From the applicable to the abstruse: an example in representation theory

Speaker: Jerry Folland, University of Washington

 

Abstract: The operations of time shift (f(t)→ f(t+1)) and frequency
shift (f(t)→ exp(2πiωt)f(t)) are fundamental ingredients of
applied Fourier analysis, and the group of operators on L²(R)
that they generate gives a unitary representation of the so-called discrete
Heisenberg group. How does this representation decompose into irreducible
representations? The answer provides illustrations of (i) some useful tools
of modern harmonic analysis, when ω is rational, and (ii) some
pathological phenomena from the dark side of representation theory, when
ω is irrational. We shall discuss these results after providing a
bit of background on unitary representation theory.

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