Geometry, calculus and Zil'ber's conjecture

In the case of the field C, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the field but this will be too coarse to give a differentiable 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 byMontgomery-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. Webeginwith a linearly ordered set 〈M,<〉 equippedwith the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of M , n = 1, 2, . . . , in some first order expansionM of 〈M,<〉. The main topological assumption is that definable functions in one variable are piecewise continuous, or equivalently thatM is order minimal (see definition in the next section). In order to get a differentiable structure we need to assume also the existence of a sufficiently complicated definable subset ofM . We then conclude that one can develop on these grounds a theory of differentiation for definable sets inM. In particular, one can find two definable functions {+, ·} such that for some open interval I ⊆ M , 〈I,+, ·〉 is a real closed field, call itR. As is already known, under these assumptions every definable subset of R can be partitioned into finitely many definable manifolds-like sets with respect to R. We discuss the theorem from fewdifferent angles. In the language ofmodel theory we formulate first a more general result, the Trichotomy Theorem,