The meeting will focus on those areas in the model theory of valued fields in which there have been important recent developments. It is expected that there will be about two scheduled lectures per day, together with informal talks in smaller groups. The aim of the workshop is to encourage discussion and collaboration among the participants. The following themes will be discussed in the main lectures, with the emphasis on new topics and open problems. A list of expected participants can be viewed from here and the timetable can be reached here.
(1) Quantifier-elimination and rigid subanalytic geometry.
Building on work of Lipshitz and Robinson, and using some ideas of Berkovich, Gardener and Shoutens have proved a quantifier elimination result for rigid subanalytic geometry, analogous to the work of Denef and van den Dries on the p-adic subanalytic case. Applications of the latter to zeta functions in group theory will be explored.
(2) Motivic integration.
Denef and Loeser have applied model-theoretic work of Pas to Kontsevich's recent theory of motivic integration.
(3) Application of minimality to VC-dimension.
van den Dries, Haskell and Macpherson have proved that the p-adic subanalytic structures considered under (1) are uniformly P-minimal, and Lipshitz and Robinson have proved the analogue in the rigid subanalytic case. Applications of results such as these to VC-dimension will be discussed.
(4) Imaginaries in valued fields.
Hrushovski has sketched two ways of proving elimination of imaginaries for algebraically closed valued fields (in a language with additional sorts for closed disks), and these are currently being worked out by Haskell, Hrushovski, and Macpherson. In the process a promising-looking theory of independence is developed. This and applications will be investigated.
(5) Valuations, derivations, automorphisms.
Scanlon has examined existentially closed structures consisting of algebraically closed fields expanded by valuation/derivation hybrids. These have relevance to work of Tate and Voloch in diophantine geometry - to finding bounds on p-adic values of points in certain Finiteness Theorems. In addition, liftings of automorphisms from residue fields to henselian fields will be considered, as will a conjecture of van den Dries, Macintyre and Marker on the valuation/derivation structure of the field of LE series.
Places on the workshop are limited and those interested in attending should contact the organisers or Tracey Dart at ICMS.
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[These pages last updated 28th April 1999]