Math Colloquium

### Speaker: Benedikt Steinar Magnússon

Title: Pluripotential theory in several complex variables explained by the Dirichlet problem in the plane

Location: V-157, VRII.

Time: Friday, February 12 at 13:20.

### Abstract:

The goal is to give a brief introduction to pluripotential theory in several complex variables using the Dirichlet problem in the plane as a starting point. We will start by looking at different solution methods for the Dirichlet problem. One of them, the Perron method, motivates our approach to similar problems in several complex variables. But when we look at the ingredients of the Dirichlet problem it turns out that not all of them generalize well to several complex variables so we will have to carefully choose what we bring with us on our adventure into \(\mathbb C^n\).

The problems in several complex variables we consider includes for example the global extremal function which is not only useful in pluripotential theory but also in holomorphic function theory.

Math Colloquium

### Speaker: Arkadiusz Lewandowski

Title: Some remarks on holomorphically contractible systems.

Location: V-157, VRII.

Time: Friday, September 25, at 15:00-16:00.

### Abstract:

We shall discuss the ideas behind the holomorphically contractible systems. As examples, we introduce systems of Carathéodory and Kobayashi pseudodistances. We shall discuss some properties of those objects with particular accent put on their behaviour under certain set-theoretical operations.

Math Colloquium

### Speaker: Ahmed Zeriahi

Title: Weak solutions to degenerate complex Monge-Ampère Flows

Location: TG-227, Tæknigarður.

Time: Friday, September 4, at 15:00-16:00.

### Abstract:

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. The purpose of this lecture is to explain how to develop a viscosity theory for degenerate complex Monge-Ampère flows on compact Kähler manifolds. Our general theory allows in particular to define and study the (normalized) Kähler-Ricci flow on varieties with canonical singularities, generalizing results of J. Song and G. Tian.

This is a joint work with P. Eyssidieux and V. Guedj (see arXiv:1407.2504).

Math Colloquium

### Speaker: Evgeny Poletsky, Syracuse University

Title: Hardy spaces on hyperconvex domains: recent advances

Location: Naustið, Endurmenntun (here)

Time: Thursday, June 11, at 15:00-16:00.

### Abstract:

In 2008 M. Stessin and the speaker introduced on a general hyperconvex domain \(D\) the spaces of holomorphic functions \(H^p_u(D)\) as analogs of the classical Hardy spaces on the unit disk. This spaces are parameterized by plurisubharmonic exhaustion functions \(u\) of \(D\). When \(D\) is strictly pseudoconvex they all are the subsets of classical Hardy spaces \(H^p(D)\) and coincide with \(H^p(D)\) when \(u\) is a pluricomplex Green function.

In my talk we will provide all necessary definitions and discuss recent advances: complete description of these spaces on the unit disk and their projective limits on strongly pseudoconvex domains.

The Icelandic Mathematical Society will hold a meeting on Thursday April 23 at 16:45 in the lecture room V-158 in the building of the School of Engineering and Natural Sciences, University of Iceland, at Hjardarhagi.

We follow our traditions by starting the meeting with a cup of coffee or tea, but at 17:15 Finnur Larusson, University of Adelaide, Australia, will give a lecture:

**Flexibility and rigidity in holomorphic geometry**

An international conference in complex analysis and complex geometry will be held at the University of Iceland April 24-26. The goal of this talk is to give a general audience some insight into new research in this area, with a focus on two fundamental themes in complex geometry: flexibility and rigidity. The talk should be accessible to anyone who has done an undergraduate course in complex analysis. The talk will be given in English.

Math Colloquium

### Speaker: Emmanuel Mazzilli, Université de Lille 1

Title: J-holomorphic curves in real analytic hypersurface.

Location: ~~Naustið, Endurmenntun~~, **VR-II, 158**.

Time: Thursday, April 23, at 15:00-16:00.

### Abstract:

In my talk, I will speak about the existence of J-holomorphic curves in real analytic hypersurface for J an real analytic almost complex structure. In particular, I will discuss some generalizations of a Freeman’s theorem and a Diederich-Fornaess’s theorem on compact hypersurface in almost complex setting.

Math Colloquium

### Speaker: Miroslav Englis, Mathematics Institute, Prague & Opava, Czech Republic

Title: High-power asymptotics of weighted harmonic Bergman kernels

Location: ~~Naustið, Endurmenntun~~, **VR-II, 158**.

