Salih Azgin

Valued Difference Fields: I have written my thesis on the model theory of valued difference fields, that is valued fields equipped with a distinguished automorphism. The thesis contains analogues of the Ax-Kochen principle in various contexts depending on the interaction between the valuation and the distinguished automorphism. It turns out that some of the formalism introduced for certain valued difference fields can be applied to ordinary valued fields of positive characteristic. This does not lead to any new results, but it provides an alternative insight to certain algebraically maximal valued fields of positive characteristic. I believe that there is more to be discovered in the connection between valued difference fields and valued fields of positive characteristic. As of April 2009, there is new hope and some preliminary success towards a better understanding of certain valued difference fields.

Extremal Valued Fields: (Joint with F-V. Kuhlmann, F. Pop - with thanks to S. Starchenko) We find an almost-complete classification of extremal fields (fields where the values of polynomials in several variables reach a maximal valuation when evaluated with tuples from the valuation ring). The notion is originally introduced by Ershov, it needs to be tweaked a little so that it makes sense. Henselian fields in residue characteristic zero were expected to be extremal, but this is far from being the case. See below for the paper.

Reverse QE for Valued Fields: I tackle the following question: Is it true that every valued field that admits relative quantifier elimination in the Denef-Pas language (3-sorted language with angular component map) is henselian? The answer is yes if the value group has archimedean rank 1, a recent result of Yimu Yin. I generalize this to finite rank value groups by assuming a slightly stronger form of relative quantifier elimination. I am also considering questions in the same spirit for different languages. Paper is in progress.

*Address:*	Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1
*Office:*	Hamilton Hall 409
*Telephone:*	(905) 525-9140 Ext. 27031
*Email:*	sazgin at math dot mcmaster dot ca
*External Links:*	Burcu Okay (My wife, she's an artist) Homepage A joint blog (Turkish)

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