Franz-Viktor Kuhlmann (UŚl, Katowice), Resolution of singularities and the decidability of Laurent series fields over finite fields

Oddział Górnośląski Polskiego Towarzystwa Matematycznego
 

W dniu 17 marca 2016 roku (CZWARTEK) o godz. 16:00 w sali 553 w Instytucie Matematyki Uniwersytetu Śląskiego w Katowicach, ul. Bankowa 14

Prof. dr hab.Franz-Viktor Kuhlmann (Uniwersytet Śląski w Katowicach)

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WYKŁAD 
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RESOLUTION OF SINGULARITIES AND THE DECIDABILITY OF LAURENT SERIES FIELDS OVER FINITE FIELDS
 

Abstract:

In 1964 Hironaka proved resolution of singularities for algebraic varieties of arbitrary dimension over fields of characteristic 0. In positive characteristic, it has only been proved for dimensions up to 3 by Abhyankar and recently by Cossart and Piltant. The general case has remained open although several working groups of algebraic geometers have attacked it. In 1965, Ax and Kochen, and independently Ershov proved that the elementary theory of the fields Q_p of p-adic numbers is decidable, that is, there exists a recursive algorithm which for every given assertion formulated in first order logic decides whether it holds in Q_p or not. Laurent series fields over finite fields are very similar to the p-adics; both of them are complete discretely valued fields. Nevertheless, it is still an open problem whether they have a decidable elementary theory, in spite of the efforts from several excellent model theorists.
At first glance there seems to be no connection between these two deep open problems. But following ideas by Zariski, one can also consider local uniformization, a local form of resolution. Already in 1940 he proved it to hold for algebraic varieties of all dimensions in characteristic 0. But again, the case of positive characteristic has remained open. By its definition, local uniformization is a problem of valuation theoretical nature. My own work has shown that the local uniformization as well as the decidability problem have a common core: the structure of valued function fields, It is horrendously complex and not well enough understood in positive characteristic. In my talk, I will give a survey on the present state of the ongoing research related to both problems.

                                                                                                                                                                                                                                                   Franz-Viktor Kuhlmann

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