Model Theory of Holomorphic Functions

This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing. We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal. We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions $\mathcal{R}$ we associate a sheaf $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ of analytic germs which is closed under application of the implicit function theorem. We prove that $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call $\mathcal{R}$-analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in $\mathcal{R}$. If this condition is satisfied then the $\mathcal{R}$-analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture

Open-closed homotopy algebra in mathematical physics

In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures ($A_\infty$-algebras) by closed strings ($L_\infty$-algebras).Comment: 38 pages, 4 figures; v2: published versio

The Angehrn-Siu Type Effective Freeness For Quasi-Log Canonical Pairs

We prove the Angehrn-Siu Type effective freeness and effective point separation for quasi-log canonical pairs. As a natural consequence, we obtain that these two results hold for semi-log canonical pairs. One of the main ingredients of our proof is the inversion of adjunction for quasi-log canonical pairs, which is established in this paper

Wheeled props in algebra, geometry and quantization

These are expanded notes of author's talk at the ECM 2008 attempting to give an elementary introduction into the main ideas of the theory of wheeled props for beginners, and also a survey of its most recent major applications (ranging from algebra and geometry to deformation theory and Batalin-Vilkovisky quantization) which might be of interest to experts.Comment: The Proceedings versio