Die folgenden Vorträge haben im Sommersemester 2011 im Oberseminar Modelltheorie stattgefunden.

 

Donnerstag, 14.04.2011 um 16.00 Uhr, Oberseminar Modelltheorie

Bijan Afshordel (Freiburg)

(Gast von Arno Fehm)

Beschränkte PAC-Unterstrukturen von stabilen Strukturen

Abstract: PAC-Körper gehen auf James Ax Ende der 1960er Jahre zurück. Später führten Hrushovski den Begriff der PAC-Unterstruktur von streng minimalen Strukturen und danach Pillay und Polkowska von stabilen Strukturen ein.

In meinem Vortrag möchte ich den Begriff einer PAC-Unterstruktur einer beliebigen Struktur einführen. Desweiteren gebe ich elementare Invarianten a la Cherlin, van den Dries und Macintyre für beschränkte PAC-Unterstrukturen von stabilen Strukturen an und diskutiere Amalgamationseigenschaften.

 

Dienstag, 19.04.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Martin Hils (Paris 7)

(Gast von Itay Kaplan)

Bad fields and uniformity results in Kummer theory

Abstract: A bad field is an algebraically closed field K together with a proper infinite subgroup U of the multiplicative group such that (K,+,x,U) is a structure of finite Morley rank. In characteristic 0, bad fields exist (joint work with Baudisch, Martin Pizarro and Wagner) and may be obtained by 'collapsing' an infinite rank analogue due to Poizat. The construction uses is via Hrushovski's amalgamation method, and some non-trivial results from algebraic geometry are needed to make this method work in the particular context:

(i) Ax's differential version of Schanuel's Conjecture (or rather a consequence thereof due to Zilber, called weak CIT), in order to get a definable control of the predimension;

(ii) a uniformity result in Kummer theory, in order to definably control multiplicities.

In the talk, we will start with an overview of the construction of the bad field. We will then focus on (ii), presenting a definability result in Kummer theory in the case of the multiplicative group. Gabber suggested a different proof which generalises to the case of semi-abelian varieties. We will explain a further generalisation to the context of definable abelian groups of finite Morley rank, and which allows a purely model-theoretic proof. This part is very recent and joint work with M. Bays and M. Gavrilovich.

Dienstag, 03.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Christoph Hanselka (Konstanz)

Sums of 2m-th Powers (Part II)

Abstract: Using the Ax-Kochen-Ershov principle for Henselian valued fields, I will talk about a suitable elementary description of the class of Laurent series in order to prove a result by Prestel on the existence of a degree bound in a criterion for the representability of polynomials over the reals as sums of 2m-th powers of rational functions.

Dienstag, 10.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Annalisa Conversano (Konstanz)

Connected components of groups and o-minimality
(joint papers with Anand Pillay)

Abstract: In the talk we first introduce the definably connected component G⁰, the type-definable connected component G⁰⁰ and the Aut-invariant connected component G⁰⁰⁰ of a group G definable in a saturated structure M. Then we specialize to the case where M is o-minimal giving a description of the quotients G/G⁰⁰, G/G⁰⁰⁰, G⁰⁰/G⁰⁰⁰ as topological groups endowed with the logic topology, in terms of a basic decomposition of G.

 

Dienstag, 17.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Karen Lange (Notre Dame)

(Gast von Salma Kuhlmann)

Limit computable integer parts

Abstract: An integer part I of a real closed field R is a discrete ordered subring  containing 1 such that  for all r∈R there exists  a unique i∈I with i ≤ r < i+1.  Mourgues and Ressayre showed  that every real closed field R has an integer part.   For a countable real closed field R, we showed that the integer part obtained by the procedure of Mourgues and Ressayre is ∆⁰_ωω(R). We would like to know whether there exists a construction that yields a computationally simpler integer part, perhaps one that is limit computable, i.e., ∆⁰₂(R).

All integer parts are Z-rings, discretely ordered rings that have  the euclidean algorithm for dividing by integers.  By a result of Wilkie, any Z-ring can be extended to an integer part for some real closed field.  We show that we can compute a maximal Z-ring I for any real closed field R that is ∆⁰₂(R), and we then examine whether this I must serve as an integer part for R.

We also show that certain subclasses of ∆⁰₂(R) are not sufficient to give integer parts for arbitrary R.

This is joint work with Paola D'Aquino and Julia Knight.

References

(1) P. D'Aquino, J. Knight, and K. Lange, "Limit computable integer parts,"  to appear in Archive for  Math. Logic.

(2) J. Knight and K. Lange, "Countable real closed fields and developments of bounded length," to appear in Proceedings of London Math. Soc.

(3) M. H. Mourgues and J.-P. Ressayre, "Every real closed field has an integer part,'' J. Symb. Logic, vol. 58 (1993), pp. 641-647.

(4) Alex Wilkie, "Some results and problems on weak systems of Arithmetic", in Logic Colloquium '77, North Holland.

