## Sérfræðingur í hagnýttri stærðfræði við Raunvísindastofnun Háskóla Íslands

Laust er til umsóknar fullt starf sérfræðings í hagnýttri stærðfræði við stærðfræðistofu Raunvísindastofnunar Háskólans.

## Sérfræðingur í stærðfræði við Raunvísindastofnun Háskóla Íslands

Laust er til umsóknar starf sérfræðings í stærðfræði við stærðfræðistofu Raunvísindastofnunar Háskólans.

## Tom Steentjes (17/03/17)

Anders Claesson, mars 13, 2017

Málstofa í stærðfræði

### Titill: Feedback stabilization of nonlinear systems: „universal“ constructions towards real-life applications

Staðsetning: Tg-227 (Tæknigarður, 2. hæð)
Tími: Föstudagur 17. mars kl. 13:20

### Ágrip:

Various feedback stabilizers based on Sontag’s „universal“ formula for stabilizing control laws are presented, incorporating restrictions inspired by real-life applications. The first main contribution is an extension of Sontag’s „universal“ formula for positive nonlinear control systems. More specifically, an auxiliary function is introduced in the feedback interconnection, such that invariance of the positive orthant is retained for the system in closed loop with the „universal“ stabilizer. We further state a „universal“ event-based stabilizer for bounded controls and develop an extension of the controller for positive systems. In a motivating case study from systems biology, the methodology is shown to provide clinically realistic control inputs, which can be used for treatment in real life. The second main contribution is the construction of continuous and piecewise affine (CPA) feedback stabilizers for nonlinear control systems affine in the input, motivated by the ease of implementation of the resulting control law. A verification procedure for „universal“ CPA stabilizers is provided, together with an alternative computational method for CPA stabilizers via linear programming. Two numerical examples are presented for illustration of the CPA method.

Short bio:
Tom Steentjes was born in Tilburg, the Netherlands, in 1993. He received his BSc degree in Electrical Engineering (Automotive) in 2014, from the Department of Electrical Engineering at Eindhoven University of Technology (TU/e). In 2016, he completed the Systems and Control master’s program at the TU/e. The MSc thesis, entitled „Feedback stabilization of nonlinear systems: ‘universal’ constructions towards real-life applications“, was supervised by dr. Alina Doban and dr. Mircea Lazar. The MSc degree was granted with distinction Cum Laude.

## Paolo Zanardi (20/02/17)

Anders Claesson, febrúar 9, 2017

Málstofa í stærðfræði

### 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

## Thomas Vallier (03/02/17)

Anders Claesson, janúar 31, 2017

Málstofa í stærðfræði

### 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.

## Benedikt Stufler (27/01/17)

Anders Claesson, janúar 24, 2017

Málstofa í stærðfræði

### 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.

## Daniel Fernandez (05/12/16)

Anders Claesson, desember 1, 2016

Málstofa í stærðfræði

### 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.

## Sara Zemljic (28/11/16)

Anders Claesson, nóvember 23, 2016

Málstofa í stærðfræði

### 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.

## Hermann Þórisson (11/10/16)

Sigurður Örn Stefánsson, október 8, 2016

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.

Benedikt Magnússon, október 6, 2016

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.