Málstofa í stærðfræði

### Fyrirlesari: Paolo Zanardi, University of Southern California

### Titill: Quantum algorithms for topological and geometric analysis of Big Data

Staðsetning: V-138 (VR-II)

Tími: Mánudagur 20. febrúar kl. 10:00

### Ágrip:

Extracting useful information from large data sets can be a daunting task. Topological methods for analysing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying topological features and for determining how such features persist as the data is viewed at different scales. I will discuss quantum machine learning algorithms for calculating Betti numbers – the numbers of connected components, holes and voids – in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speed-up over the best currently known classical algorithms for topological data analysis.

Reference: Seth Lloyd, Silvano Garnerone e Paolo Zanardi, Quantum algorithms for topological and geometric analysis of data, Nature Communications 7, 10138 (2016). See also: http://news.mit.edu/2016/quantum-approach-big-data-0125

Málstofa í stærðfræði

### Fyrirlesari: Thomas Vallier, University of Iceland

### Titill: Discussion on Bootstrap percolation on a random graph coupled with a lattice

Staðsetning: Tg-227 (Tæknigarður, 2. hæð)

Tími: Föstudagur 3. febrúar kl. 13:20

### Ágrip:

will give an informal seminar on the paper by Janson, Kozma, Ruszinko and Sokolov. In the paper „Bootstrap percolation on a random graph coupled with a lattice“, the authors consider a set of vertices \(N\)x\(N\) on a lattice. On top of that, every vertex shares a link with any other vertex with a probability inversely proportional to there block distance and independently of any other link. That means for two vertices \(u\) and \(v\) at distance \(d P(u,v) = c/(Nd)\) where \(c\) is a constant.

The authors derive the diameter of the of the graph in a very elegant way which we will unfortunately not focus on. Instead, we will discuss the other part which deals with the following cellular automaton with threshold \(k\):

1. Start with a set of active (or infected depending on the terminology you want to use) vertices at time \(0\), \(A(0)\).

2. Any vertex which has at least k active vertices in its closed neighbourhood (including itself) at time \(t\) is set as active at time \(t+1\). Notice that two vertices are neighbours if they share an edge in common either from the lattice or from the random connections.

Otherwise, if it has less than \(k\) active vertices in the closed neighbourhood then the vertex is set as inactive at time \(t+1\).

3. We repeat the process over time.

The authors study the „mean field approximation“ which simplifies the problem by averaging the links over space. Under this assumption, the problem simplifies into a 1-dimensional dynamical system. The authors derive the fixed points of the dynamical system for

\(k \leq 3\).

There are good indications that the process becomes more interesting (that means more complicated) for larger \(k\). I would like to discuss that subject with you all if you’re curious about it. Anyone who’s interested in talking and sharing ideas is very welcome.

Málstofa í stærðfræði

### Fyrirlesari: Benedikt Stufler, École normale supérieure de Lyon

### Titill: The asymptotic geometric shape of random combinatorial trees

Staðsetning: TG-227 (Tæknigarður, 2. hæð)

Tími: Föstudagur 27. janúar kl. 13:20

### Ágrip:

In his pioneering papers in the early 90s, Aldous established the continuum random tree (CRT) as the scaling limit of random labelled trees. He conjectured that the CRT also arises as scaling limit of trees considered up to symmetry. The convergence of random Pólya trees, that is, unlabelled rooted trees, was confirmed around 20 years later by Haas and Miermont, and we discuss an alternative proof by Panagiotou and the speaker. The second part of the talk treats random unlabelled unrooted trees and discusses a very general result that allows for a transfer of asymptotic properties of rooted trees to unrooted trees, in particular the convergence toward the CRT.

Málstofa í stærðfræði

### Fyrirlesari: Daniel Fernandez, University of Iceland

### Titill: Information geometry and Holography

Staðsetning: V-158, VRII

Tími: Mánudaginn 5. desember kl. 15:00

### Ágrip:

Information Geometry allows for families of probability distributions to be imbued with a natural measure: the Fisher metric. In particular, it is possible to define a Fisher metric on the instanton moduli space of field theories, which in many cases leads to hyperbolic geometries. Furthermore, Euclidean AdS emerges naturally as the metric of the SU(2) Yang Mills instanton moduli space. These are the first steps towards understanding the emergence of holographic dualities.

Málstofa í stærðfræði

### Fyrirlesari: Sara Zemljic, University of Iceland

### Titill: Generalized Sierpiński graphs

Staðsetning: V-158, VRII

Tími: Mánudaginn 28. nóvember kl. 15:00

### Ágrip:

(Generic) Sierpiński graphs are two parametric family of graphs, one parameter tells us what complete graph is the main building block for the graph, and the other parameter tells us in which iteration we are. A generalization of these graphs was proposed such that instead of taking a complete graph we start with an arbitrary graph G as our base graph and build the graph \(S_G^n\) in the same iterative manner as generic Sierpiński graphs. We will examine many basic graph properties and standard graph invariants for generalized Sierpiński graphs.

