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SUMMARY:Jan Dobrowolski "Sets\, groups\, and fields definable in vector sp
aces with a bilinear form"
DTSTART;VALUE=DATE-TIME:20211125T100000Z
DTEND;VALUE=DATE-TIME:20211125T110000Z
DTSTAMP;VALUE=DATE-TIME:20211208T012619Z
UID:indico-event-7262@indico.math.cnrs.fr
DESCRIPTION:Sets\, groups\, and fields definable in vector spaces with a\n
bilinear form.\n\nThere is a long history of study of algebraic objects de
finable in\nclassical mathematical structures. As a prominent example\, by
results of Weil\, Hrushovski\, and van den Dries\, it is known that the g
roups definable in an algebraically closed field K are precisely the algeb
raic groups over K\, and the only infinite field definable in K is the fie
ld K itself. The talk will be a report on my work on dimension\, definable
groups\, and definable fields in vector spaces over algebraically closed
[real closed] fields equipped with a non-degenerate alternating bilinear f
orm or a non-degenerate [positive-definite] symmetric bilinear form. The m
ain result states that every definable group is (algebraic-by-abelian)-by-
algebraic [(semialgebraic-by-abelian)-by-semialgebraic]\, which\, in parti
cular\, answers a question of N. Granger. It follows that every definable
field is definable in the field of scalars\, hence either finite or defina
bly isomorphic to it [finite or algebraically closed or real closed]. If t
ime permits\, I will very briefly discuss some model-theoretic phenomena i
n the considered structures\, including an observation from a recent joint
work with D. Hoffmann describing dividing of formulas in the algebraicall
y closed case\, which answers\nanother question of Granger.\n\nhttps://ind
ico.math.cnrs.fr/event/7262/
LOCATION:Braconnier Fokko
URL:https://indico.math.cnrs.fr/event/7262/
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