## Hermann Þórisson (30/11/2018)

Valentina Giangreco, nóvember 3, 2018

Málstofa í stærðfræði

### Titill: What is typical?

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 30. nóvember kl. 11.40

### Ágrip:

The word “typical” is often used in a loose sense for events in a stationary random process (such as occurrences of heads in repeated coin tosses). This concept can be made precise using so-called Palm version of the process. It is however not well known that there are in fact two Palm versions and that it is the less known version that captures the typicality property. So using the well known version is flawed except in the special case when the two versions coincide.

In this talk the elementary example of repeated coin tosses (indexed by the integers) will be used to make transparent what the issue is.

In the latter half of the talk we shift drastically to random measures on a rather general class of Abelian groups (in the coin tossing example the  group is the integers under addition and the random measure is formed by mass points of size one at the heads). After giving the formal definition of the two Palm versions we present a theorem motivating the claim that it is the less known Palm version that captures the typicality property. If time allows we skim through the proof which relies on the concepts of shift-coupling and mass-stationarity.

## Watse Sybesma (23/11/2018)

Valentina Giangreco, október 29, 2018

Málstofa í stærðfræði

### Titill:

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 23. Nóvember kl. 11.40

## Hjalti Þór Ísleifsson (16/11/2018)

Valentina Giangreco, október 25, 2018

Málstofa í stærðfræði

### Titill: Veik grannmynstur á Banach rúmum.

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 16. Nóvember kl. 11.40

### Ágrip

Við byrjum á að skilgreina veika grannmynstrið á normrúmum og veika* grann- mynstrið á tvírúmum þeirra. Grundavallar eiginleikar þessarra grannmynstra verða kynntir til en sérstök áhersla verður á að skoða þjappleika. Við sönnum setningu Banachs og Alaoglu um að lokaða einingarkúla tvírúms sé þjöppuð í veika* grannmynstrinu. Svo fjöllum við um sjálfhverf Banach rúm, kynnumst grundvallareiginleikum þeirra og skoðum setningu Milman og Pettis sem gefur nægjanlegt rúmfræðilegt skilyrði fyrir því að Banach rúm sé sjálfhverft. Við sönnum setningu Kakutani um að einingarkúla Banach rúms sé þjöppuð í veika grannmynstrinu þá og því aðeins að rúmið sé sjálfhverft. Loks skoðum við setningu Eberlein og Smulian um að þrjár gerðir þjappleika séu jafngildar í normrúmum með veika grannmynstrinu.

## Þórður Jónsson (02/11/2018)

Valentina Giangreco, október 24, 2018

Málstofa í stærðfræði

### Titill: The structure of the spatial slices of 3-dimensional causal triangulations

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 2. Nóvember kl. 11.40

### Ágrip:

We show that there is a bijection between the spatial slices of 3-dimensional causal triangulations and a class of two-dimensional cell complexes satisfying some simple conditions.  The talk will be preceded by a short introduction to the subject.

## Eggert Briem (19/10/2018)

Valentina Giangreco, október 12, 2018

Málstofa í stærðfræði

### Titill: Gelfand Theory for Real Banach Algebras

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 19. Óktober kl. 11.40

### Ágrip:

A real Banach algebra is a Banach algebra over the reals. We will only consider commutative Banach algebras with unit. An example is the algebra of continuous functions, f , on the unit disc, analytic in the interior of the disc, satisfying f (z) = f (z). The norm on the algebra is the sup-norm.

Another example is the algebra of continuously differentiable real-valued functions on the unit interval with the norm given by

∥f∥ = ∥f∥∞ +∥f′∥∞

According to Gelfand theory, a commutative Banach algebra A with unit, over the complex numbers, can be represented as an algebra of continuous complex valued functions on a compact Hausdorff space X, with

sup_{x ∈ X}|ã(x)| = r(a) := lim∥a^n∥^1/n

for a ∈ A. Here X is the space of multiplicative linear functionals on A, equipped with the w∗-topology, and ã(x) = x(a) for x ∈ X.

This result also holds for real Banach algebras. Furthermore, the representati- on consists of real valued functions if and only if

r(a^2) ≤ r(a^2 +b^2)      a,b ∈ A.
We will prove this using only real Banach space theory. If there is time we

will also talk about the general case where there is no condition on A.

