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Benjamin Vejnar got his PhD at Charles University in Prague in 2013. After that, he worked as an assistant professor at the same university and in the early 2021 he became an associate professor. In 2018 he was awarded with the Neuron prize for young promising scientists. He serves as an advisor of students of all degrees. His research interests cover continuum theory, descriptive set theory and some parts of topological dynamics.
CURSO:
Descriptive set theoretical properties of continua
The aim of this talk is to explain several basic concepts of Descriptive set theory which can help with understanding some common features of continua. For example, the notions of analytic and coanalytic sets are helpful when disproving the existence of universal elements.
In the first part of this talk, we deal with the complexity of sets. At the lower level, we study sets of points in compacta in the following sense: Components are known to be closed sets; composants are of type F-sigma (countable unions of closed sets); the set of end points of a dendrite is of type G-delta (countable intersection of open sets); path component of a continuum is an analytic set; end points of a dendroid form a coanalytic set. At the upper level, we consider the complexity of collections of continua in the hyperspace of the Hilbert cube (e.g. Peano continua, dendrites or dendroids).
In the second part, we study the complexity of equivalence relations. The lower level is formed by relations between points in compacta (e.g. decomposition into components, path components or composants) whereas the upper level is focused on the homeomorphism relation between compacta considered as a subset of the squared hyperspace of the Hilbert cube.
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