## Mitja Nedić (16/04/15)

Benedikt Magnússon, apríl 15, 2015

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.

Benedikt Magnússon, október 27, 2014

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.

## Auðunn Skúta Snæbjarnarson (01/10/14)

Benedikt Magnússon, september 22, 2014

Meistaraprófsfyrirlestur

### Auðunn Skúta Snæbjarnarson Titill: Approximation of Holomorphic Functions in the Complex Plane

Tímasetning: Miðvikudagur 1. október, klukkan 16:40.

### Ágrip:

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.

Leiðbeinandi: Ragnar Sigurðsson
Prófdómari: Reynir Axelsson

## Benedikt Magnússon (22/09/14)

Benedikt Magnússon, september 17, 2014

Málstofa í stærðfræði

### Fyrirlesari: Benedikt Magnússon Titill: Carleman approximations

Staðsetning: V-157, VRII
Tími: Mánudagur 22. september, frá 15:00 til 16:00.

### Ágrip:

I will introduce Carleman’s remarkable extension of the Weierstrass approximation theorem. In its simplest form it states that if $$f$$ and $$\epsilon$$ are continuous functions on the real line $$\mathbb R \subset \mathbb C$$, and $$\epsilon > 0$$ then there exists an entire function $$F$$ such that |f(x)-F(x)| < \epsilon(x)[/latex], for all $x\in \mathbb R$. I will show what the obstructions are for doing this kind of approximations, and, most importantly, how all this generalizes to several complex variables.

## Dan Popovici (26/06/14)

Benedikt Magnússon, júní 25, 2014

Málstofa í stærðfræði

### Fyrirlesari: Dan Popovici, Institut de Mathématiques de Toulouse Titill: Positivity cones of the Aeppli chomologoy of compact complex manifolds

Staðsetning: V-155, VRII
Tími: Fimmtudaginn 26. júní, frá 15:00 til 16:00.

### Ágrip:

We define the Gauduchon cone of a compact complex $n$-dimensional manifold $X$ as the open convex cone consisting of Aeppli cohomology classes of powers $\omega^{n-1}$ of Gauduchon metrics $\omega$, while the sG (strongly Gauduchon) cone is defined as the intersection of the Gauduchon cone with a certain vector subspace. We will discuss the roles that these two cones play in describing fundamental geometric properties of $X$ as well as in the geometry of holomorphic deformations of the complex structure of $X$.

## Stephane Rigat (27/05/14)

Benedikt Magnússon, maí 22, 2014

Málstofa í stærðfræði

### Fyrirlesari: Stephane Rigat Titill: Fokas Method applied on Boundary Value Problems for axisymmetric potentials

Staðsetning: V-158, VR-II.
Tímasetning: Þriðjudagur 27. maí , 2014, klukkan 14:00.