arXiv subjects

Denis Gorodkov

Publications and source records attributed to Denis Gorodkov.

3 recordsLinked to original sources

arXiv2026

Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces

We prove that if $M$ is a closed Riemannian surface of diameter $d$ and area $v$ with sectional curvature in the $[-1,1]$ interval, then a closed geodesic net of length $l$ has at most $f(l,d,v)$ branch points, where $f(l,d,v)=(400\bar{l})^{(180\bar{l})^4}$ for $\bar{l}= \max\{l, \frac{\exp(d)}{\min\{1, \frac{v}{4}\}}\}$ This answers a question posed by S. Becker-Kahn. We also prove that for each geodesic net in the Euclidean plane with at most n unbalanced (boundary) vertices such that all its unbalanced vertices have degree 1, the number of balanced vertices of degree $\ge 3$ (=branch points) does not exceed $(25n)^{2n^2}$. This answers a question posed in [GM] and [NP].

math.DG↗

arXiv2024

Hurwitz numbers for reflection groups $G(m,1,n)$

We are extending results from \cite{B-Hurwitz} by building a parallel theory of simple Hurwitz numbers for the reflection groups $G(m,1,n)$. We also study analogs of the cut-and-join operators. An algebraic description as well as a description in terms of ramified covering of Hurwitz numbers is provided. An explicit formula for them in terms of Schur polynomials are provided. In addition the generating function of $G(m,1,n)$-Hurwitz numbers is shown to give rise to $m$ independent variables $\tau$-function of the KP hierarchy. Finally we provide an ELSV-formula type for these new Hurwitz numbers.

math.CO↗

arXiv2016

A 15-vertex triangulation of the quaternionic projective plane

In 1992, Brehm and K\"uhnel constructed a 8-dimensional simplicial complex $M^8_{15}$ with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold "like a projective plane" in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to $\mathbb{H}P^2$. This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of $M^8_{15}$. As a result, we obtain that it is indeed a minimal triangulation of $\mathbb{H}P^2$.

math.AT↗