## Sigurður Freyr Hafstein

Carlos Argaez Garcia, November 11, 2019

Math colloquium

#### Title: Lyapunov functions for stochastic differential equations and their computation

Room: HB5 (Háskólabíó)
Time: Friday 15th November, 11:40hrs

#### Abstract:

Attractors and their basins of attraction in deterministic dynamical systems are most commonly studied using the Lyapunov stability theory.  Its centerpiece is the Lyapunov function, which is an energy-like function from the state-space that is decreasing along all solution trajectories.
The Lyapunov stability theory for stochastic differential equations is much less developed and, in particular, numerical methods for the construction of Lyapunov functions for such systems are few and far between.  We discuss the general problem and present some novel numerical methods.

Carlos Argaez Garcia, October 30, 2019

Math colloquium

#### Title: A new method for computation and verification of contraction metrics

Room: HB5 (Háskólabíó)
Time: Friday 8th November, 11:40hrs

#### Abstract:

The determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases.
The Riemannian metric can be expressed by a matrix-valued function on the phase space.
The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF).
Then, we combine the RBF method (to compute a contraction metric) with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification.
This presentation is based on a joint work with Prof. Sigurdur Hafstein (University of Iceland), and Prof. Peter Giesl (University of Sussex, UK).

Carlos Argaez Garcia, September 24, 2019

Math colloquium

#### Title: Pattern avoidance in various domains

Room: HB5 (Háskólabíó)
Time: Friday 25th October, 11:40hrs

#### Abstract:

When one searches the web for “pattern avoidance” most of the results are about pattern avoiding permutations and their variants, such as colored, partial, multi-, affine, signed, and poset permutations. However there are definitions and results about similar concepts in other objects, such as graphs and topological spaces. We will survey these examples of pattern avoidance as well as highlighting more recent variants, such as polyominoes, integer partitions and alternating sign matrices.

This talk will be accessible to any student who has walked past a room where discrete mathematics was being taught.

## Björn Birnir

Carlos Argaez Garcia, August 23, 2019

#### Title: When can we expect the Greenland glacier to melt?

Room: VR-II,V-258
Time: Tuesday 27nd August, 11:00hrs

#### Abstract:

It was suggested by Rose (2005) that because of the migratory and responsive nature of the capelin, a small pelagic fish that is key to the ecology and fisheries of the North Atlantic, it can be viewed as the “canary in the coalmine” to detect signals of environmental changes in the Arctic Ocean. In this talk we will combine analysis of data and extensive simulations of the migrations of the capelin and its physiology to analyze the changes in the ocean environment taking place over the last half-century. The environmental data for the last thirty year is obtained from a database called Copernicus, constructed by the European Union. Our goals will be to understand and predict the migrations of the capelin and its interactions with the ocean environment. We will explain how these have changed over time and how they are likely to change in the future. Then we will explain how our simulations can be compared with data, with the aim of finding out the rate of the temperature changes in the Arctic Ocean and when thresholds for major disruptions in Arctic environments are likely to be reached. The recent changes in the spawning routes of the capelin lead to a startling prediction.

Carlos Argaez Garcia, August 20, 2019

#### Title: Continuous Quantum Walks

Room: VR-II, V-158
Time: Thursday 22nd August, 11:00hrs

#### Abstract:

Continuous quantum walks arise naturally as quantum analogues of continuous random walks, but in contrast to their classical counterparts, they exhibit some curious and counter-intuitive properties. I will give an introductory talk on continuous quantum walks and present some of these exciting properties.
The motivation for studying quantum walks largely comes from quantum physics and quantum computing, however, the emphasis of this talk will be on the mathematics. In particular, no prior knowledge of anything quantum is assumed.

## Finnur Lárusson

Valentina Giangreco, June 7, 2019

Math Colloquium

### Title: Chaotic holomorphic automorphisms of Stein manifolds with the volume density property

Location: VR-II, V-158
Time: Tuesday July 9 at 11.00 am

### Abstract:

I will report on joint work with Leandro Arosio. Let $X$ be
a Stein manifold of dimension $n\geq 2$ satisfying the volume density
property with respect to an exact holomorphic volume form. For example,
$X$ could be $\C^n$, any connected linear algebraic group that is not
reductive, the Koras-Russell cubic, or a product $Y\times\C$, where $Y$
is any Stein manifold with the volume density property. We prove that
chaotic automorphisms are generic among volume-preserving holomorphic
automorphisms of $X$. In particular, $X$ has a chaotic holomorphic
automorphism. Forn\ae ss and Sibony proved (but did not explicitly
state) this for $X=\C^n$ in 1997. We follow their approach closely.
Peters, Vivas, and Wold showed that a generic volume-preserving
automorphism of $\C^n$, $n\geq 2$, has a hyperbolic fixed point whose
stable manifold is dense in $\C^n$. This property can be interpreted as
a kind of chaos. We generalise their theorem to a Stein manifold as above.

## Guðmundur Magnússon

Valentina Giangreco, June 7, 2019

Math Colloquium

### Title: Multi Dispatcher Systems and interacting policies

Location: VR-II, V-258
Time: Tuesday June 18 at 11.00 am

### Abstract:

This project explores the performance of parallel server system. The focus of the project is a system using multiple heuristic policies to route jobs to a server. To achieve this there is a simulator created that uses the Monte Carlo method to numerically simulate the performance of a system. It displays with figures the result of some simulations under different conditions and answers how different policies interact in this multi dispatcher system.

BSc thesis under the supervision of  Esa Olavi Hyytiä.

## Sylvain Arguillère

Valentina Giangreco, June 7, 2019

Math Colloquium

### Title: Shape analysis through flows of diffeomorphisms

Location: VR-II, V-258
Time: Thursday June 13 at 11.00 am

### Abstract:

The goal of shape analysis is to compare shapes in a way that takes into account their geometric properties. The end goal is to give an adapted framework for the statistical analysis of medical data, in order to identify sick patients automatically for example. In this talk, I will describe a method introduced by Alain Trouvé, which allows to compare shapes through flows of diffeomorphisms with minimal energy, using tools from differential geometry and optimal control.

## Wolfgang Woess

Valentina Giangreco, May 31, 2019

Math Colloquium

### Title: THE LANGUAGE OF SELF-AVOIDING WALKS

Location: VR-II, V-155
Time: Tuesday June 4 at 11.00 am

### Abstract:

Let X = (VX, EX) be an infinite, locally finite, connected graph without
loops or multiple edges. We consider the edges to be oriented, and EX is equipped with
an involution which inverts the orientation. Each oriented edge is labelled by an element
of a finite alphabet Σ. The labelling is assumed to be deterministic: edges with the same
initial (resp. terminal) vertex have distinct labels. Furthermore it is assumed that the
group of label-preserving automorphisms of X acts quasi-transitively. For any vertex o
of X, consider the language of all words over Σ which can be read along self-avoiding
walks starting at o. We characterize under which conditions on the graph structure this
language is regular or context-free. This is the case if and only if the graph has more
than one end, and the size of all ends is 1, or at most 2, respectively. (joint work with Christian Lindorfer).

## Primoz Potocnik

Valentina Giangreco, May 20, 2019

Math Colloquium

### Title: Symmetries of finite graphs – a personal overview

Location: VR-II, V-158
Time: Tuesday May 28 at 11.00 am

### Abstract:

Whether a graph is more or less symmetric is typically measured in terms of its automorphism group consisting of all permutation of its vertices which preserves the adjacency relation. The highest level of symmetry is achieved when the automorphism group has only one orbit on the vertices and/or edges of the graph. I will give a personal and gentle overview of the problems and the results about this class of graphs.