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SUMMARY:Tirthankar Bhattacharyya (IISc Bangalore)
DTSTART:20200930T113000Z
DTEND:20200930T130000Z
DTSTAMP:20260423T021247Z
UID:WOTOA/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WOTOA/6/">On
  the geometry of the symmetrized bidisc</a>\nby Tirthankar Bhattacharyya (
 IISc Bangalore) as part of Webinars on Operator Theory and Operator Algebr
 as\n\n\nAbstract\nWe study the action of the automorphism group of the $2$
  complex dimensional manifold symmetrized bidisc $\\mathbb G$ on itself. T
 he automorphism group is $3$ real dimensional. It foliates $\\mathbb G$ in
 to leaves all of which are $3$ real dimensional hypersurfaces except one\,
  viz.\, the royal variety. This leads us to investigate Isaev's classifica
 tion of all Kobayashi-hyperbolic $2$ complex dimensional  manifolds for wh
 ich the group of holomorphic automorphisms has real dimension $3$ studied 
 by Isaev. Indeed\, we produce a biholomorphism between the symmetrized bid
 isc and the domain\n\n	\\[\\{(z_1\,z_2)\\in \\mathbb{C} ^2 : 1+|z_1|^2-|z_
 2|^2>|1+ z_1 ^2 -z_2 ^2|\, Im(z_1 (1+\\overline{z_2}))>0\\}\\]\n\nin Isaev
 's list. Isaev calls it $\\mathcal D_1$. The road to the biholomorphism is
  paved with various geometric insights about $\\mathbb G$.	\n\nSeveral con
 sequences of the biholomorphism follow including two new characterizations
  of the symmetrized bidisc and several new characterizations of $\\mathcal
  D_1$. Among the results on $\\mathcal D_1$\, of particular interest is th
 e fact that $\\mathcal D_1$ is a ``symmetrization''. When we symmetrize (a
 ppropriately defined in the context) either $\\Omega_1$ or $\\mathcal{D}^{
 (2)} _1$ (Isaev's notation)\, we get $\\mathcal D_1$.  These two domains $
 \\Omega_1$ and $\\mathcal{D}^{(2)} _1$ are in Isaev's list and he mentione
 d that these are biholomorphic to $\\mathbb D \\times \\mathbb D$. We prod
 uce explicit biholomorphisms between these domains and $\\D \\times \\D$.\
 n
LOCATION:https://researchseminars.org/talk/WOTOA/6/
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