Armen Sergeev
Quantization of the universal Teichmueller space
Universal Teichm\"uller space $\mathcal T$ consists of
quasisymmetric homeomorphisms of the circle $S^1$ (i.e.
homeomorphisms of $S^1$, preserving its orientation and extending to quasiconformal homeomorphisms of the disc), normalized modulo M\"obius transformations. It has a natural K\"ahler structure and contains all classical Teichm\"uller spaces as complex submanifolds.
Moreover, the space $\mathcal T$ includes also the space
$\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ of normalized diffeomorphisms of the circle, which may be considered as a "smooth" part of $\mathcal T$. The space $\text{Diff}_+(S^1)/\text{M\"ob}(S^1)$ may be quantized, using its embedding into an infinite-dimensional Siegel disc. However, this method does not apply to the whole
universal Teichm\"uller space $\mathcal T$, for which quantization we use the "quantum calculus" of Connes--Sullivan.
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