A real analytic function can always be continued holomorphically to some domain. However, the holomorphic continuations of definable functions in an o-minimal structure may not be definable. I will present joint work with P. Speissegger in which we study holomorphic continuations of functions definable in two o-minimal expansions of the real field. I will also discuss how to apply these results to the complex Gamma function and Riemann zeta function.[-]

We study the relationship between metric and algebraic structures on the section ring of a projective manifold and an ample line bundle over it. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^{2}$-norm. We then show that, in fact, those algebraic properties characterise $L^{2}$-norms and describe some applications of this classification. The semiclassical version of Ohsawa-Takegoshi theorem lies at the heart of our approach.[-]