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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Generalizing a question of Gromov\, Part I - Julia
F. Knight (University of Notre Dame\, None / Othe
r)
DTSTART;TZID=Europe/London:20220606T143000
DTEND;TZID=Europe/London:20220606T153000
UID:TALK174785AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/174785
DESCRIPTION:This talk is Part I of an account of joint work wi
th Johanna Franklin and Meng-Che (Turbo) Ho.
\;Johanna Franklin's talk is Part II. \;Gromo
v asked\, ``What is a typical group?'' \;He w
as thinking of finitely presented groups\, and he
proposed an approach involving limiting density. &
nbsp\;In 2013\, I conjectured that for presentatio
ns with $n\\geq 2$ generators and a single relator
\, the elementary first order sentences true in th
e typical group are those true in the free group.
\;The conjecture is still open\, but there ar
e partial positive results by Kharlampovich and Mi
asnikov\, and by Ho and Logan. \;In our joint
work\, Franklin\, Ho\, and I consider other algeb
raic varieties\, in the sense of universal algebra
\, asking the analogue of Gromov's question.
\;We have examples illustrating different possible
behaviors. \;For varieties with finitely man
y unary function symbols\, we have a general resul
t with conditions sufficient to guarantee that the
analogue of the conjecture holds. \;The proo
f uses a version of Gaifman's Locality Theorem\, p
lus ideas from random group theory. \;Part I
will describe Gromov's original question and its e
xtension to other algebraic varieties\, with some
examples. \; \; \; \;
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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