Homogenization of iterated singular integrals with applications to random quasiconformal maps

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Kari Astala, Steffen Rohde, Eero Saksman and I have (finally!) uploaded to the arXiv our preprint “Homogenization of iterated singular integrals with applications to random quasiconformal maps“. This project started (and was largely completed) over a decade ago, but for various reasons it was not finalised until very recently. The motivation for this project was to study the behaviour of “random” quasiconformal maps. Recall that a (smooth) quasiconformal map is a homeomorphism $latex {f: {bf C} rightarrow {bf C}}&fg=000000$ that obeys the Beltrami equation

$latex displaystyle frac{partial f}{partial overline{z}} = mu frac{partial f}{partial z}&fg=000000$

for some Beltrami coefficient $latex {mu: {bf C} rightarrow D(0,1)}&fg=000000$; this can be viewed as a deformation of the Cauchy-Riemann equation $latex {frac{partial f}{partial overline{z}} = 0}&fg=000000$. Assuming that $latex {f(z)}&fg=000000$ is asymptotic to $latex {z}&fg=000000$ at infinity, one can (formally, at least) solve for $latex {f}&fg=000000$ in terms of $latex {mu}&fg=000000$…

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Published by Submathematics

Undergraduate at the Chennai Mathematical Institute

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