MAT6932 – Spec Topics in Logic: Invariant Descriptive Set Theory
Course: Spec Topics in Logic MAT6932, section 3F90, course number 13183
Meeting time and place: LIT 205, MWF 7th period
Instructor: Jindrich Zapletal
Office: 456 Little Hall, office hours MWF 4th period
Contact: zapletal@math.ufl.edu, 352-294-2343
Course Description: Descriptive set theory of analytic equivalence relations. Basics of Polish spaces, Borel and analytic sets, Borel reducibility between analytic equivalence relations on Polish spaces. 3 credits
Course Pre-Requisites / Co-Requisites: Elements of topology/ Introduction to topology 1, or the corresponding first-year test, or instructor’s permission.
Course Objectives: Students who successfully complete the course will understand the chart of analytic equivalence relations ordered by Borel reducibility, and they will be able to place many classification problems in mathematics on the chart.
Textbook: Su Gao, Invariant Descriptive Set Theory, Chapman & Hall/CRC 2008, ISBN 9780429146152
Provisional timeline of the course:
Each of the following topics will take approximately 8 lectures.
1. Polish spaces.
Polish space is a completely metrizable separable metric space. Constructions of Polish spaces. Borel and analytic sets, and their comparison. Universal analytic sets.
2. Analytic equivalence relations.
Borel and analytic equivalence relations. Borel reducibility and the chart of all analytic equivalence relations. Examples from mathematical practice. Universal analytic equivalence relations. Silver dichotomy: every Borel equivalence relation either has countably many classes or a perfect set of pairwise incompatible elements. Glimm–Effros dichotomy.
3. Countable Borel equivalence relations.
Every countable Borel equivalence relation (cBer) is an orbit equivalence relation of a Borel action of a countable group. Universal cBer. Amenability and a cBer not reducible to a Vitali equivalence relation. Examples from mathematical practice.
4. Polish groups, their continuous actions, and orbit equivalence relations.
Polish groups. Examples from mathematical practice. Universal Polish groups. Universal orbit equivalence relations. E1 conjecture: there is a simplest Borel equivalence relation not reducible to an orbit equivalence relation.
5. Equivalence relations classifiable by countable structures.
Isomorphisms of countable structures and permutation groups. Model theory and infinitary languages. Turbulence. Examples from mathematical practice.
Grading: There will be five brief homework assignments, one per each of the topics mentioned above, each worth 20 pts. The total grade will be figured according to the standard flat curve: A above 93%, A- 90-92%, B+ 87-89%, B 83-86%, B- 80-82%, C+ 77-79%, C 73-76%, C- 70-72%, D+ 67-69%, D 63-66%, D- 60-62 %, E, I, NG, WF 59%.
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