply connected (more generally, nilpotent) spaces, can be suppressed into a commutative differential graded algebra (Dennis Sullivan’77). In general case, there are many results, contained, e. g., in the books “Rational Homotopy Theory I, II” by Y. Felix, S. Halperin, J.-C. Thomas. Read more

The Brieskorn Spheres And Their Fundamental Groups

8 minute read

Published:

The Brieskorn spheres $\Sigma(p, q, r)$ are important examples of homology 3-dimensional spheres, i. e., manifolds with the homology being like one of a 3-dimensional sphere. They were extensively studied by Egbert Brieskorn and John Milnor. Henry Poincaré described the first one, $\mathbb{S}^3 / 2I\cong \Sigma(2, 3, 5)$ (see the proof below) in his “Cinquiéme complément à l’Analysis situs” (1904), where

\begin{equation*} 2I := \langle a, b \ | \ a^2 = b^2 = (ab)^5 \rangle\cong \mathrm{SL}(2, \mathbb{Z}/5) \end{equation*}
is the binary icosahedral group. Now, the manifold $\mathbb{S}^3 / 2I$ is known to be the Poincaré sphere. Read more

Just A Fun Pic

less than 1 minute read

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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