Málstofa í stærðfræði

### Fyrirlesari: Hermann Þórisson

Titill: On the Skorohod Representation

Staðsetning: Naustið, Tæknigarði

Tími: Þriðjudaginn 11. október kl. 12:30-14:00

### Ágrip:

According to the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s. in the metric. A density analogue of this theorem says that a sequence of probability densities on a general measurable space has a probability density as a pointwise lower limit if and only if there exists a coupling with elements converging a.s. in the discrete metric. In this talk the discrete-metric theorem is extended to stochastic processes considered in a widening time window. The extension is then used to prove the Skorohod representation theorem.

Meistaraprófsfyrirlestur

### Arnbjörg Soffía Árnadóttir

Titill: Grúpuverkanir á óendanleg stefnd net og hlutbrautafallið

Staðsetning: Naustið, Endurmenntun.

Tímasetning: Fimmtudagur 6. október 2016, klukkan 16:00.

### Ágrip:

Við notum grúpuverkanir til þess að skoða óendanleg stefnd net. Við byrjum á því að skilgreina grúpumótun sem við köllum hlutbrautafallið. Við notum svo þessa mótun til þess að skoða ýmsa eiginleika óendanlegra stefndra neta, þ.á.m. myndir netamótana, háörvavegagegnvirkni, Cayley-Abels net og vöxt neta.

Leiðbeinendur: Rögnvaldur G. Möller og Jón Ingólfur Magnússon, báðir prófessorar við Raunvísindadeild Háskóla Íslands.

Prófdómari: Peter M. Neumann, emerítus við The Queen’s College, Oxford University.

Málstofa í stærðfræði

### Fyrirlesari: Séverine Biard, University of Iceland

### Titill: On pseudoconvex domains and some of their applications

Staðsetning: V-158, VRII

Tími: Mánudaginn 10. október kl. 15:00

### Ágrip:

One of the most commonly studied object in several complex variables is pseudoconvex domains, which I will introduce. Domains of holomorphy in \(\mathbb{C}^n\), they are studied in more general complex manifolds thanks to a powerful notion: plurisubharmonicity. Those domains are the support of the famous inhomogeneous Cauchy-Riemann equation or also called \(\bar\partial\)-equation. Combined with the existence of a bounded plurisubharmonic function, I will also talk about some applications to complex geometry and complex dynamics.

Málstofa í stærðfræði

### Fyrirlesari: Erik Broman, Chalmers University of Technology and Gothenburg University

### Titill: Infinite range continuum percolation models

Staðsetning: V-158, VRII

Tími: Mánudaginn 3. október kl. 15:00

### Ágrip:

In the classical Boolean percolation model, one distributes balls in \(R^d\) in a random, homogeneous way. The density of these balls is controlled by a parameter \(\lambda.\) Depending on this density, the collection of balls then either form an infinite cluster (\(\lambda\) large) or consists of only small components (\(\lambda\) small).

In the talk I will discuss two variants of this model, both which are infinite range. In the first case, the balls are replaced by bi-infinite cylinders with radius 1. We then investigate what the connectivity structure of the resulting set is, and how this depends on \(\lambda\) as well as on the underlying geometry (Euclidean vs hyperbolic).

In the second case, we replace the balls with attenuation functions. That is, we let \(l:(0,\infty) \to (0,\infty)\) be some non-increasing function and for every \(y\in R^d\) we define \(\Psi(y):=\sum_{x\in \eta}l(|x-y|)\). We study the level sets \(\Psi_{\geq h}\), which is simply the set of points where the random field \(\Psi\) is larger than or equal to \(h.\) We determine for which functions \(l\) this model has a non-trivial phase transition in \(h.\) In addition, we will discuss some classical results and whether these can be transferred to this setting.

The aim is that the talk should be accessible to anyone with a mathematical, but not necessarily probabilistic, background.

Málstofa í stærðfræði

### Fyrirlesari: Rögnvaldur Möller, University of Iceland

### Titill: Highly-arc-transitie digraphs of prime out-valency

Staðsetning: V-158, VRII

Tími: Mánudaginn 19. september kl. 15:00

### Ágrip:

Joint work with Primoz Potocnik, Ljubljana, and Norbert Seifter, Leoben.

The concept of a highly-arc-transitive digraph was defined by Cameron, Praeger and Wormald in a paper that appeared in 1993. Examples constructed by various people have shown that suggestions put forward in that paper are wrong. But if it assumed that the highly-arc-transitive digraph has prime out-valency then some of the suggestions of Cameron, Praeger and Wormald are correct. The second part of the talk is about a general method to construct k-arc-transitive digraphs that are not (k+1)-arc-transitive. This construction gives examples that limit the possibilities of extending the results in the first part and also give examples of digraphs with polynomial growth that are k-arc-transitive but not (k+1)-arc-transitive