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CATEGORIES:DPMMS info aggregator
SUMMARY:Distorting Banach spaces - Professor Kevin Beanlan
d (Washington &\; Lee University\, Virginia\, U
SA)
DTSTART;TZID=Europe/London:20170308T140000
DTEND;TZID=Europe/London:20170308T150000
UID:TALK71504AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/71504
DESCRIPTION:A Banach space $X$ with a norm $\\|\\cdot\\|$ is c
alled D-distortable if\nthere is an equivalent nor
m $|\\cdot|$ on $X$ so for each\ninfinite-dimensio
nal subspace $Y$ of $X$ there are vectors $x\,y \\
in Y$\nwith $\\|x\\|=\\|y\\|=1$ and $|x|/|y|>D$. A
space is arbitrarily distortable\nif it is D-dist
ortable for every $D>1$. A result of R.C. James fr
om the\n1960s shows that the Banach spaces $\\ell_
1$ and $c_0$ are not\ndistortable for any $D>1$. S
hortly after this V. Milman showed that if a\nBana
ch space does not contain any $\\ell_p$ or $c_0$ i
t must have a\nsubspace that is $D$-distortable fo
r some $D>1$. In the 1990s it was\nshown explicitl
y by Odell and Schlumprecht that Tsirelson's famou
s space\nwas itself $D$-distortable for each $D<2$
. In the 1990s there were\nseveral surprising\, dr
amatic results concerning distortion including the
\nconstruction\, by Schlumprecht\, of the first kn
own arbitrarily\ndistortable Banach space and the
very unexpected proof that $\\ell_p$ is\narbitrari
ly distrotable for $p$ in the reflexive range. It
is still an\nopen question as to whether there is
a Banach space that is distortable\nbut not arbitr
arily distortable. In particular\, it is not known
if\nTsirelson's space satisfies bounded distortio
ns. Recently there has been\nsome renewed attentio
n to this and other problems related to\ndistortio
n. On the website MathOverFlow\, W.T. Gowers and P
. Dodos\nsuggested a set of problems that quantify
distortion in a subtle\ncombinatorial way. In thi
s talk\, we will explain the solution to some of\n
these problems and how the problems relate to desc
riptive set theory and\npotentially some deep comb
inatorial principles. Some of the work we will\nme
ntion is joint with Ryan Causey and Pavlos Motakis
.
LOCATION:CMS\, MR14
CONTACT:HoD Secretary\, DPMMS
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