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SUMMARY:Distorting Banach spaces - Professor Kevin Beanland (Washington &a
 mp\; Lee University\, Virginia\, USA)
DTSTART:20170308T140000Z
DTEND:20170308T150000Z
UID:TALK71504@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:A Banach space $X$ with a norm $\\|\\cdot\\|$ is called D-dist
 ortable if\nthere is an equivalent norm $|\\cdot|$ on $X$ so for each\ninf
 inite-dimensional subspace $Y$ of $X$ there are vectors $x\,y \\in Y$\nwit
 h $\\|x\\|=\\|y\\|=1$ and $|x|/|y|>D$. A space is arbitrarily distortable\
 nif it is D-distortable for every $D>1$. A result of R.C. James from the\n
 1960s shows that the Banach spaces $\\ell_1$ and $c_0$ are not\ndistortabl
 e for any $D>1$. Shortly after this V. Milman showed that if a\nBanach spa
 ce does not contain any $\\ell_p$ or $c_0$ it must have a\nsubspace that i
 s $D$-distortable for some $D>1$. In the 1990s it was\nshown explicitly by
  Odell and Schlumprecht that Tsirelson's famous space\nwas itself $D$-dist
 ortable for each $D<2$. In the 1990s there were\nseveral surprising\, dram
 atic results concerning distortion including the\nconstruction\, by Schlum
 precht\, of the first known arbitrarily\ndistortable Banach space and the 
 very unexpected proof that $\\ell_p$ is\narbitrarily distrotable for $p$ i
 n the reflexive range. It is still an\nopen question as to whether there i
 s a Banach space that is distortable\nbut not arbitrarily distortable. In 
 particular\, it is not known if\nTsirelson's space satisfies bounded disto
 rtions. Recently there has been\nsome renewed attention to this and other 
 problems related to\ndistortion. On the website MathOverFlow\, W.T. Gowers
  and P. Dodos\nsuggested a set of problems that quantify distortion in a s
 ubtle\ncombinatorial way. In this talk\, we will explain the solution to s
 ome of\nthese problems and how the problems relate to descriptive set theo
 ry and\npotentially some deep combinatorial principles. Some of the work w
 e will\nmention is joint with Ryan Causey and Pavlos Motakis.
LOCATION:CMS\, MR14
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