Complexity of classification problems in topology
| Název práce v češtině: | Složitost klasifikačních problémů v topologii |
|---|---|
| Název v anglickém jazyce: | Complexity of classification problems in topology |
| Klíčová slova: | borelovská redukce|relace homeomorfismu|metrizovatelný kompaktní prostor|Peanovo kontinuum |
| Klíčová slova anglicky: | Borel reduction|homeomorphism relation|metrizable compact space|Peano continuum |
| Akademický rok vypsání: | 2019/2020 |
| Typ práce: | disertační práce |
| Jazyk práce: | angličtina |
| Ústav: | Katedra matematické analýzy (32-KMA) |
| Vedoucí / školitel: | doc. Mgr. Benjamin Vejnar, Ph.D. |
| Řešitel: | skrytý - zadáno a potvrzeno stud. odd. |
| Datum přihlášení: | 20.09.2019 |
| Datum zadání: | 20.09.2019 |
| Datum potvrzení stud. oddělením: | 04.10.2019 |
| Datum a čas obhajoby: | 29.05.2024 09:00 |
| Datum odevzdání elektronické podoby: | 27.02.2024 |
| Oponenti: | Pawel Krupski |
| doc. RNDr. Miroslav Zelený, Ph.D. | |
| Zásady pro vypracování |
| 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. |
| Seznam odborné literatury |
| 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 |
| Předběžná náplň práce v anglickém jazyce |
| 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. |