# Welcome!

## Introduction

I'm a PhD student in mathematics at the Department of Mathematics of Uppsala University, where I'm also a member of the CAPA group. In addition to my PhD studies I was previously a board member and secretary of the Mathematical Society.

In addition to my academical engagements I am also a co-organizer of the Uppsala Big Data Meetup and a member of the Uppsala Linux User Group.

My area of research is called *dynamical systems*. In particular I study
renormalization of exact symplectic twist maps of the plane and their invariant
Cantor sets. Outside of dynamical systems I also have an interest in geometry,
topology, mathematical physics and how computers can be used to make
mathematically rigorous computations, something called *rigorous numerics*.

Outside of mathematics I also have an interest in various topics of computer science. Aside from the previously mentioned interest in rigorous numerics my current interests are mainly focused on data science, machine learning and cryptography. When it comes to programming languages I have a particular fondness for Haskell and Python though I also program in C++. I also hope to learn Rust in the near future. As part of my excursions in data science I have also learned to use Apache Hadoop and Apache Spark as well as program in Scala.

## Dynamical Systems

Dynamical systems describe how objects and their properties change with respect to time. For example it can describe the motion of a particle under the influence of a force such as gravity, how the concentrations of various substances change during a chemical reaction or how the number of fish in a lake changes from year to year.

Dynamical systems are usually divided into two categories depending on how time
is modeled in the system: *continuous* and *discrete*. The first two examples
above are examples of continuous dynamical systems since time is considered as a
continuum. There are no holes. The last example is a discrete dynamical system
since the number of fish is only given at discrete times, namely once a year.
Mathematically a dynamical system consists of a space \(M\) called the
*phase space* and a function \(F\) taking as its arguments a time \(t\) and a
point \(x\) of phase space and outputs another point \(y=F(t,x)\) of phase
space, the point to which \(x\) has moved after time \(t\). We therefore also
require that \(F(0,x)=x\) for every point \(x\), i.e. if no time passes then the
points do not move. Discrete dynamical systems are usually just considered to be
a function \(f\) taking points of phase space to other points of phase space and
time is considered as integers giving the number of iterations of this function.
If the point \(x\) is mapped to the point \(y\) after time \(t=n\) we write
\(y=f^{n}(x)\) where \(f^{n}\) denotes \(n\) iterations of \(f\).

## Chaos

Many nonlinear dynamical systems, for example most dynamical systems coming from
applications, display a phenomenon called *chaos*. Examples of dynamical systems
exhibiting chaotic behaviour include the Lorenz system describing atmospheric
convection, the Belousov-Zhabotinsky chemical reaction, Chua's circuit, the
kicked rotator and the double pendulum.

One common mathematical definition of a chaotic dynamical system states that a dynamical system is chaotic if it has three properties called sensitive dependence on initial conditions, topological mixing and if periodic points are dense.

*Sensitive dependence on initial conditions* means that no matter how close you
look at any particular point you will always find other points that eventually
move a fixed distance away from the point you're looking at. In this way it is a
mathematically precise definition of the famous butterfly effect. As an
illustration consider the circle in the plane. We can describe any point on this
circle by giving the angle between the point and the \(x\)-axis counted
counterclockwise. We can then consider the discrete dynamical system given by
doubling the angle, called the *doubling map*. Then no matter how close two
points are to each other the angle between them will double with each iteration.
So for any fixed angle, say 90 degrees, the angle between the two points will be
larger than 90 degrees.

*Topological mixing*, also called *topological transitivity*, means that if we
pick two nonempty, open sets and apply the dynamical system to one of them then
it will eventually intersect the other set. This condition is a formal way of
saying that any collection of points will eventually be spread throughout phase
space. Our example of the doubling map of the circle is topologically mixing. If
we start with any arc along the circle it will double in size with each
iteration of the doubling map. No matter how small the arc we choose is it will
eventually cover the entire circle and will therefore intersect any other arc as
well.

Lastly, *periodic points being dense* requires some explanation. First, a point
\(x\) of phase space is called *periodic* if there is some time \(T\)
such that \(F(T,x)=x\). This means that the point will repeatedly move
around for a while in a specific orbit and then come back to its starting point.
That such points are *dense* means that whichever point of phase space we pick
we can always find a periodic point arbitrarily close to it. No matter how much
we zoom in on our chosen point we will always see periodic points as well.
Compare this with how the rational numbers are distributed within the real
numbers. The periodic points of the doubling map are dense in the circle. To see
this requires first that rational numbers are dense in the real numbers and that
a rational number has an eventually periodic binary expansion in the same way as
their decimal expansion is always eventually periodic. The doubling map
corresponds to shifting the binary expansion by one place and so that every
point on the circle with a rational angle is eventually periodic. In conclusion
we have informally proven that the doubling map is a chaotic dynamical system,
giving us yet another example.

## Further Reading

Here are some links for those that are interested in reading more about dynamical systems, chaos, renormalization, or my field of research.