Model Theory of Fields (WS 2015/16)

Lectures: Tuesday 12-14, SR 403
Exercise sessions: Thursday 12-14, SR 403

Lectures

1. ACF, ACF_p, Quantifier elimination.
2. Strong minimality of ACF_p, basic examples of strongly minimal theories, Zariski closed sets, geometric interpretation of QE, Nullstellensatz.
3. ACF_p: acl and dcl, definable functions are definably piecewise rational, dimension: RM X, acl-dim X, dim V, trd_K K(V).
4. ACF_p: Elimination of imaginaries.
5. ACF_p: Non local modularity.
6. Derivations, extension lemmas, DCF_0.
7. DCF_0: Quantifier elimination.
8. DCF_0: Geometric Axioms, linear differential equations.
9. DCF_0: Differential algebra.
10. DCF_0: Differential algebra.
11. DCF_0: 1-types.
12. DCF_0: Morley rank.
13. DCF_0: acl and non-forking independence.
14. Differential Galois theory for Picard-Vessiot extensions.
15. Differential Galois theory for Picard-Vessiot extensions.

Exercises

1. Week 1
2. Week 2
3. Week 3
4. Week 4
5. Week 5
6. Week 6
7. Week 7
8. Week 8 (No exercises, extra lecture.)
9. Week 9
10. Week 10: DCF_0 is not finitely axiomatizable./a>
11. Week 11: Differential Nullstellensatz, isolated 1-types.
12. Week 12: More on isolated 1-types.
13. Week 13: DCF_0 eliminates imaginaries.
14. Week 14: Non-forking independence.
15. Week 15 (No exercises, extra lecture.)

References

• [K] I. Kaplansky, Introduction to differential algebra, 1957.
• [M.mtf] D. Marker, An Introduction to the Model Theory of Fields.
• [M.dcf] D. Marker, Model Theory of Differential Fields.
• [P.acf] A. Pillay, Model Theory of algebraically closed fields, in E. Bouscaren (ed.) Model Theory and Algebraic Geometry, LNM 1696, 1999.
• [TZ] K. Tent und M. Ziegler, A course in Model Theory, Cambridge University Press, 2012.
• [Z.dcf] M. Ziegler, Anfänge der Differentialalgebra.