Ahmed Yekta Ökten

Estimates of the Kobayashi distance of strongly pseudoconvex domains

We will present sharp upper and lower bounds (of the same type) for the Kobyashi distance of a C^2,α-smooth strongly pseudoconvex domain. These bounds come from the study of the behaviour of complex geodesics and extend the well-known Balogh-Bonk estimate (2000). The talk is based on a joint work with L. Kosinski and N. Nikolov.

Presentation

Andrew Zimmer

Bounded geometry in several complex variables

I will discuss several different bounded geometry conditions for a complex manifold. Then describe some applications.

Presentation

Armen Edigarian

Isometries on the symmetrized bidisc

We study Caratheodory isometries in the symmetrized bidisc. We show that any C¹-isometry is holomorphic or anti-holomorphic. By-pass we give also a new proof of the description of automorphisms of the symmetrized bidisc, using „only” formulas for the distance and infinitesimal metric, but not using W. Kaup’s result on the orbit of the origin under automorphisms (as it was done by Jarnicki-Pflug).

Filippo Bracci

Visibility and geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains

In this talk I will summarise the recent results (and open questions) about visibility and geodesic loops in bounded domains of the complex space which are both complete Kobayashi hyperbolic and Gromov hyperbolic. In particular, I will give a characterization of visibility and the existence of geodesic loops in terms of “geodesic frames” and show some applications.

Presentation

Gautam Bharali

The squeezing function: exact computations and a new application

In recent years, work on the squeezing function has focused largely on a diagnostic for detecting (local) strong Levi pseudoconvexity. Comparatively less attention has been focused on what the squeezing function can say about the complex geometry of a domain and on the very amusing — but rather hard — problem of computing the squeezing function exactly. In this talk, we shall revisit the latter aspects of the squeezing function. This examination of the squeezing function will rely on an old idea, suitably adapted, of “fitting” large polydiscs into domains, which has been useful in the past in the non-strongly pseudoconvex context. The talk is based on a joint work with Diganta Borah and Sushil Gorai.

Presentation

Herve Gaussier

Invariant metrics – old and more recent results

Invariant metrics under the action of biholomorphisms, such as the Kobayashi and Carathéodory metrics, or the Bergman and Kähler-Einstein metrics, are important metrics that can encode certain properties of the complex manifolds on which they are defined. In this talk, I will address some problems concerning their study. I will present some classical results establishing links between the geometric properties of complex manifolds and the properties of their invariant metrics. I will also present more recent studies in this direction.

Presentation

Łukasz Kosinski

Extension property

We study sets in convex domains that have the polynomial extension property.

Matteo Fiacchi

The Julia-Wolff-Carathéodory theorem in convex domains of finite type

The classical Julia-Wolff-Carathéodory theorem says that, if f is a holomorphic self-map of the unit disk D and ξ∈∂D such that liminf_z→ξ(1−|f(z)|)/(1−|z|)<∞ then the derivative f′ has non-tangential limit at ξ and the limit value can be computed in terms of the Poincaré distance. The theorem has been generalized to several complex variables by Rudin in the unit ball, by Abate in strongly convex and by Abate-Tauraso in convex domains of finite type, under some technical assumptions. In this talk I will present a version of the Julia-Wolff-Carathéodory theorem in convex domains of finite type in full generality, using metric geometry techniques. This is a joint work with L. Arosio. Presentation

Sławomir Dinew

Complex nonlinear equations, weak solutions and admissible domains

We report on recent progress in the study of nonlinear elliptic equations in domains in Cⁿ. Existence of weak solutions to those, if suitable boundary behavior is assumed, forces geometric properties of the domain.

Włodzimierz Zwonek

2-proper holomorphic images of the Cartan domains

The ideas that generalize the construction of the symmetrized bidisc and the tetrablock as 2-proper holomorphic images of the Cartan domains will be presented. The new domains have many geometric properties that will be presented in the lecture. That will contain description of automorphisms, properties of the Bergman kernel, (non)-convexity of the domains. The talk is based on the common research with Gargi Ghosh.

Presentation

Zywomir Dinew

Aspects of B-regularity

The notion of B-regularity was introduced by Sibony in his famous Duke paper from 1987. For compact sets it means that one can approximate uniformly an arbitrary continuous function by continuous plurisubharmonic functions defined on (varying) neighborhoods. We will present some new properties of B-regular compacts and domains and provide some examples. This is an ongoing research with Slawomir Dinew.

Presentation