(in Polish) Topics in Descriptive Dynamics and Combinatorics 1000-1M26TDD

The following is a fairly ambitious list of potential topics we hope to cover. Depending on participants' interests, we may place more or less emphasis on connections to logic, measure-theoretic aspects, or Borel graph theory.
- Review of descriptive set theory: Polish and standard Borel spaces, Borel and analytic sets, measures and Baire category, Wadge reducibility
- Borel and analytic equivalence relations: basic examples, uniformization theorems and smoothness, Borel reducibility
- Polish groups and actions: Pettis's theorem, Effros's theorem, Vaught transforms, the Becker–Kechris theorems
- Countable model theory: infinitary first-order logic, spaces of countable structures, the Lopez-Escobar theorem, examples of classification problems, generalized frameworks including compact/metric structures (if time permits)
- Countable Borel equivalence relations: the Lusin–Novikov and Feldman–Moore theorems, properties of homomorphisms and reductions, essential countability (if time permits)
- Structurings of equivalence relations: connection with infinitary logic, examples including smoothness, hyperfiniteness, linear orderings, graphings, treeings, free actions
- Invariant measures: ergodicity and ergodic decomposition, compressibility and Nadkarni's theorem, quasi-invariant measures and Radon–Nikodym cocycles (if time permits)
- Measurable reducibility and structurability: amenability and the Connes–Feldman–Weiss theorem, examples of cocycle rigidity and non-reducibility results
- Measurable combinatorics and group theory (selected topics depending on time): measurable colorings, measurable matchings and flows, measurable equidecompositions (Banach–Tarski and circle squaring), connection to random graphs and graph limits, cost, measure equivalence of groups
- Other topics in Borel graph theory (selected topics depending on time): G₀ dichotomy, Borel vs measurable vs Baire-measurable chromatic numbers, Marks's determinacy method, homomorphism problems and CSPs, ends of graphs