Complexity of classification problems in topology
| Thesis title in Czech: | Složitost klasifikačních problémů v topologii |
|---|---|
| Thesis title in English: | Complexity of classification problems in topology |
| Key words: | borelovská redukce|relace homeomorfismu|metrizovatelný kompaktní prostor|Peanovo kontinuum |
| English key words: | Borel reduction|homeomorphism relation|metrizable compact space|Peano continuum |
| Academic year of topic announcement: | 2019/2020 |
| Thesis type: | dissertation |
| Thesis language: | angličtina |
| Department: | Department of Mathematical Analysis (32-KMA) |
| Supervisor: | doc. Mgr. Benjamin Vejnar, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 20.09.2019 |
| Date of assignment: | 20.09.2019 |
| Confirmed by Study dept. on: | 04.10.2019 |
| Date and time of defence: | 29.05.2024 09:00 |
| Date of electronic submission: | 27.02.2024 |
| Opponents: | Pawel Krupski |
| doc. RNDr. Miroslav Zelený, Ph.D. | |
| Guidelines |
| The student will study the general background of invariant descriptive set theory and major recent results dealing with classification problems in topology. He will try to find some nontrivial Borel reductions dealing with topological spaces or with topological structures. |
| References |
| S. Gao, Invariant Descriptive Set Theory
A. S. Kechris, Classical descriptive set theory S. B. Nadler, Continuum theory, an introduction P. Krupski, B. Vejnar; The complexity of homeomorphism relations on some classes of compacta, arXiv 2018 J. Zielinski, The complexity of the homeomorphism relation between compact metric spaces. Adv. Math., 291: 635-645, 2016 |
| Preliminary scope of work in English |
| In recent decades the notion of a Borel reduction becomes to be an extremely useful tool for comparing the complexities of classification problems. Using this notion one can formally state that for example the classification problem of compact metrizable spaces up to homeomorphism is the same as the classification problem of Polish metric spaces up to isometry.
There are two things that need to be preserved so that we can deal with a classification problem in that way and so that most of the theory works well. The objects under consideration have to be encapsulated in a natural way by a Polish space and the equivalence relation which express the sameness has to be analytic. This is the case for majority of classification problems studied in mathematics. |