Selected topics in set theory

General data

Course ID:	1000-1M09WZM
Erasmus code / ISCED:	11.104 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Selected topics in set theory
Name in Polish:	Wybrane zagadnienie teorii mnogości
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective monographs
Short description:	This advanced course will be devoted to selected topics of Descriptive Set Theory. It is a branch of Set Theory which studies "definable" subsets of the reals (and similar spaces like R^n, the Cantor set and other spaces of infnite sequences or trees). "Definable" sets include, in particular, Borel sets and their continuous images. Such sets have many regular properties: they are Lebesgue measurable, have the Baire property (which is a topological analogue of measurability) and the Continuum Hypothesis restricted to their class is valid. The notions and results from Descriptive Set Theory have applications in various branches of Mathematics and also in Theoretic Computer Science.
Full description:	This advanced course will be devoted to selected topics of Descriptive Set Theory. It is a branch of Set Theory which studies "definable" subsets of the reals (and similar spaces like R^n, the Cantor set and other spaces of infnite sequences or trees). "Definable" sets include, in particular, Borel sets and their continuous images. Such sets have many regular properties: they are Lebesgue measurable, have the Baire property (which is a topological analogue of measurability) and the Continuum Hypothesis restricted to their class is valid. The notions and results from Descriptive Set Theory have applications in variousbranches of Mathematics and also in Theoretic Computer Science. During the course we will present some partition theorems, i.e., descriptive counterparts of the Ramsey theorem such as the Galvin-Prikry, Silver and Ellentuck theorems. We will show some uniformization results concerning Borel and projective sets. We will consider questions concerning Borel games and their determinacy. We are also planning to present connections between Descriptive Set Theory and Automata Theory, more precisely: applications of Descriptive Set Theory to problems concerning sets of infinite sequences or trees acceptable by finite automata. We assume familiarity with basic Set Theory (including the transfinite induction, ordinal and cardinal numbers) and Topology.
Bibliography:	1. A.S. Kechris, Classical descriptive set theory, Graduate Texts in Math. 156, Springer-Verlag, 1995.
Assessment methods and assessment criteria:	(in Polish) Zaliczenie ćwiczeń na podstawie aktywności w rozwiązywaniu zadań (w tym domowych). Egzamin ustny, sprawdzający znajomość wszystkich pojęć i twierdzeń wraz z (niektórymi) dowodami.

This course is not currently offered.

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