Logic 3: Abstract Topological Dynamics and Descriptive Set Theory
The lecture takes place every Tuesday and Friday 10:15 - 11:50.
The excersise session is every Tuesday 12:00 - 1:00.
The first lecture will be on 10.10 and the first exercise session on 17.10.
Topics that we plan to cover:
In the first part of the course, we will cover basic concepts from descriptive set theory (Polish spaces, Borel pointclasses, analytic and coanalytic sets, coanalytic ranks) and in the second part of the course, we will focus on abstract topological dynamics and on selected topics connecting logic, topological dynamics, and Ramsey theory. In particular, we will prove Hindman's partition theorem via ultrafilters and the Furstenberg structure theorem for distal minimal flows (and we will see that the ordinal ranks associated to distal minimal flows are coanalytic ranks studied in descriptive set theory).
Literature:
1) J. Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, 153, 1988. xii+265 pp.
2) F. Beleznay, M. Foreman, The collection of distal flows is not Borel, Amer. J. Math. 117 (1995), no. 1, 203–239.
3) A. Blass, Ultrafilters: where topological dynamics = algebra =combinatorics, Topology Proc. 18 (1993), 33–56. https://arxiv.org/abs/math/9309203
4) H. Furstenberg, The structure of distal flows, Amer. Jour. of Math. Vol. 85, No. 3 (1963), pp. 447-515.
5) S. Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), 293. CRC Press, Boca Raton, FL, 2009. xiv+383 pp.
6) A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.
7) T. Tao, blog, 254A lecture 6 and lecture 7
8) A. Tserunyan, lecture notes on descriptive set theory, available here: https://www.math.mcgill.ca/atserunyan/Teaching_notes/dst_lectures.pdf
Problem sheets:
Problem Sheet 1 pdf (due 17.10 at 12pm)
Problem Sheet 2 pdf (due 24.10 )
Problem Sheet 3 pdf (due 31.10)
Problem Sheet 4 pdf (due 7.11)
Problem Sheet 5 pdf (due 14.11)
Problem Sheet 6 pdf (due 21.11)
Problem Sheet 7 pdf (due 28.11)
Problem Sheet 8 pdf (due 5.12)
Problem Sheet 9 pdf (due 12.12)
Problem Sheet 10 pdf (due 19.12)
Problem Sheet 11 pdf (due 9.01)
Problem Sheet 12 pdf (due 23.01)
Lecture notes:
Lecture 1 pdf
Lecture 2 pdf
Lecture 3 pdf
Lecture 4 pdf
Lecture 5 pdf
Lecture 6 pdf
Lecture 7 pdf
Lecture 8 pdf
Lecture 9 pdf
Lecture 10 pdf
Lecture 11 pdf
Lecture 12 pdf
Lecture 13 pdf
Lecture 14 pdf
Lecture 15 pdf
Lecture 16 pdf
Lecture 17 pdf
Lecture 18 pdf
Lecture 19 pdf
Lecture 20 pdf
Lecture 21 pdf
Lecture 22 pdf
Lecture 23 pdf
Lecture 24 pdf
Lecture 25 pdf
Lecture 26 pdf
Lecture 27 pdf
Here are also unofficial lecture notes (I did not read them carefully) pdf