Time: Thursday, April 23, at 11:00-12:00.

### Abstract:

The asymptotics of the weighted Bergman kernels with respect to the weight \(|r|^\alpha\), where \(r\) is a defining function for a smoothly bounded strictly pseudoconvex domain and \(\alpha\to+\infty\), play prominent role in mathematical physics (Berezin quantization) as well as in complex geometry (Donaldson’s balanced metrics); the standard tool for their derivation is the famous description of the boundary singularity of the Bergman kernel due to Fefferman, combined with a construction due to Forelli and Rudin. The talk will describe why it is noteworthy to study the analogous asymptotics also for the Bergman kernels for harmonic functions, and will give a complete answer for the case of radial weights on the ball and horizontal weights on the upper half-space. The proofs actually proceed by relating the problem to the holomorphic case mentioned above, but on a different domain.

Math Colloquium

### Speaker: Mitja Nedić, Stockholm University

Title: q-conevxity

Location: Interactive room, 2 floor in Tæknigarður.

Time: Thursday, April 16, at 11:00-12:00.

### Abstract:

We will begin the talk by recalling the notion of the Levi form and list some of its basic properties. With the help of the Levi form we then define plurisubharmonic functions and show the equivalence of this definition with others. We will also quickly look at some examples and list some properties of plurisubharmonic functions. We will then define the notion of q-convexity in the case of functions, manifolds and bundles and explore some examples and properties.

We list the theorems that connect metric q-convexity of holomorphic bundles with the q-convexity of complex manifolds.

Finally, we state the Morse lemma for plurisubharmonic function and use it to prove the Morse lemma for q-convex functions. If time, we will state the theorems on the CW decomposition of 1-complete and q-complete manifolds and their consequences.

Math Colloquium

### Speaker: Arkadiusz Lewandowski, University of Iceland

Title: Separate vs joint regularity of functions

Location: V-157, VRII

Time: Monday, November 3 at 15:00-16:00.

### Abstract:

Consider the following problem:

Given two domains \(D \subset K^p, G \subset K^q\), where \(K\) equals either \(\mathbb R\) or \(\mathbb C\), and a function \(f\) on the product \(D \times G\), taking complex values, and such that:

1. \(f(a,-)\) is in \(F(G)\), for any \(a\) in \(D\),

2. \(f(-,b)\) is in \(F(D)\), for any \(b\) in \(G\),

we ask whether \(f\) is in \(F(D\times G)\).

Here for any open set \(U\) in any \(K^n, F(U)\) is some abstract family of functions.

We shall discus the cases \(F \in \{\mathcal{C,O,H,SH}\}\), where \(\mathcal C\) denotes the family of continuous functions, \(\mathcal O\) is the family of holomorphic functions, \(\mathcal H\) stands for the family of harmonic functions, and \(\mathcal{SH}\) – the family of subharmonic functions.

Masters thesis presentations

Location: Askja, 130.

Time: Wednesday October 1., at 16:40.

### Abstract:

Fjallað er um nálganir á fáguðum föllum, með margliðum, ræðum föllum eða heilum föllum. \(L^2\)-tilvistararsetning Hörmanders fyrir Cauchy-Riemann virkjann er notuð til þess að sanna alhæfingu á setningu Bernstein-Walsh, sem lýsir jafngildi milli mögulegrar fágaðar framlengingar á falli \(f\) á opinni grennd við þjappað hlutmengi \(K\) og runu bestu nálgana \((d_n(f,K))\) á \(f\) með margliðum af stigi minna eða jöfnu \(n\). Alhæfingin notar bestu nálganir á \(f\) með ræðum föllum með skaut í gefnu mengi. Fjallað verður um setningu Vitushkins, en hún lýsir hvernig fáguð rýmd mengis er notuð til þess að auðkenna þau þjöppuðu mengi \(K\) með þann eiginleika að sérhvert fall \(f\), samfellt á \(K\) og fágað á innmengi \(K\), megi nálga í jöfnum mæli á \(K\) með margliðum. Að lokum er setning Vitushkins beitt til þess að sanna alhæfingu á setningu Arakelians, sem lýsir nálgun í jöfnum mæli á fáguðum föllum á ótakmörkuðum mengjum með heilum föllum.

Advisor: Ragnar Sigurðsson

Examiner: Reynir Axelsson