Oberseminar mathematische Logik/ Graduate Seminar in Set Theory (S4A5)
Organizers
- Dr. Philipp Lücke
- Dr. Philipp Schlicht
Time and location
Monday 16.00-18.00 in room 0.011, Endenicher Allee 60.
The participants of the seminar are welcome for coffee and tea in the Plückerraum 1.012 at 16.00 before the talks.
Contents
An introduction to Shelah's pcf-theory following a tutorial by James Cummings.
Plan
06 OctoberFrank Stephan (Singapore) The complexity of verbal and pattern languages over groups
The talk presents the complexity of verbal languages and pattern
languages of Thurston automatic groups in terms of the Chomsky hierarchy.
Here the language generated by a pattern is taken as the set of
representatives of all strings obtained when chosing values for the
various variables. For noncommutative free groups,
it is shown that the complexity of the verbal and pattern languages
(in terms of level on the Chomsky hierarchy) does not depend on the
Thurston automatic representation, that verbal languages cannot
be context-free (unless they are either the empty word or the full group)
and that pattern languages cannot be regular (unless they are either a
singleton or the full group). Verbal languages and pattern languages can,
however, be indexed languages.
Furthermore, it is shown that in the general case, it might depend on
the exactly chosen Thurston automatic representation which level a
verbal language takes in the Chomsky hierarchy. There are
examples of groups where, in an appropriate representation, all pattern
languages are regular or context-free, respectively.
This is joint work with Sanjay Jain and Alexei Miasnikov; the talk builds
on and extends work presented at LICS 2012.
20 October
Tomas Silveira Salles On second-countable Hausdorff quotients of \beta\omega\setminus\omega and lifting homeomorphisms between them
After a short introduction to the space omega*=\beta\omega\setminus\omega, the algebra P(omega)/fin, and the duality between them, we will characterise the second-countable Hausdorff quotients of omega*, showing how they can be constructed, and use this to lift any homeomorphism between two such quotients.
- 10.11.14/17.11.14 Reduced powers of ON (Part 1), Finn Schmieter and Felix Chopra
- 24.11.14/01.12.14 Reduced powers of ON (Part 2), Ana Njegomir and Regula Krapf (Notes)
References
- [1] James Cummings: PCF. Tutorial, Young Set Theory Workshop 2013, Santuario di Oropa.
- [2] Uri Abraham and Menachem Magidor: Cardinal arithmetic. Handbook of set theory. Vols. 1, 2, 3, 1149-1227, Springer, Dordrecht, 2010.
- [3] Maxim R. Burke and Menachem Magidor: Shelah's pcf theory and its applications. Ann. Pure Appl. Logic 50 (1990), no. 3, 207-254.
- [4] M. Holz, K. Steffens and E. Weitz: Introduction to cardinal arithmetic. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 1999. viii+304 pp. ISBN: 3-7643-6124-7
- [5] Saharon Shelah: Cardinal arithmetic. Oxford Logic Guides, 29. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1994. xxxii+481 pp. ISBN: 0-19-853785-9