Convex polytopes

Abstract: The talk demonstrates numerous examples of convex polytopes such that their characteristic $V - E + F$, which establishes the connection between the number of vertices $V$, edges $E$ and faces $F$, can be any integer number and is not necessarily equal to $2$ as required by Leonhard Euler's theorem on convex polyhedra in $E^3$. One of the methods for obtaining non-Euler convex polytopes is considered; it is the construction of the convex hull of the Diophantine equation $x_1^2 + x_2^2 + x_3^2 = m$, where $x_k, m \in \mathbb{Z}$.

Ukrainiangeneral mathematicsgeometric topologynumber theory

Audience: researchers in the discipline

Семінар з фрактального аналізу / Fractal analysis seminar

Series comments: Weekly research seminar on fractal analysis (online)

Topics:

• theory of fractals (fractal geometry and fractal analysis)
• Hausdorff–Besicovitch dimension, techniques and methods for its calculation and estimation
• functions and transformations preserving fractal (Hausdorff–Besicovitch, entropic, box-counting, packing, etc.) dimension
• sets of metric spaces that are essential for functions, sets, and dynamical systems
• self-similar, self-affine properties of mathematical objects
• systems of encoding for real numbers (numeral systems) and their applications
• metric number theory and metric theory of representations of numbers
• probabilistic number theory and probabilistic theory of representations of numbers
• measures supported on fractals, particularly singular measures and probability distributions
• nowhere monotonic and nowhere differentiable functions, functions with fractal properties
• theory of groups determined by invariants of transformations preserving tails of representation of numbers, digit frequencies, etc.

The talks are mostly in Ukrainian but English is also acceptable