## Thomas Selig (12/10/2018)

Valentina Giangreco, október 2, 2018

Málstofa í stærðfræði

### Titill: The Abelian sandpile model on permutation graphs

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 12. Óktober kl. 11.40

### Ágrip:

A permutation graph is a graph whose edges are given by the inversions of a permutation. The Abelian sandpile model (ASM) is a Markov chain on the set of so-called configurations of a graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the long-time running of the model. We exhibit a bijection between the set of recurrent configurations for the ASM on permutation graphs and the set of tiered trees, introduced by Duggan et al. This provides a new bijective proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We also show a link between the minimal recurrent configurations and the set of complete non-ambiguous binary trees, introduced by Aval et al.

## Þorsteinn Jónsson (02/07/18)

Anders Claesson, júní 29, 2018

Málstofa í stærðfræði

### Titill: Hnikaaðferðir til að læra dreififöll gagna

Staðsetning: V-147 (VR-II)
Tími: Mánudagur 2. júlí kl. 10:30

### Ágrip:

Á þessari málstofu mun ég kynna safn aðferða sem að leyfir okkur að skilgreina tölfræðileg líkön sem lýsa líkindadreifingum sem búa til gögn.
Til þessa kynnum við tauganet sem hraða, en ótrúlega áhrifaríka leið til að leysa hnikaverkefni fyrir vel valið kostnaðarfall.
Ég mun ræða tvær mismunandi aðferðir til þess að skilgreina þetta kostnaðarfall ásamt því að sýna áhugaverðar niðurstöður.

## Thomas Selig (27/6/18)

Sigurður Örn Stefánsson, júní 25, 2018

Málstofa í stærðfræði

### Titill: EW-tableaux, permutations and recurrent configurations of the sandpile model on Ferrers graphs.

Staðsetning: VRII, V-147
Tími: Miðvikudagur 27. júní kl. 10:30

### Ágrip:

The Abelian sandpile model (ASM) is a dynamic process on a graph. More specifically, it is a Markov chain on the set of configurations on that graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the long-time running of the model. We study the ASM on Ferrers graphs, a class of bipartite graphs in one-to-one correspondence with Ferrers diagrams. We show that minimal recurrent configurations are in one-to-one correspondence with a set of certain 0/1 fillings of the Ferrers diagrams introduced by Ehrenborg and van Willigensburg. We refer to these fillings as EW-tableaux, and establish a bijection between the set of EW-tableaux of a given Ferrers diagram and a set of permutations whose descent bottoms are given by the shape of the Ferrers diagram. This induces a bijection between these permutations and minimal recurrent configurations of the ASM. We enrich this bijection to encode all recurrent configurations, via a decoration of the corresponding permutation. We also show that the set of recurrent configurations over all Ferrers graphs of a given size are in bijection with the set of alternating trees of that size.

## Delphin Sénizergues (11/06/18)

Sigurður Örn Stefánsson, júní 11, 2018

Málstofa í stærðfræði

### Titill: Random metric spaces constructed using a gluing procedure

Staðsetning: VRII-V147
Tími: Mánudagur 18. júní kl. 10:50

### Ágrip:

I will introduce a model of random trees which are constructed by iteratively gluing an infinite number of segments of given length onto each other. This model can be generalized to a gluing of „blocks“ that are more complex than segments. We are interested in the metric properties of the limiting metric space, mainly its Hausdorff dimension. We will show that its Hausdorff dimension depends in a non-trivial (and surprising !) manner on the different scaling parameters of the model and the dimension of the blocks.

## Tony Guttmann (28/05/18)

Anders Claesson, maí 25, 2018

Málstofa í stærðfræði

### Titill: On the number of Av(1324) permutations

Staðsetning: V-147 (VR-II)
Tími: Mánudagur 28. maí kl. 10:50

### Ágrip:

We give an improved algorithm for counting the number of 1324-avoiding permutations, resulting in 14 further terms of the generating function, which is now known to length 50.
We re-analyse the generating function and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of 1324-avoiding permutations of length n behaves as $$B\cdot \mu^n \cdot \mu_1^{\sqrt{n}} \cdot n^g$$.
We estimate $$\mu = 11.600 \pm 0.003.$$ The presence of the stretched exponential term $$\mu_1^{\sqrt{n}}$$ is an unexpected feature of the conjectured solution, but we show that such a term is present in a number of other combinatorial problems.
(A.J. Guttmann with A.R. Conway and P. Zinn-Justin)