## Closed ordered differential fields

#### Marcus Tressl (Manchester)

An ordered differential field is an ordered field $$(K,\le )$$ equipped with a derivation $$\delta :K→K$$; no interaction of $$\le$$ and $$\delta$$ is assumed. A closed ordered differential field (CODF) is an existentially closed ordered differential field. Michael Singer has shown that CODFs are axiomatizable and constitute the model completion with quantifier elimination of ordered differential fields in the first order language $$\{+,−,⋅,≤,0,1,\delta\}$$. Recently, some progress was made in the model theoretic study of CODFs and I will review this (after presenting CODF in a geometric way). I will then focus on an intermediate value theorem for continuous semi-algebraic functions in the CODF context and on a canonical representation of definable sets with the aid of closed semi-algebraic sets.