21-624 Descriptive Set Theory
12 units, Carnegie Mellon University
The central theme of the course is the study of definable subsets of Polish spaces a class of topological spaces containing many spaces studied across mathematics. In some sense, restricting attention to definable sets makes life nicer for instance, properly interpreted the continuum hypothesis becomes true And weird BanachTarski stuff doesnt happen. However, it comes at a price equivalence relations can have more classes than elements, and acyclic graphs can become hard to properly color. For the first portion of the course we will work through the basic theory of Borel and analytic subsets of Polish spaces, highlighting their interaction with measure and Baire category. We will then focus on the modern theory of equivalence relations, paying special attention to orbit equivalence relations of group actions and connectedness relations of graphs. Along the way we will establish several classical dichotomy theorems. Three 50 minute lectures
Prerequisites: (21-355 (min B) or 21-235 (min B) or 21-455 (min B)) and (21-373 (min B) or 21-237 (min B)) and 21-329 (min B)
Offered: Spring 2027
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