An essential tool for understanding the world

By Christiane Rousseau

It is often said that mathematics is everywhere. And so it is. A GPS receiver calculates its position by using the time it takes for satellite signals to reach it. Secure communications are encrypted so that they are opaque to anyone other than the recipient. The JPEG format of photos taken by a phone camera is a mathematical compression of the information contained in the photo. Without this clever compression, the photo files would be enormous.  

And since any transmission of information introduces errors, correction codes are all-pervading in telecommunications networks, including those of mobile phones and television. Without them, remote-controlled robots on Mars would not be able to execute commands received from Earth. This omnipresence of mathematics explains why the theme of the first International Day of Mathematics in 2020 was precisely “Mathematics is everywhere”. 

Mathematics is indeed omnipresent in understanding our planet and the organisation of our civilization. As early as 1623 Galileo wrote that “The book of the universe is written in mathematical language”. Four centuries later, environmental challenges have become one of humanity's priorities. The population of the planet continues to grow – according to United Nations projections it reached eight billion people in November 2022 and should stabilize at around 11 billion, while at the same time climate change is affecting agricultural yields. 

Earth Overshoot Day, which marks the moment when humanity has consumed all the resources the planet can generate in a year, is occurring earlier and earlier. While it came at the end of December in 1970, in 2021 it occurred on 29 July. The Sustainable Development Goals of the United Nations Agenda 2030 are the international community's response to these challenges, and the mathematical sciences have a key role to play. 

Climate modelling involves putting into equations the interactions between various climate agents – sun, atmosphere (including greenhouse gases), oceans, soils, glaciers, plant systems, etc. It is essential to collect data if we want to predict the evolution of climate systems, and this prediction is the preserve of mathematics. Simulating these systems and establishing major trends is a tour de force that requires enormous computing power, ever more powerful algorithms and the optimal use of data. The work, carried out on several fronts, includes identifying long-term trends and highlighting regional trends. 

It is also important to accurately quantify the degree of uncertainty. Techniques exist to improve weather forecasting and to anticipate seasonal trends. This is particularly useful, as climate change increases the frequency and magnitude of extreme events. 

Hurricane track forecasting is an area where spectacular progress has been made – it is now possible to predict the path of hurricanes almost seven days ahead. A better knowledge of the risks makes it possible to anticipate and guard against their consequences in the coming decades. How high should a dyke be built? Should a neighbourhood be rebuilt or relocated after a flood? How often will droughts threaten water availability? How can cities be modified to reduce the impact of heat waves?

These forecasts are based on mathematical modelling, in other words a drastic simplification of reality. A good model allows you to see the big picture that may be hidden by too much detail. Let's take the case of an epidemic. The simplest model, called the SIR model, classifies individuals into three compartments: the Susceptible, the Infected and the Removed (recovered or deceased). 

In this model, the basic rule explains the movements between compartments from one day to the next. It is then possible to calculate how the populations of the compartments evolve over a long period of time. 

This model, simplistic as it is, reveals the main laws of an epidemic: the exponential growth, the peak of the epidemic, the herd immunity phenomenon by which the epidemic dies out before the whole population has been infected, the flattening of the curve of cumulative cases (orange curve), the basic reproduction number (R number), which measures the average number of infections generated by a primary infection and which characterizes the contagiousness of a disease. 

These general laws inform decision-makers about the evolution of the epidemic. The model can then be improved for more accurate predictions. For example, the basic rule for day-to-day movement can be adjusted to take account of health measures or the appearance of new, more contagious variants. The compartments can also be divided into sub-compartments (age groups, social classes, gender, recovered and deceased), etc.  

Another field of action of mathematics is that of optimization. How to organize the transport and distribution of mail or goods? How can train timetables be planned to facilitate connections, minimize the number of trains, and optimize staff working hours? The question also arises for urban public transport and aviation companies. These problems are part of what is known as operations research and, while they are easy to state, the number of possibilities is too large to find an optimal solution by trial and error. 

Finding the best solutions requires the development of very clever and effective algorithms. These same techniques can be applied to the ecological transition, which requires moving from over-consumption of resources to optimization of their use. How can we save energy or water, reduce food waste, make the most of limited resources by using them equitably? These are areas where mathematics has a role to play. 

Artificial intelligence (AI) is breaking new ground in mathematics and statistics. The breakthrough comes from the fact that we can now programme computers or robots to learn. For example, a human can recognize a cat. A computer is taught to do the same. To do this, it is trained with hundreds of thousands of images and corrected when it makes mistakes. The computer continuously improves its programme and learns to recognize a cat, even in a position it has never seen before. In image and sound recognition, the successes of artificial intelligence are spectacular and are more than a match for humans.

Applications are multiplying. Artificial intelligence can be used to produce low-cost maps of poverty, for example, using public domain satellite images. Daytime images reveal man-made infrastructure. Combined with night-time images, AI identifies inhabited areas that are not lit at night – a sign of poverty. Artificial intelligence is also being used in semi-arid regions of northern Kenya to secure access to water. Using the data, it is possible to predict areas where water shortages might occur and develop mitigation strategies.

With countless and highly varied applications, mathematics constitutes an exceptional toolbox.  It is no coincidence, then, that it is all around us.