My math research

Functional Analysis

  1. Arens-Michael envelopes of Laurent Ore Extesnions

    Abstract: For an Arens-Michael algebra $A$ we consider a class of $A$-$\hat{\otimes}$-bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over $A$. Given a Fr\'echet-Arens-Michael algebra $A$ and an topologically invertible Fr\'echet $A$-$\hat{\otimes}$-bimodule $M$, we construct an Arens-Michael algebra $\widehat{L}_A(M)$ which serves as a topological version of the Laurent tensor algebra $L_A(M)$. Also, for a fixed algebra $B$ we provide a condition on an invertible $B$-bimodule $N$ sufficient for the Arens-Michael envelope of $L_B(N)$ to be isomorphic to $\widehat{L}_{\widehat{B}}(\widehat{N})$. In particular, we prove that the Arens-Michael envelope of an invertible Ore extension $A[x, x^{-1}; \alpha]$ is isomorphic to $\widehat{L}_{\widehat{A}}(\widehat{A}_{\widehat{\alpha}})$ provided that the Arens-Michael envelope of $A$ is metrizable.

    Remarks: This paper originated from my bachelor thesis, which was written and successully defended back in 2016. The idea is that Arens-Michael envelope is a conveient way to "complete" an algebra in such a way that the result becomes a locally convex algebra such that its topology can be generated by submultiplicative seminorms. Keep in mind that the resulting algebra need not be a Banach algebra. The most important fact one needs to know about Arens-Michael envelopes is that the Arens-Michael envelope of $\mathbb{C}[x]$ is $\mathcal{O}(\mathbb{C}) = \mathcal{H}(\mathbb{C})$. As it turns out, if the algebra $A$ is non-commutative, the structure of $\widehat{A}$ can be quite weird. There is no algorithm which allows one to efficiently describe $\widehat{A}$ if $A$ is described via generators/relations data.

    P.S.A few months after publishing the paper, I asked the following question. As far as I am concerned, it is still open!

  2. Homological dimensions of analytic Ore extensions

    Abstract: If $A$ is an algebra with finite right global dimension, then for any automorphism $\alpha$ and $\alpha{\text{-derivation }} \delta$ the right global dimension of $A[t; \alpha, \delta]$ satisfies \[ \text{rgld} \, A \le \text{rgld} \, A[t; \alpha, \delta] \le \text{rgld} \, A + 1. \] We extend this result to the case of holomorphic Ore extensions and smooth crossed products by $\mathbb{Z}$ of $\hat{\otimes}$-algebras.

    Remarks: This paper is the first one dedicated to homological dimensions of locally convex algebras.

  3. Homological dimensions of smooth crossed products

    Abstract: In this paper we provide upper estimates for the global projective dimensions of smooth crossed products $\mathscr{S}(G, A; \alpha)$ for $G = \mathbb{R}$ and $G = \mathbb{T}$ and a self-induced Fr\'echet-Arens-Michael algebra $A$. In order to do this, we provide a powerful generalization of methods which are used in the works of Ogneva and Helemskii.

    Remarks: This paper is my master's thesis, written and succesfully defended in 2018. In this work I generalize the results of Helemskii and Ogneva to the noncommutative setting.

Number Theory

  1. Orthorecursive expansion of unity

    Abstract: We study the properties of a sequence $c_n$ defined by the recursive relation \[ \frac{c_0}{n+1}+\frac{c_1}{n+2}+\ldots+\frac{c_n}{2n+1}=0 \] for $n \geq 1$ and $c_0=1$. This sequence also has an alternative definition in terms of certain norm minimization in the space $L^2([0,1])$. We prove estimates on the growth order of $c_n$ and the sequence of its partial sums, infinite series identities, connecting $c_n$ with the harmonic numbers $H_n$ and also formulate some conjectures based on numerical computations.

    A few years ago my friend and I came up with a natural polynomial construction: let us define a sequence of real polynomials $p_n \in \mathbb{R}[x]$ as follows: \[ p_0(x) = 1, \] \[ p_n(x) = p_{n-1}(x) + c_n x^n, \quad \int_0^1 p_n(x) x^n dx = 0. \] This is not a novel construction, and it is known in the litrature as an "orthorecursive expansion". As one can see, we only use the inner product sturcture to define $p_n$, therefore, this construction makes sense in any Hilbert space. However, as it turns out, if we consider this expansion in particular, the coefficients $c_n$ exhibit some extremely weird properties, which we attempted to investigate in this paper.

    Geometric Group Theory

  1. Fundamental inequality for hyperbolic Coxeter and Fuchsian groups equipped with geometric distances

    Abstract: We prove that the hitting measure is not equivalent to the Lebesgue measure for a large class of nearest-neighbour random walks on hyperbolic reflection groups and Fuchsian groups.

  2. The fundamental inequality for cocompact Fuchsian groups

    Abstract: We prove that the hitting measure is singular with respect to Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.