Mathematical Models in the Physical, Natural and Social Sciences

I am teaching this course for the first time in academic year 2020/21. It is a new course in a strange setting: a challenge, but also an opportunity. Given that this course is compulsory in the Teaching and Scientific Communication curriculum, I have tried to weave this perspective in the course’s plan.

The course’s material can be found on its Moodle page. Here I will just briefly discuss the Syllabus.


Knowledge and understanding

Students of this course should show good understanding of some mathematical theories and their applications in the real world. They must be able to navigate the relevant mathematical literature and related popular science websites, as should manage to connect theoretical materials and practical applications.

In other words, by the end of the course, students should know the contents of the course (see below, for more details about them), but there are also meta-skills that should be nurtured: students should be able to find additional topics in the literature or through more informal channels, expand on them and adapt them to the target audience. This should happen starting from a theoretical topic, for which to find examples in the real world applications, as well as the other way around, starting with a non trivial real world problem to solve, abstracting it, and designing a (simple) mathematical model or framework to study it. This leads us to the next point.

Applying knowledge and understanding

Students of this course should be able to formalize some of prototypical real world problems, and to recall the historical and theoretical development that led to their solution. They must be able to provide examples of application of mathematics in some of these cases.

So, for the topics touched in class, students should be able to retrace the steps made, both from a mathematical and a historical perspective. At the same time, they should be able to move beyond the explicit examples give in class, seeing similarities with known models as a starting point for developing new ones in new contexts.

Making judgments

Students of this course must be able to search the literature for examples and results to be used when discussing the proposed themes. They must be able to recognize the role of mathematics in the real world problems.

When searching the literature (and other resources), students should be able to assess whether the presentation is mathematically sound, relevant to the problem, at the right level for the intended audience. When they see a new model, they should be able to understand the mathematics behind each part of the model, and discuss critically the modelling choices made.

Communication skills

Students of this course must be able to present, in a complete and appealing, as well as suitable for the target audience, some typical real-world problems that are solved with the help of mathematics, providing their own examples of application of the related mathematical techniques.

This point has already been touched: students should not just passively absorb content, but given in particular the flavour of the chosen curriculum, should be able to elaborate and the adapt the examples introduced to their target audience. They should also learn meta-skills, in order to be able to recover information and material from sources, and cook it into their own presentations, be they class lectures, popularisation talks, web pages, books.

Learning ability

Students must be able to follow the teacher’s suggestions, and to integrate them with examples obtained through their own study of mathematical literature and websites devoted to the popularization of mathematics.

This is a very important point: for those students attending the course, an active participation in very important. The lectures won’t be one directional, but want to be a discussion with contributions from everyone. There will be reading and writing assignments to complete from one week to the next one, which will play an important role in the course.


We will choose, among many possible applications, some examples of mathematics in everyday life: from CD players to the stock exchange, from computer tomography to transport planning. We will discuss the role of probability in forensic science and, through captivating paradoxes, in gambling, finance and day-to-day life.

As the previous paragraph from the Syllabus suggests, there are several possible topics we might want to touch, and a choice has to be made. I would like the input of all participants on this matter. I will list here some models and topics I find interesting (and suitable) for the course.

Actually, let me start with two meta-topics. March 14 ( day) has been named by UNESCO the International Day of Mathematics. In 2020 the theme chosen has been Mathematics is Everywhere, which fits quite well within this course. The theme for 2021 has already been announced, Mathematics for a Better World, so we could use this as an inspiration in the choice of topics we want to consider. Let me now give the promised list of potential contents:

  • Epidemiology: this is quite the actual topic, so it might be worth discussing. In particular we can also debate the way such models have been (mis)represented in newspapers and other media outlets.
  • Forensic science: there are several mathematical models when we get into courthouses, for example fraud detection, algorithms to set bail money, DNA testing, prosecutor’s fallacy. Some of these have a common thread, the misunderstanding of probability theory, and Bayes’ theorem.
  • Communications: this is also a broad topic, starting with Shannon’s theorem and other early results and moving on to correction codes and their applications, as well as compression algorithms. The applications range from zipped files, to mp3, to CDs, to space probes.
  • Astrodynamics: talking about space probes, how do you send a spacecraft from a point A to a point B in space? How does a rocket work?
  • Tilings and packings: which are the shapes you can use to tile a surface? Which is the best way to transport apples? And what about 24-dimensional apples?
  • Graph theory: from comics to bridges to topology and back. And then further to coloring maps.
  • Game theory: can you take all the fun out of a game by analysing it mathematically? Or does it make the game even better? Can it help in real non-gaming life? What about elections?

As you can see, there are several options, and this list is definitely not complete.


From the previous parts, you should have by now gotten the gist of this course: the focus is not so much on the specific models presented, although they are definitely of interest, but on the “how to” parts. We want to see how to develop and refine these models, how to understand and criticise them, how to present them and the mathematics they use, how to search for them in the literature and online.

I would like this course to be very “hands on”, with active contribution from the students. In particular, I would like to use our fortnightly class meetings to have an informed discussion on the different topics. So I will introduce the topics in the online classes, provide some references and suggest some reading, then we can have a fruitful debate.