Math colloquium

**Speakers:
**Sangmin Lee, Seoul
National University

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

Room: VHV-007 (Veröld)

Time: Monday 24^{th} 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.

Math
colloquium

**Speaker**: Ragnar
Sigurðsson, University of Iceland

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

Room: VRII-258

Time: Thursday February 6^{th}, 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.

Staðsetning: HB5 (Háskólabíó)

Tími: Föstudag 29.Nóvember kl.11:40

#### Math
colloquium

**Speakers: Lukas
Schneiderbauer, University of Iceland**

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

Room:
HB5 (Háskólabíó)

Time: Friday 29^{th} 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.

Math colloquium

**Speakers: Anna
Helga Jónsdóttir and Benedikt Steinar Magnússon, University of Iceland**

**Title 1: **Student evaluations of teaching at the University of Iceland –
analysis of data from 2013 – 2017.

#### 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.

**Title 2: **Online course notes in Edbook and the role of the textbook

#### 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 22^{th} November, 11:40hrs

Math colloquium

**Speaker**: **Sigurður
Freyr Hafstein****, University of Iceland**

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

Room:
HB5 (Háskólabíó)

Time: Friday 15^{th} 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.

Math colloquium

**Speaker**: **Iman
Mehrabinezhad, University of Iceland**

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

Room: HB5 (Háskólabíó)

Time: Friday 8^{th} 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).

Math colloquium

**Speaker**: **Henning Arnór
Úlfarsson, University of Reykjavík**

**Title: **Pattern avoidance in various domains

Room: HB5 (Háskólabíó)

Time: Friday 25^{th} 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.

**Speaker:
Björn Birnir, Center for Complex and Nonlinear Science at the
University of California at Santa Barbara (UCSB)**

**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.

**Speaker:** Arnbjorg Soffia Arnadottir**,University
of Waterloo**

**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.

Math Colloquium

### Speaker: Finnur Lárusson, Adelaide University

### 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.