### Seminars

### on knot theory and related topics

# Discontinuously basic sets and the 13th problem of Hilbert

Date:

Time:

2021 05 03

15:30 (GMT+0)

Speaker:

Ivan Rechetnikov

Knots and representation theory

A subset $M \subset \textbf{R}^3$ is called a \emph{discontinuously basic subset}, if for any funciton $f \colon M \to \textbf{R}$ there exist such functions $f_1; f_2; f_3 \colon \textbf{R} \to R$ that $f(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ for each point $(x_1, x_2, x_3)\in M$. We will prove a criterion for a discontinuous basic subset for some specific subsets in terms of some graph properties. We will also introduce several constructions for mimimal discontinuous non-basic subsets.

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