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.

Málstofa í stærðfræði

Fyrirlesari: Arkadiusz Lewandowski
Titill: Some remarks on holomorphically contractible systems.

Staðsetning: V-157, VRII.
Tími: Föstudagur 25. september, klukkan 15:00-16:00.

Ágrip:

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.

Málstofa í stærðfræði

Fyrirlesari: Ahmed Zeriahi
Titill: Weak solutions to degenerate complex Monge-Ampère Flows

Staðsetning: TG-227, Tæknigarður.
Tími: Föstudagur 4. september, klukkan 15:00-16:00.

Ágrip:

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).

Málstofa í stærðfræði

Fyrirlesari: Evgeny Poletsky, Syracuse University
Titill: Hardy spaces on hyperconvex domains: recent advances

Staðsetning: Naustið, Endurmenntun (hér)
Tími: Fimmtudagur 11. júní, klukkan 15:00-16:00.

Ágrip:

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.

Fundur verður haldinn í Íslenska stærðfræðafélaginu fimmtudaginn 23. apríl kl 16:45 í stofu VR-158 í HÍ. (Húsi Verkfræði-og náttúruvísindasviðs við Hjarðarhaga.)

Fundurinn hefst með hefðbundnum kaffiveitingum, en kl 17:15 heldur Finnur Lárusson stærðfræðingur við Adelaide-háskóla í Ástralíu fyrirlestur sem ber yfirskriftina:

Sveigjanleiki og stjarfi í fágaðri rúmfræði
Alþjóðleg ráðstefna um tvinnfallagreiningu og fágaða rúmfræði verður haldin dagana 24.-26. apríl í Háskóla Íslands, Nordan 2015. Þessum fyrirlestri er ætlað að gefa breiðum áheyrendahópi örlitla innsýn í nýjustu rannsóknir á þessu sviði. Fjallað verður um sveigjanleika og stjarfa, grundvallarfyrirbæri sem togast á í fágaðri rúmfræði. Fyrirlesturinn ætti að vera aðgengilegur öllum sem lokið hafa fyrsta námskeiði í tvinnfallagreiningu. Af tillitssemi við erlenda gesti hefur fyrirlesarinn verið beðinn að tala ensku.

Málstofa í stærðfræði

Speaker: Emmanuel Mazzilli, Université de Lille 1
Title: J-holomorphic curves in real analytic hypersurface.

Staðsetning: Naustið, Endurmenntun VR-II, 158
Tímasetning: Fimmtudaginn 23. apríl, klukkan 15:00-16:00.

Ágrip:

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.

Málstofa í stærðfræði

Speaker: Miroslav Englis, Mathematics Institute, Prague & Opava, Czech Republic
Title: High-power asymptotics of weighted harmonic Bergman kernels

Staðsetning: Naustið, Endurmenntun, VR-II, 158.
Tímasetning: Fimmtudaginn 23. apríl, klukkan 11:00-12:00.

Ágrip:

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.

Málstofa í stærðfræði

Speaker: Mitja Nedić, Stockholm University
Title: q-conevxity

Staðsetning: Interactive room, 2. hæð í Tæknigarði.
Tími: Fimmtudagur 16. apríl, klukkan 15:00-16:00.

Ágrip:

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.

Málstofa í stærðfræði

Fyrirlesari: Arkadiusz Lewandowski, University of Iceland
Titill: Separate vs joint regularity of functions

Staðsetning: V-157, VRII
Tími: Mánudagur 3. nóvember, frá 15:00 til 16:00.

Ágrip:

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.