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SUMMARY:Integrals over unitary groups\, maps on surfaces\, and Euler chara
 cteristics - Michael Magee\, Durham
DTSTART:20180131T160000Z
DTEND:20180131T170000Z
UID:TALK93970@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:For a positive integer r\, fix a word w in the free\ngroup on 
 r generators.  Let G be any group.  The word\nw gives a `word map' from G^
 r to G: we simply replace the\ngenerators in w by the corresponding elemen
 ts of G.  We\nagain call this map w.  The push forward of Haar measure und
 er\nw is called the w-measure on G.  We are interested in\nthe case G = U(
 n)\, the compact Lie group of n-dimensional\nunitary matrices.  A motivati
 ng question is: to what extent do the\nw-measures on U(n) determine algebr
 aic properties of the\nword w? For example\, we have proved that one can d
 etect the\n'stable commutator length' of w from the w-measures on\nU(n).  
 Our main tool is a formula for the Fourier\ncoefficients of w-measures\; t
 he coefficients are rational\nfunctions of the dimension n\, for reasons c
 oming from\nrepresentation theory.\n\nWe can now explain all the Laurent c
 oefficients of these\nrational functions in terms of Euler\ncharacteristic
 s of certain mapping class groups.\nI'll explain all this in my talk\, whi
 ch should be broadly accessible and of general\ninterest.  Time permitting
 \, I'll also invite the audience to consider some\nremaining open question
 s. This is joint work with Doron Puder (Tel Aviv University).
LOCATION:MR13
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