## Sangmin Lee, Seoul National University

Carlos Argaez Garcia, February 19, 2020

Math colloquium

#### Title: Complete 1st post-Minkowskian potential from scattering amplitudes.

Room: VHV-007 (Veröld)
Time: Monday 24th January  10:00hrs

#### Abstract:

Building upon recent progress in applying amplitude techniques to perturbative general relativity, we propose a closed-form formula for the conservative Hamiltonian of a spinning binary system at the 1st post-Minkowskian order. It is applicable for general spinning bodies with arbitrary spin multipole moments. It is linear in gravitational constant by definition, but exact to all orders in momentum and spin expansions. At each spin order, our formula implies that the spin-dependence and momentum dependence factorize completely. We compare our formula to a similar one derived in 2017 from a spinning test-body near a Kerr black hole and find perfect agreement.

## Ragnar Sigurðsson, University of Iceland

Carlos Argaez Garcia, February 1, 2020

Math colloquium

#### Title: Norms on complexifications of real vector spaces.

Room: VRII-258
Time: Thursday February 6th, 10:50 hrs.

#### Abstract:

The subject of this lecture is of general interest and it only requires knowledge of elementary linear algebra.

The complexification V_C of a real vector space
V is the smallest complex vector space which contains V
as a real subspace. If V is a normed space, then it is
of interest to know how norms may extend from V to V_C.

I will look at a real normed space V and give formulas
for the smallest and largest extension of a general norm
on V to a norm on V_C. These formulas are not explicit
so it is of interest to find explicit formulas in particular
examples. This is possible for extentions of norms induced
by inner products. The Lie norm is the largest
extension of the Euclidean norm on R^n to a complex norm
on C^n.

In complex analysis we deal a lot with plurisubharmonic
functions and an important source for examples are
functions of the form log||f||, where f is a holomorphic
map from a complex manifold into C^n and ||.|| is a norm
on C^n. In his thesis, Auðunn Skúta Snæbjarnarson, studied
the Lie norm on C^n and calculated interesting formulas for
the so called Monge-Ampere measure of log||f||, which is
indeed not an easy task.

## Lukas Schneiderbauer

Carlos Argaez Garcia, November 22, 2019

Staðsetning: HB5 (Háskólabíó)
Tími: Föstudag 29.Nóvember kl.11:40

#### Title: Non-Commutative Geometry: An introduction.

Room: HB5 (Háskólabíó)
Time: Friday 29th November, 11:40hrs

#### Abstract:

This is my attempt to introduce non-commutative geometry to mathematicians. After putting forward the main ideas and main theorem(s), I will concentrate on the construction of simple examples in the context of fuzzy spaces (special cases of non-commutative geometries). In case time still allows it, I shall tell you about my past research in this area.

## Anna Helga Jónsdóttir och Benedikt Steinar Magnússon

Valentina Giangreco, November 19, 2019

Math colloquium

#### Abstract 1:

Student evaluations of teaching (ísl. kennslukönnun) is administrated at the end of each and every course at the University of Iceland with the purpose of improving teaching and learning. In the talk, analysis of data from student evaluations from 2013 to 2017 at the UI will be presented. Mixed effect models were used to investigate possible relationships between the grades students give courses and several variables, such as the age and gender of the student and the teacher, number of students taking the course and the average final grade in the course.

#### Abstract 2:

In the last years I, with the help of many good people, have been developing a platform for online course notes called Edbook (http://edbook.hi.is). It consists of using Sphinx, which was developed for and in Python, along with specialized extension suited for teaching material in Mathematics. We have been using these notes in a few courses, mostly big calculus courses. The students have overall been very happy with them but in the spring of 2019 I had students in Mathematical Analysis II (STÆ205G) answer a more detailed survey about their usage of the teaching material they used. I will introduce the Edbook platform and the results of the survey which raises some questions about the role of the textbook today.

Room: HB5 (Háskólabíó)
Time: Friday 22th November, 11:40hrs

## 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).

## Henning Arnór Úlfarsson

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.

## Arnbjorg Soffia Arnadottir

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.