As the name suggests, in this module there will be some mathematics. No prior knowledge of mathematical modelling, dynamical systems theory, probability, and statistics is assumed. However, if you haven’t done any mathematics in a while and feel like you may find it useful to review some of the basics, this is the right place for that!

Additionally, if you feel quite comfortable with the maths content included in this webpage, we have also included more advanced resources that you may find helpful throughout the course and some (very nice) examples of mathematical modelling and complex systems. Although many of these won’t be covered in depth, at the end of the course you will have all the necessary mathematical and computational tools to model and explore these systems and appreciate some cutting edge research questions in complexity science.

Mathematical Foundations

Some Algebra

Of course, Khan Academy offers some great algebra introductory material.

This is a long playlist (92 videos), you are not expected to be completely familiar with all the topics included. However, you may find quite useful reviewing the following topics:

• Introduction to Algebra;
• Equation of a Line;
• Solving Linear Equations;
• Complex Numbers (basics only!);
• Quadratic Equations;
• Logarithms.

In the playlist you will also find topics such as:

• Definition of a Function;
• Systems of Equations;
• Proof by Induction,

a more in depth analysis of these topics will be included in the lectures and the seminars!

Some Statistics and Probability

This part is a bit more advanced. All the basics of statistics and probability theory will be covered in the module.

Again, Khan Academy offers some great extra introductory material. You may find useful the following topics from this playlist:

• The average, mean, median, and mode;
• Population vs Sample mean

To conclude, here is a nice visualization of the most important formula in probability theory: Bayes Theorem, by Grant Sanderson, in art 3Blue1Brown.

All of these topics will be included in the module, but in a more formal way!

Course Tasters

Some Linear Algebra

During the seminars/tutorials we will spend quite some time familiarizing with linear algebra as this is one of the best (and arguably most fun) ways to learn how to code. In particular, week 3 will be focussing on linear algebra and you will work through some of the topics included in the (now a classic) 3B1B series on “The Essence of Linear Algebra”.

Some Mathematical Modelling & Dynamical Systems

Here I include some mathematical modelling introductory examples.

• Once again Khan Academy is your best friend here:

  • Example with DE

• check out Steven Strogatz’s famous articles (Dynamical Systems):

  • “The Two Lovers” New York Times article;
  • “Love Affairs” short article).

• a more fun example: How to model a population of rabbits and foxes: by Numberphile with Tom Crawford.

Finally, some Complex Systems!

“I think the next [21st] century will be the century of complexity” – Stephen Hawking

Attribution: photo by TDUB303, https://tdubphoto.com/photography/

So what are complex systems?

Quoting Simon (1962) “The Architecture of Complexity”:

I shall not undertake a formal definition of ‘complex systems'

I certainly won’t attempt to define what complex systems are in this short page! However, if fireflies drew your attention and you are curious to explore this a little bit more prior to the lectures, here are some of my favourite examples that you can explore:

• Complexity Explorables, curated by Dirk Brockmann, includes dozens of interactive examples of complex systems. I selected below a couple of my favourites, but I highly reccommend exploring more at Complexity-Explorables.org:
  • how do thousands of fireflies synchronize along the river banks? Check this video, and then have a look at The Kuramoto Model Complexity Explorables
  • How would you model an epidemic? Check the Susceptible, Infected, Recovered, Susceptible (SIRS) model by Janina Schöneberger;
• If you’re unsure whether or not to go out tonight, check the famous “El Farol Bar Problem”, Medium article written by Jørgen Veisdal.

Note: all these examples inlcude what you will learn in this module: Mathematical Modelling, Dynamical Systems, Statistics, and Probability!

Here, I selected a few nice introductions to complexity science you may want to read and other resources that may be of interest:

• What is Complexity Science?, curated by Manlio De Domenico and Hiroki Sayama;
• Introduction to Complexity Science, from the recent book “Complexity science: the study of emergence” by Henrik Jeldtoft Jensen;
• a bit of history:
  • 50 years of More is Different;
  • The Architecture of Complexity (1962) Herbert A. Simon;
  • What is Complexity by Murray Gell-Mann;

If you are particularly passionate about music, I highly suggest checking out the work of Max Cooper: Emergence. Max Cooper (audio-visual artist with a PhD in computational biology) was also a guest at the workshop on emergence organized by the University of Sussex.

Extra work

So, you like the course and want to go further. How? This depends a lot on your specific interests. One nice course that covers a lot of bases is MIT’s Intro to computational thinking course. The pluto notebook format is the same, so nothing is stopping you!