Abstract: The classical Watts' theorems identify functors which are tensor products or Hom functors by internal properties. We extend these theorems to homological algebra and algebraic topology. So, in the easiest case, we characterize all functors from the unbounded derived category $D(R)$ of a ring $R$ to $D(S)$ which are given by the derived tensor product with a complex of bimodules (recovering a result of Keller's in this case). We draw conclusions about Brown representability of homology and cohomology functors.

Note room change: MC107.

Abstract: O-minimal structures are categories of sets and mapping having nice geometrical properties. To each o-minimal expansion of a real closed field, one can associate the set of germs at infinity of its unary functions, which form a Hardy field. Valuational properties of these Hardy fields give good information about the initial structure. After a lengthy introduction of all the previously named objets and motivated by a conjecture of L. van den Dries and a result of F.-V. and S. Kuhlmann, I will discuss whether an o-minimal expansions of the field of the reals is, in general, fully determined by its associated Hardy field. I will also relate this question to the Hilbert's 13th Problem.