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Department of Mathematical Sciences


This week's seminars

Geometry and Topology Seminar: Integrals over unitary groups, maps on surfaces, and Euler characteristics

Presented by Michael Magee, Durham University

18 January 2018 13:00 in CM221

This is joint work with Doron Puder (Tel Aviv University).
For a positive integer r, fix a word w in the free
group on r generators. Let G be any group. The word
w gives a `word map' from G^r to G: we simply replace the
generators in w by the corresponding elements of G. We
again call this map w. The push forward of Haar measure under
w is called the w-measure on G. We are interested in
the case G = U(n), the compact Lie group of n-dimensional
unitary matrices. A motivating question is: to what extent do the
w-measures on U(n) determine algebraic properties of the
word w?

For example, we have proved that one can detect the
'stable commutator length' of w from the w-measures on
U(n). Our main tool is a formula for the Fourier
coefficients of w-measures; the coefficients are rational
functions of the dimension n, for reasons coming from
representation theory.

We can now explain all the Laurent coefficients of these
rational functions in terms of Euler
characteristics of certain mapping class groups.
I'll explain all this in my talk, which should be broadly accessible and of general
interest. Time permitting, I'll also invite the audience to consider some
remaining open questions.

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