mgr Kamil Orzechowski

Kamil Orzechowski M.A. - ORCID: 0000-0003-0786-1463

The topic of my dissertation is: “The Banach–Tarski paradox and its variants for subsets of normed spaces over non-Archimedean valued fields“.

In 1924, S. Banach and A. Tarski proved one of the most shocking theorems in mathematics, called the Banach–Tarski paradox. It states that if we take two bounded sets A and B (with nonempty interiors) in a three-dimensional Euclidean space, each of them can be divided into a finite number (the same for A and B) of pairwise disjoint subsets in such a way that the corresponding subsets of A can be transformed into corresponding subsets of B using rotations and translations.

So far, theorems related to the Banach–Tarski paradox have been proved for some subsets of Euclidean spaces, which are normed spaces over the field of real numbers. The goal of my research is to obtain similar results for subsets of normed spaces over non‑Archimedean valued fields. Such fields seem to be a natural alternative to the field of real numbers, with very different topological properties.

The research problem is a meeting point of several branches of mathematics: algebra (group theory and field theory), non-Archimedean functional analysis, theory of ultrametric spaces, elements of topology and measure theory.

My other research interests include some aspects of group theory (especially combinatorial and geometric group theory, the problem of amenability), as well as the theory of the asymptotic dimension of metric spaces.

Publications:

  1. Orzechowski, K. (2020). APD profiles and transfinite asymptotic dimension. Topology and its Applications, 283, 107394. https://doi.org/10.1016/j.topol.2020.107394
  2. Orzechowski, K. (2019). Characterization of the Haagerup property for residually amenable groups. Colloquium Mathematicum, 155(2), 215–226. https://doi.org/10.4064/cm7502-3-2018
  3. Dronka, J., Wajnryb, B., Witowicz, P., Orzechowski, K. (2017). Growth functions for some uniformly amenable groups. Open Mathematics, 15(1), 502–507. https://doi.org/10.1515/math-2017-0049