Fasten your seat belts!

Recently I was flying back from New York to London and as soon as we took off, I heard the magical phrase: “Please keep your seatbelt fastened during the whole flight. We expect a bumpy ride”. The pilot was right – it was so bad that I couldn’t sleep, watch any movies, not to mention complete any work I had planned to do. To be honest, I was sure we would crash, so I’m happy just because I can write this blog post today.

This adventurous trip reminded me of one of seminars I attended during my first year of Mathematics of Planet Earth program. I should have paid more attention to Dr Paul Williams from the University of Reading, who claimed that due to the climate change we can expect more turbulence while flying over the Atlantic Ocean.

Most of us associate global warming with increased temperatures on the ground. However, as the above mentioned atmospheric scientist reported, it also makes the jet stream even stronger.

According to the Met Office, jet streams are ribbons of strong winds around 9 to 16 km above the Earth’s surface (so right below the tropopause). They move weather systems with the speed of up to 200 mph. The temperature difference between tropical and polar air masses is their main cause. Meteorologists care about jet streams a lot because waves and ripples formed along them can dramatically deepen Atlantic depressions while moving towards Europe.

Jet streams make flights from America to Europe faster than westbound journeys. Indeed, my flight ticket to the USA states that the journey lasted 8 hours 27 minutes while on the way back it took 7 hours 10 minutes. The pilot could have done even better, because the record on this route belongs to Boeing 777 operated by British Airways that in January 2015 landed at Heathrow after 5 hours and 16 minutes. They took advantage of the jet stream that brought heavy rainfalls and winds to the UK.

While jet streams work in favour of passengers travelling to the capital of the UK, they also make flights towards Big Apple longer. Especially because these winds are getting stronger due to the climate change causing increased differences between temperatures of troposphere and stratosphere. The stronger the jet streams become, the shorter the eastbound and the longer the westbound flights. The problem is that quicker journeys from America won’t compensate for the increased flight time against the wind. Williams estimates that each airplane flying over Atlantic will spend extra 2000 hours in the air, which means millions of gallons of jet fuel burnt. This will lead to the emission of 70 million kilograms of carbon dioxide, about as much as annual emission from 7100 average British households. It’s a vicious cycle: climate change causes more carbon dioxide burnt, which causes climate change, which causes…

The increased time spent in the air isn’t the only unfavourable effect of the climate change on aviation. Research shows that passengers should expect more turbulence incidents. Every year hundreds of people suffer injuries due to unexpected “bumps” during the flight. In 2016 videos such as the one taken on the flight from Abu Dhabi to Jakarta went viral. During this flight turbulence was so strong that 31 passengers and crew members had to seek medical help after landing in Indonesia. Such incidents make me think that my flight wasn’t as traumatic as I believed!

Jet stream is one of the common causes of the clear-air turbulence, a turbulence not associated with a cloud. This type of turbulence can be dangerous because radars aren’t able to detect it; this is why it’s usually unexpected not only by passengers, but also by pilots. And Dr Paul Williams with Dr Manoj Joshi (University of East Anglia) pointed out that we have to prepare for more such surprises as the level of carbon dioxide in the atmosphere increases.

Apart from obvious discomfort and dangers, increased turbulence leads also to considerable financial problems. Williams’ report states that airlines spend millions of dollars to repair damage caused by turbulence. Moreover, sometimes airlines have to find longer routes avoiding places notorious for occurring turbulences, which leads to even more money spent and more pollutants emitted. For us, passengers, it means delays as well as longer flights.

So fasten your seat belts – just in case. And have a safe flight!

Image: http://www.nytimes.com/2007/06/12/business/12turbulence.html?_r=0

Article originally published on my personal blog.

More Than Just Symbols

 

When you think of a mathematics textbook you probably imagine a series of intimidating pages with a few words and a bunch of strange (often Greek) symbols. I don’t think I’m alone in thinking that the fact that a lot of modern mathematics is only presented in this form is a bit of a crime. For instance, Tristan Needham expresses a similar feeling in the pre-amble to his text “Visual Complex analysis.” Professional mathematicians can usually get some idea of what is going on in the pages of a paper or textbook. However, anyone who hasn’t had as much training in their past loses out. Particularly since, to the untrained eye, there is no way to associate these abstract symbols with anything visual or otherwise.

Reading a mathematics textbook is not like reading a novel–it can be a slow and arduous process. Despite this, someone with mathematical knowledge might eventually be able to understand what is going on. To draw an analogy with computers – it is as if you need the “correct software” installed in your mind to process the text. The same concept applies to reading novels written in other languages – if you don’t have the “correct software” installed in your mind then all that you will see is a series of random symbols. An education in mathematics allows you obtain this “software”, once you have this you can “speak” about things you have never spoken about before.

The important thing to draw from above is that the symbols are simply placeholders for various ideas and concepts just as they are in any other written language. In my opinion adding visuals or graphics to a piece of mathematics significantly helps us to tie down what the symbols are trying to suggest (even if it is just a crude analogy). On the other hand, it is probably true that even illustrations and graphs on their own are probably not enough. Without the proper context, graphs or illustrations may simply appear as static creations with no further meaning. I think additional understanding can be achieved by playing around with the image in your head (or by sketching variations with a pen and paper). This playful approach to imagining visualising mathematical concepts no doubt inspired artists such as M. C. Escher (M. C Escher was a Dutch graphic artist who is well known for his often mind boggling and mathematically inspired work).

