### Critical measures, S-curves and G-functions

Curves with a special symmetry property (S-property) in a harmonic external eld play an important role in

several elds of modern analysis such as approximation theory, orthogonal polynomials and nonlinear PDE.

Such curves are also connected to a number of classical problems in geometric theory of analytic functions

(moduli problems, quadratic dierentials and so on).

Equilibrium measures of S-curves are critical measures which connect them to a large class of equilibrium

problems for logarithmic potential. On the other hand, potentials of those measures (g-functions) may be

also dened in terms of certain boundary value problems.

The lecture will review some of above mentioned connections and, in particular, some of the old and

new result on the existence problem for S-curves.

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