Winter 2008

Descriptive Set Theory

**Room and Time**

MWF 11:00-11:50 in MS 5217

*Note:* If it is convenient for the students, I would like
to change to a two-day schedule.

**Instructor**

D.A. Martin

Offices: MS 7935 and Dodd 355

Office Hours TBA

Email Address: dam@math.ucla.edu

**Content**

Descriptive set theory is usually characterized as the study of
definable sets in * Polish * spaces (separable, completely
metrizable spaces). The subject classifies definable sets into
hierarchies based on the complexity of their definitions. For each
hierarchy, one studies such things as the properties shared by all
sets in the hierarchy and the structural properties of the levels of
the hierarchy. Examples of hierarchies are the * Borel *
hierarchy, the * projective * hierarchy, and the * Wadge*
hierarchy, the last of these being an ultra-fine hierarchy that
refines and extends most or all the others. An example of a property
shared by all sets in a hierarchy is universal measurability, which is
a property of all Borel sets (i.e., of all sets in the Borel hierarchy).

**Text**

There is no text for the course, but
Alexander S. Kechris's book, * Classical Descriptive Set Theory*,
published by Springer-Verlag, contains all material the course will
cover. Moreover the course will have almost the same prerequisites as
the book (but fewer) and and will use pretty much the same terminology
and notation. The course will also follow the book in dealing (almost
exclusively) with *classical* descriptive set theory as opposed to
* effective* descriptive set theory, which refines the classical
theory using the concept of recursive (computable) function. Finally,
the course will follow the book in making considerable use of infinite
games.

**Prerequisites**

What's needed is familiarity with elementary set theory, including the notions of wellordering, ordinal number, and transfinite induction, and a little acquaintance with the basics of general topology and measure theory.

**Course work**

One or two homework problems will be assigned every week or two.