[mk-duyuru] Valuation theory Franz-Victor Kuhlman

Ali Nesin nesin at bilgi.edu.tr
26 Nis 2009 Paz 02:27:18 EEST

Prof. Franz-Victor Kuhlmann Istanbul Bilgi Universitesi Dolapdere kampusunde
"Valuation Theory and its Model Theory" uzerine bes konusma verecek.
Tanimlardan baslayacak konusmalar bir yil cebir alan her ogrencinin
anlayabilecegi duzeyde olacak.

Tarihler: 29 ve 30 Nisan ve 4, 5 ve 6 Mayis.
Saat 17,00.
Konusmalarin ozeti asagida.

1) A quick introduction to Valuation Theory
I will introduce the notions of valuation, valuation ring, place, value
group and residue field. Examples for valued fields will be given: power
series fields, p-adic fields, non-archimedean ordered fields, valued
function fields.
Further topics: extensions of valuations, the fundamental inequality,
Hensel's Lemma and henselizations.

2) Two open problems
There are two deep open problems connected with valuation theory in positive
characteristic. Resolution of singularities and its local form, local
uniformization, are both still open in positive characteristic, despite the
effort of many first class algebraic geometers (including Hironaka!). The
latter problem is about valued function fields. Another deep open problem is
the model theory of Laurent series fields over finite fields (which has
resisted the attacks of several excellent model theorists). I will describe
these problems and show their connection with the valuation theoretic
phenomenon of "defect", which has turned out to be the main enemy.

3) The defect
This talk will give examples of extensions of valued fields with non-trivial
defect and give a survey on what is known and what is not known about the

4) F_p((t))
This talk will take a closer look at the positive and negative results we
know about the model theory of the Laurent series field F_p((t)) over the
field F_p with p elements. It turns out that additive polynomials play an
important role here. There are important open probems about the
interconnection of additive polynomials over non-perfect fields and
valuation theory.

5) Local uniformization
This talk surveys the existing theorems giving partial results on local
uniformization in positive characteristic and will sketch some basic ideas
of their proofs. There are many very challenging open problems in this area
which connects valuation theory with algebraic geometry.

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