On Moebius and conformal maps between boundaries of CAT(-1) spaces by Kingshook Biswas

20 March 2017 to 25 March 2017

VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru

Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions between mathematics and theoretical physics, especially string theory, channels through complex analytic geometry. Some of the high points of research in this topic are: Yau’s proof of Calabi’s conjecture, Donaldson-Uhlenbeck-Yau’s theorem that poly stable vector bundles are precisely the solutions of the Hermitian-Einstein equation, Demailly’s work of Kobayashi hyperbolicity.

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0:00:00 Complex Geometry
0:00:07 On Moebius and conformal maps between boundaries of CAT(-1) spaces
0:00:20 Moebius group in dimensions
0:02:00 Isometrics of (real) hyperbolic space:
0:05:23 Mostow rigidity
0:09:21 Marked length spectrum rigidity
0:11:22 CAT(-1) spaces
0:15:22 Boundary at infinity
0:18:19 Examples
0:19:37 Visual metrics
0:24:31 Gromov inner product:
0:25:27 Moebius and conformal maps between metric spaces
0:29:50 Cross-ratio on deltaX
0:35:06 Marked length spectrum and Moebius maps
0:37:59 Conformal maps and geodesic conjugacies
0:43:49 The integrated Schwarzian of a conformal map
0:46:09 Integrated Schwarzian and cross-ratio distortion
0:48:58 Almost-isometric extension of Moebius maps
0:50:21 Complete, simply connected, negatively curved manifolds: infinitesimal rigidity
0:53:06 Proof of almost-isometric extension
0:57:47 Proof of infinitesimal rigidity