On the construction of regular maps to Grassmannians

A continuous map $f:\mathbb{C}^n\longrightarrow \mathbb{C}^N$ is called $k$-regular, if the image of any $k$ distinct points in $\mathbb{C}^N$ are linearly independent. The study of the existence of regular maps was initiated by Borusk 1957, and later attracted attention due to its connection with the existence of interpolation spaces in approximation theory, and certain inverse vector bundles in algebraic topology.

In this talk, based on a joint work with Joachim Jelisiejew, we consider the general problem of the existence of regular maps to Grassmannians $\mathbb{C}^n\longrightarrow Gr(\tau,\mathbb{C}^N)$. We will discuss the tools and methods of algebra and algebraic geometry to provide an upper bound on $N$, for which a regular map exists.