Dienstag, 24.05.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Karen Lange (Notre Dame)

(Gast von Salma Kuhlmann)

Generalized power series and  real closed fields

Abstract: Mourgues and Ressayre showed that every real closed field R has an integer part by constructing a special embedding of R into a field k⟨⟨G⟩⟩ of generalized power series.   Let k be the residue field of R, and let G be the value group of R. The field k⟨⟨G⟩⟩ consists of elements of the form  Σg∈Sa_gg where a_g ∈ k and the support of the power series S ⊆ G is well ordered.   Julia Knight and I previously analyzed an algorithmic version of their construction for countable R and showed that  generalized power series in the image of R are of length less than (ω^ω)^ω and that R has an integer part that is ∆_ω^ω(R).  Ressayre showed that every real closed exponential field has an integer part I that is closed under 2^x for positive elements of I  using the same approach as in M. H. Mourgues and J.-P. Ressayre, "Every real closed field has an integer part''. However, he had to  choose more carefully the value group G and the embedding of R into k⟨⟨G⟩⟩.  We demonstrate that  these alterations cause Ressayre's construction in the exponential case to be much more complex than Mourgues and Ressayre's original construction.

This is joint work with Paola D'Aquino, Julia Knight, and Salma Kuhlmann.

References

(1) J. Knight and K. Lange, "Countable real closed fields and developments of bounded length," to appear in Proc. of London Math. Soc..

(2)  M. H. Mourgues and J.-P. Ressayre, "Every real closed field has an integer part,'' J. Symb. Logic, vol. 58 (1993), pp. 641-647.

(3) J.-P. Ressayre, "Integer parts of real closed exponential fields," in Arithmetic, Proof Theory, and Computational Complexity, Oxford Logic Guides, vol. 23 (1993), pp. 278-288. 

Dienstag, 14.06.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Itay Kaplan (Konstanz)

Groups in dependent theories

Abstract: In this talk I will first introduce dependent theories, and then discuss some results about group in them. I hope to prove the existence of the type definable connected component theorem due to Shelah.

Dienstag, 21.06.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Martin Bays (Paris 7)

(Gast von Margaret Thomas)

Categoricity of Exponential Maps

Abstract: Exponential maps of semi-abelian varieties, of which the familiar complex exponential map and the Weierstrass p-functions are special cases, are some of the most natural non-algebraic complex analytic functions in existence. I will discuss the model theory of some associated structures, with an emphasis on categoricity/classification (higher amalgamation) concerns. Zilber, Shelah, Kummer, Faltings and others will make their appearances, as model theoretic and number theoretic techniques mesh.

 

Dienstag, 28.06.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Will Anscombe (Oxford)

(Gast von Arno Fehm)

Aspects of definability in henselian fields of positive characteristic

Abstract: We investigate definability in fields of power series over finite fields, motivated by open problems in the model theory of these fields. Inevitably, many of the difficulties are due to the positive characteristic. Using Prestel-Ziegler's t-henselian fields, we will look at existential definability; first looking locally and then drawing conclusions about existentially definable sets, subsets, and substructures. Specialising the context to power series fields K((t)) but not restricting to existential formulas, we study subsets definable with parameters from K. We will use the well-known automorphism group together with a Hensel-like property to find that the orbits of elements are existentially definable (although using the parameter t). Finally, we use these results to find an existential definition of the valuation ring which uses parameters only from K.

 

Dienstag, 05.07.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Katharina Dupont (Konstanz)

Definable Valuations

Abstract: Given a field F we are interested if there exists a formula φ(x) in the language of rings L_ring ring = (0, 1; +, −, ·), possibly with parameters, such that {x ∈ F | φ(x)} is a non-trivial valuation ring on F.

We will give examples of fields on which there is a definable valuation and of fields for which we can proof that there is no definable valuation. For some of the positive examples we will show that the valuation is definable without parameters.

The talk will be mainly based on the preprint "Definable Valuations" of Jochen Koenigsmann.

Dienstag, 12.07.2011 um 16.15 Uhr, Oberseminar Modelltheorie

Mikael Matusinski

Surreal numbers as exp-log series

Abstract: In his monograph [Gon86], H. Gonshor showed that Conway’s real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed in [vdDE01] that it is a model of the theory of Rexp. Our aim is to show that surreal numbers form a field of exp-log series and transseries. This would help us, applying previous results of ours [KM11][Sch01], to endow the field of surreal numbers with a natural derivation.

After giving a quick overview of these results, we will present in this talk our first step in this direction : the description of what we call the exponential equivalence classes of surreal numbers. This notion takes place naturally between the Conway’s omega map and the generalized epsilon numbers.

This is a joint work with J. van der Hoeven and S. Kuhlmann.

References

[Gon86] H. Gonshor, An introduction to the theory of surreal numbers, London Mathematical Society Lecture Note Series, vol. 110, Cambridge University Press, Cambridge, 1986.

[KM11] S. Kuhlmann and M. Matusinski, Hardy type derivations on fields of exponential logarithmic series., to appear in J. Algebra, arxiv 1010.0896 (2011).

[Sch01] M.C. Schmeling, Corps de transséries, Ph.D. thesis, Université Paris-VII, 2001.

[vdDE01] Lou van den Dries and Philip Ehrlich, Fields of surreal numbers and exponentiation, Fund. Math. 167 (2001), no. 2, 173–188.