Presents Results from a Very Active Area of Research Exploring an
active area of mathematics that studies the complexity of
equivalence relations and classification problems, Invariant
Descriptive Set Theory presents an introduction to the basic
concepts, methods, and results of this theory. It brings together
techniques from various areas of mathematics, such as algebra,
topology, and logic, which have diverse applications to other
fields. After reviewing classical and effective descriptive set
theory, the text studies Polish groups and their actions. It then
covers Borel reducibility results on Borel, orbit, and general
definable equivalence relations. The author also provides proofs
for numerous fundamental results, such as the Glimm-Effros
dichotomy, the Burgess trichotomy theorem, and the Hjorth
turbulence theorem. The next part describes connections with the
countable model theory of infinitary logic, along with Scott
analysis and the isomorphism relation on natural classes of
countable models, such as graphs, trees, and groups. The book
concludes with applications to classification problems and many
benchmark equivalence relations. By illustrating the relevance of
invariant descriptive set theory to other fields of mathematics,
this self-contained book encourages readers to further explore this
very active area of research.
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