Course details
Semester
 Autumn 2021
 Monday, October 4th to Friday, December 10th
Hours
 Live lecture hours
 10
 Recorded lecture hours
 0
 Total advised study hours
 40
Timetable
 Fridays
 13:05  13:55
Course forum
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Description
In broad brushstrokes, we can generally describe classification programmes in mathematics as follows. You have some mathematical objects that you want to classify, up to isomorphism or some other such equivalence relation. You also have some mathematical objects that you want to use as invariants, again possibly up to some equivalence relation. And you want a reasonably definable map from the objects to be classified to the invariants, in such a way that the objects to be classified are equivalent if and only their images are equivalent.
It is often the case that the mathematical objects we are interested in, both those to be classified and the intended invariants, can be encoded by real numbers. In this case the question of classifiability boils down to the existence of a "reasonably definable" map from R to R respecting the equivalence relations. Taking "Borel" as a suitably liberal interpretation of "reasonably definable", we can sometimes prove that no such map exist, ruling out the classification. This has been successfully used to prove that some old classification programmes, in areas such as C*algebras and ergodic theory, were impossible tasks.
This course will introduce students to this area, developing the theory from theground up, culminating in Hjorth's notion of turbulence, which is the central tool for proving unclassifiability results.
It is often the case that the mathematical objects we are interested in, both those to be classified and the intended invariants, can be encoded by real numbers. In this case the question of classifiability boils down to the existence of a "reasonably definable" map from R to R respecting the equivalence relations. Taking "Borel" as a suitably liberal interpretation of "reasonably definable", we can sometimes prove that no such map exist, ruling out the classification. This has been successfully used to prove that some old classification programmes, in areas such as C*algebras and ergodic theory, were impossible tasks.
This course will introduce students to this area, developing the theory from theground up, culminating in Hjorth's notion of turbulence, which is the central tool for proving unclassifiability results.
Prerequisites
Basic topology (open and closed sets).
Syllabus
Motivation: examples. Polish spaces, Borel functions and Borel reductions. The logic action and orbit equivalence relations. Standard Borel spaces, Polish groups and Borel groups. Smooth relations, E_{0}, and the Effros Borel space. Baire category machinery, building to proof that the Effros Borel space is a universal Borel Gspace. Borel completeness. Generic ergodicity. Isomorphism of rank1 torsion free abelian groups is not smooth. Turbulence. Examples of turbulence and unclassifiability.
Lecturer

Dr Andrew BrookeTaylor
 University
 University of Leeds
Bibliography
No bibliography has been specified for this course.
Assessment
Attention needed
Assessment information will be available nearer the time.
Lectures
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