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\centerline{\bf Alexei Karlovich (Lisbon, Portugal)}
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{\bf SPECTRUM OF THE CAUCHY SINGULAR INTEGRAL\\ 
OPERATOR IN WEIGHTED \\
REARRANGEMENT-INVARIANT SPACES}
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The Cauchy singular integral operator $S$ is one of the main actors 
in the theory of Toeplitz operators, Riemann-Hilbert problems, 
Wiener-Hopf and singular integral equations and other fields 
of Harmonic and Complex Analysis. We are going to describe 
the spectrum of the operator $S$ in weighted 
rearrangement-invariant spaces. These spaces are wide generalizations 
of classical Lebesgue, Orlicz and Lorentz spaces. During the last few 
years it was discovered that, in depence on the curves the operators 
acts on and on the weights involved rearrangement-invariant spaces, 
there is a surprising metamorphosis of the (local) spectra of $S$ from 
circular arcs via horns and logarithmic double spirals to so-called 
logarithmic leaves with a halo. Technical details will be omitted, 
but many beautiful pictures of local spectra will be shown.
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