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Geometry, Calculus and Zil'ber's Conjecture

Published online by Cambridge University Press:  15 January 2014

Ya'acov Peterzil  and
Sergei Starchenko
Ya'acov Peterzil
Affiliation:
Department of Mathematics and Computer Science, Haifa University, Haifa, Israel.E-mail: [email protected]
Sergei Starchenko
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA.E-mail: [email protected]
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Extract

§1. Introduction. By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ (or any real closed field) where algebra alone determines the ordering and hence the topology of the field:

In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but this will be too coarse to give a diferentiable structure.

A celebrated example of how partial algebraic and topological data (G a locally euclidean group) determines a differentiable structure (G is a Lie group) is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason.

The main result which we discuss here (see [13] for the full version) is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨M, <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of Mn, n = 1, 2, …, in some first order expansion ℳ of ⟨M, <⟩.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 2 , Issue 1 , March 1996 , pp. 72 - 83
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1] Hrushovski, E., The Mordell-Lang conjecture for function fields, preprint.Google Scholar
[2] Hrushovski, E., Contributions to stable model theory, Ph.D. thesis , Berkeley, 1986.Google Scholar
[3] Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.Google Scholar
[4] Hrushovski, E. and Pillay, A., Weakly normal groups, Logic Colloquium, vol. 85.Google Scholar
[5] Hrushovski, E. and Pillay, A., Groups definable in local fields and pseudo-finite fields, Israel Journal of Mathematics, vol. 85 (1994), pp. 203–262.Google Scholar
[6] Hrushovski, E. and Zil'ber, B., Zariski geometries, to appear.Google Scholar
[7] Hrushovski, E. and Zil'ber, B., Zariski geometries, Bulletin (New Series) of the AMS, vol. 28 (1993), no. 2, pp. 315–323.Google Scholar
[8] Knight, J., Pillay, A., and Steinhorn, C., Definable sets in ordered structures II, Transactions of the AMS, vol. 295 (1986), pp. 593–605.Google Scholar
[9] Loveys, J. and Peterzil, Y., Linear o-minimal structures, Israel Journal of Mathematics, vol. 81 (1993), pp. 1–30.Google Scholar
[10] Marker, D., Peterzil, Y., and Pillay, A., Additive reducts of real closed fields, Journal of Symbolic Logic, vol. 57 (1992), no. 1, pp. 109–117.Google Scholar
[11] Peterzil, Y., Constructing a group-interval in o-minimal structures, Journal of Pure and Applied Algebra, vol. 94 (1994), pp. 85–100.Google Scholar
[12] Peterzil, Y., Pillay, A., and Starchenko, S., A classification of simple groups in o-minimal structures, in preparation.Google Scholar
[13] Peterzil, Y. and Starchenko, S., A trichotomy theorem for o-minimal structures, preprint.Google Scholar
[14] Pillay, A., An introduction to stability theory, Clarendon Press, Oxford, 1983.Google Scholar
[15] Pillay, A. and Steinhorn, C., Definable sets in ordered structures I, Transactions of the AMS, vol. 295 (1986), pp. 565–592.Google Scholar
[16] Rabinovich, E., Definability of a field in sufficiently rich incidence systems, Maths Notes 14, Queen Mary and Westfield College, University of London, 1986.Google Scholar
[17] van den Dries, L., Tame topology and o-minimal structures, preliminary version, 1991.Google Scholar
[18] Wilkie, A., Model completeness results for expansions of the real field II: The exponential function, to appear.Google Scholar
[19] Zil'ber, B., Structural properties of models of ℵ1-categorical theories, Logic methodology and philosophy of science VII (Marcus, R. Barcon et al., editors), North Holland, Amsterdam, 1986, pp. 115–128.Google Scholar