Math is in everything including football, but it takes a mathematician like Dr. Peter Markowich to explain how football equates to life.

"I really love football. I actually grew up with football. My father took me to the pitch when I was a young boy and even my mother came along. It was a family thing, and since that time I've gone to see matches," said Dr. Peter Markowich, Distinguished Professor of Applied Mathematics at KAUST.

But while the game is fun to watch, there are various aspects of football where mathematics are involved. The first example is one that Markowich says is a more philosophical concept.

FOOTBALL AND THE MEANING OF LIFE

"If you think of football as a whole – any football match that is 90 minutes – it is really a great example of a very complicated random process. I would say it's a random process with a drift," he said.

The randomness Markowich is talking about comes from everyone involved in the game. It includes the physical shape of the players, how the leading players slept during the night, how the referee sees things and the quality of the pitch. Even the weather plays a role. "Maybe it rains; maybe it doesn't. It depends on very minor things – so in some sense this is a simplified version of life. I mean, life is a random process with a drift and football is likewise," Markowich explained.

The drift is there because the better team should win, but this doesn't always happen and the "not always" is when the randomness becomes too strong and takes over the drift. "In a football match, you have 11 players on each side – the more players the more complexity you have to the game. You have external factors, internal factor and it's all random, but in many cases the drift beats the randomness," Markowich said.

Take a look at the world champions in the last 30 years. It wasn't always the favorites that won, but it was always good teams – and that is what Markowich says is the drift. There may be some consistency overall, but in a single match anything is possible.

"I think that's what makes football so exciting. It is just not predetermined. You never know when you get in the game what will happen. If you compare it with tennis, which is a game that I like very much, there are only two players so it's much less complex," said Markowich. "There is still some randomness, but most often in tennis the favorite wins. With 11 players many more things can happen."

AND THEN THERE'S THE CROWDS

Crowds are another mathematical issue. Take for example, the motion created when spectators do the Ola-wave through the stadium. There is an interesting mathematical model that goes along with it which was developed by Gueant, Lasry and Lions in their "Mean Field Games and Applications" framework. The idea begins with a large number of players. In this case, it's 80,000 spectators and a few control aspects. "The model is based on that you want to behave similar to your neighbors. So if they get up, you also tend to get up. On the other hand, when you sit, you are comfortable and when you stand you are also comfortable. You don't want to be in the middle position with bent knees and uncomfortable – but at the same time, you want to be coordinated with what your neighbor does," said Markowich.

The Mean Field Game developers have written down a mathematical equation for the Ola-wave and have tested it – and it does the job. Markowich says the reason it works is because these models are based on an averaging principle. In other words, you have a large number of people. What each one does is not really relevant, but what is relevant is the emerging – averaged -pattern. "The averaging process is based on complicated individual behavioral patterns. It may not be a world-changing thing to describe the Ola wave in a soccer stadium, but it is interesting that using mathematics you can describe such a social phenomenon," said Markowich.

While the wave equation is interesting, crowd motion is a practical and serious mathematical problem. Crowd motion in a stadium is an unusual case as it is a 3D geometry when compared to a traditional room, which is two dimensional. A stadium has different entrances and levels – and of course, an important question is how do you evacuate a stadium in a fast and organized manner?

"We have examples in football stadiums where people were killed like in the Heysel Stadium disaster in 1985. It was a catastrophic event in the crowd, started by the follow the leader behavior, that led to the deaths of 39 people and 600 more injured," said Markowich. "Managing an evacuation is a complicated mathematical process."

Using simulations, mathematicians try to come up with ways to organize the behavior of people. One idea to do this is to put a column in the middle of exits. "You might think this is going to be detrimental as it decreases the area of the exit, but in many cases it leads to the organization of the crowd," said Markowich.

He says when watching simulations, you can see that crowds exit faster when there is an obstacle in the middle or a double door. In these cases, an obstacle has a positive effect in organizing the crowd motion.

ADDING PANIC TO THE EQUATION

Panic in crowd modeling is an important factor, as when setting up the model equations, you have to discard other strategies and predefined patterns. Markowich says you may begin with a model based on averages, but in a crisis, patterns change catastrophically and these can be described by so-called mathematical panic models.

"It's not based on physics and the principles that are written in stone, but based on our behavioral instincts which are much more complicated," he explained. "In the end we still write equations which are physics-based, although the constitutive equations come from non-physical considerations like how do humans behave."

Which brings us back to football and why the game is so exciting to watch. "We are enticed by the game because it's a small model of life. It is full of random processes just like life," Markowich said.

It's a random process with a drift.

By Michelle A. Ponto, KAUST News