Geometries, Hrushovski constructions, and ampleness
Mondays 10-12 SR1d
WiSe2021/22 seminar run by Katrin Tent, assisted by Martin Bays and Blaise Boissonneau.
The motivation for the seminar is Zilber's Trichotomy Conjecture which stated that any strongly minimal set is *essentially* either a trivial structure, a vector space or an algebraically closed field where the dividing lines depend on the underlying geometry of the strongly minimal set.
We will introduce these geometries, define the different classes and then study the counterexample to this conjecture constructed by Hrushovski.
These new geometries arising from Hrushovski's counterexample give rise to a new hierarchy of a notion of ampleness which makes sense in general stable structures. We will study the ampleness hierarchy both in the context of free groups and in the context of omega-stable theories and theories of finite Morley rank.
If you are interested in participating, please contact Martin Bays <baysm@wwu.de> or Katrin Tent.
Bibliography
- [TZ]: K. Tent, M. Ziegler, A course in model theory. Lecture Notes in Logic, 40. Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012.
- [Pi-GST]: A. Pillay, Geometric Stability Theory, Section 2.2
- [Hr]: E. Hrushovski: "A new strongly minimal set" Ann. Pure Appl. Logic 62 (1993), 147-166
- [Z]: M. Ziegler: "An exposition of Hrushovski's new strongly minimal set" Ann. Pure Appl. Logic 164 (2013), 1507-1519.
- [B]: M. Bays, Geometric Stability Theory, in Lectures in Model Theory, Münster Lectures in Mathematics, 2018
- [Te1]: K. Tent, Very homogeneous generalized n-gons of finite Morley rank. J. London Math. Soc. (2) 62 (2000), no. 1, 1–15.
- [Pi-CM]: Pillay, Anand , A note on CM-triviality and the geometry of forking. J. Symbolic Logic 65 (2000), no. 1, 474–480.
- [Ev]: D. Evans, Ample dividing. J. Symbolic Logic 68 (2003), no. 4, 1385–1402.
- [Te2]: K.Tent, The free pseudospace is n-ample, but not (n+1)-ample, J. Symbolic Logic, 79, (2014) 410-428.
- [MT]: I. Müller, K. Tent, Katrin, Building-like geometries of finite Morley rank. J. Eur. Math. Soc. (JEMS) 21 (2019), no. 12, 3739–3757.
- [Sk]: R. Sklinos, On ampleness and pseudo-Anosov homeomorphisms in the free group. Turkish J. Math. 39 (2015), no. 1, 63–80.
Talks