The talk will describe recent work, joint with Barry Simon, on how
spectral theoretic ideas can be applied in the study of boundary behavior
of power series.

In particular, we describe how the notions of right-limit and
reflectionless operators from spectral theory can be used to obtain
results for power series with bounded Taylor coefficients.

We recover and (within the class of bounded coefficients) improve
various classical results.

The only known sufficient condition for the existence of a Kahler-Einstein metric on a Fano manifold can be formulated in terms of so-called alpha-invariant introduced by Tian and Yau more than 20 years ago. This invariant can be naturally defined for log Fano varieties with log terminal singularities using purely algebraic language. Using a global-to-local results of Shokurov, one can define a similar invariants for a germ of log terminal singularity. We describe the role played by alpha-invariants in birational geometry and singularity theory. We prove the existence of Kahler-Einstein metrics on many quasismooth well-formed weighted del Pezzo hypersurface and compare this result with new obstructions found by J.Gauntlett, D.Martelli, J.Sparks and S.-T.Yau. We apply our technique to classify weakly-exceptional quasismooth well-formed weighted del Pezzo hypersurface using the classification of isolated rational quasihomogeneous three-dimensional singularities obtained by S.S.T.Yau and Y.Yu.