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Monday, October 19, 2015

**Abstract:** Descriptive set theory combines techniques from set theory, topology, analysis, recursion theory, and combinatorics, to study definable subsets of Polish spaces. Examples of such sets include Borel, analytic (projection of Borel), co-analytic (complement of analytic), etc. For the past 25 years, a major focus of descriptive set theory has been the study of equivalence relations on Polish spaces that are definable when viewed as sets of pairs; e.g. orbit equivalence relations induced by continuous actions of Polish groups are analytic. This study provides appropriate framework and tools for understanding the nature of classification of mathematical objects (measure-preserving transformations, unitary operators, Riemann surfaces, etc.) up to some notion of equivalence (isomorphism, conjugacy, conformal equivalence, etc.), and measuring the complexity of such classification problems. Due to its broad scope, it has natural interactions with other areas of mathematics, such as ergodic theory and topological dynamics, functional analysis and operator algebras, representation theory, topology, etc. In this talk, I will give an introduction to this fascinating subject.