Misha Kornev

Misha Kornev

PhD student in mathematics

Blog posts

2024

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

2023

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

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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