Nowadays we can go significantly further than Escher with the power of computer graphics. As an example I have listed a few of my the coolest looking pages and blogs related to visualising mathematics and mathematical concepts below:

  • http://www.graphonaute.fr/ A selection of animations and images created by French engineering student Hugo Germain. Makes me think of what Escher’s work may have been like if he were born into the digital age!
  • http://blog.matthen.com/ Lots of nice visualisations and a few cool visual proofs of well known mathematical theorems as well. The code for the visualisations is also available allowing anyone to play around/ learn how to create their own!
  • http://imaginary.org/ Imaginary is an interactive platform which designed to showcase mathematical media content. The site contains plenty of pictures, videos and interactive demonstrations!
  • http://visualizingmath.tumblr.com/ A tumblr page full of math related visualisations.
  • http://geometric-aesthetic.tumblr.com/archive Tons of geometrical patterns and fractals.

In summary, the “beauty” of mathematics may be something “cold and austere” (as Bertrand Russell puts it), however, I believe everyone can gain if we do more to visualize the concepts involved. As well as helping our understanding, it allows us to think up strange new worlds (such as those depicted in other M.C Escher’s work – and potentially Einstein’s general relativity). Given the amount of maths out there I’m sure there is a lot of potential for mathematically inclined artists out there!

Matt
References and Further Reading
[1] This post is a continuation on themes I previously wrote about in about Maths and Visualization

Cover Image:

http://i.vimeocdn.com/video/458881089_1280x720.jpg

Programming is a piece of cake

When I think about famous mathematicians born in previous centuries, I imagine a handsome young man (well, my gender-biased imagination probably confirms the statements from my last article). He is sitting under a blooming tree with a quill and a scroll of parchment covered with equations. I must have lumped together mathematicians and poets, with the romantic charm.

With this image in mind, eighteen-year-old Paula went to her first lecture at the Faculty of Mathematics, Informatics and Mechanics (University of Warsaw). Full of expectations that I would do great things in the field of mathematics, I started copying down the words from the blackboard. “Baking a cake – the algorithm” – this is what they said. Was that a joke? No, just Introduction to Informatics.

You might wonder why such a course was obligatory for first year mathematics students. We just need a pen, a piece of paper and a great mind, right? Not quite. We can stay away from computers if we want to focus on abstract algebra or number theory. However, things get much more complicated if we would prefer to actually apply maths to other sciences. Like to weather prediction.

Lewis Fry Richardson tried to predict the weather on 20 May 1910 by hand. Don’t get me wrong, I admire his attempt, he was one of the pioneers of the weather prediction. Having said that, I am happy that scientists have developed more sophisticated techniques; Richardson failed quite badly.

Other examples? The model I am working on involves simulating the behaviour of population of worms. Without a computer I would have to give up, it is just impossible to calculate by a human being, not to mention more complicated systems, such as climate models.

The world of Four Colour Theorem

Computers even help us prove actual theorems from pure maths, for example the famous Four Colour Theorem. One of versions of this theorem states that any political map can be coloured using only four colours (or even less). The rule is that no bordering countries can have the same colour. The statement is very simple but the only way to prove it was to find all the possible configurations of countries (we are not talking about the existing countries but every map you could draw). I am pretty sure that without computers authors of the proof, Kenneth Appel and Wolfgang Haken, would be still checking all the possibilities.

I hope you understand now why the computers are so essential in mathematicians’ work. But what do they actually do for us and what does the cake mentioned in my first lecture have to do with maths?

Computers are like dogs and programming is just instructing the puppy to sit/stay/stop-stealing-my-favourite-socks-so-that-I-have-to-chase-you. You must know what you are given (input or your dog’s personality) and what you want to achieve (output or yay-I-can-wear-my-lovely-blue-socks). Anything in between is called the algorithm. Believe me or not, you are using algorithms every day. A recipe to bake a cake is one of them. Or your morning routine. When you forget one of the points or switch the order, you might have to eat a slack-baked cake or be late at work. Algorithms structure how we think about the problem.

How do we tell the computer what it is supposed to do? First we need to know in which language we are comfortable communicating. We can learn programming languages in similar way as we learn German or Spanish. The all have specific words, syntax etc. It takes practice to become fluent in one of them. And even if you are almost a native, you still need to consult a dictionary once in a while.

The problem with programming is that the computers are, paradoxically, very stupid. This is my personal view and I hope my laptop won’t take revenge on me for saying this (I still have to finish my project!). Why do I think so? If you confuse words in a foreign language, there are good chances that you will still be understood (unless you ask someone to turn you on, confusing it with turn around, which happened to me once). Computers just take what you give them and do not think. If your code is not working, it is entirely your fault. They will not try to guess what you wanted to do — do not count on their empathy. Life is brutal (and computers definitely are).

The first time you see a code – the instructions for the computer – you might be very surprised. It took me a while to accept that it is ok to write “a=a+1”. It means that we add 1 to a variable called “a”. Although if you accept “taking something with a pinch of salt”, why would you struggle with programming? We do not have to take everything literally!

I definitely think programming is a useful skill to acquire – and an essential one when you are a mathematician or a scientist. It can be hard, it can be exhausting. But trust me, the day when I managed to persuade my computer that I know what I want – and when I got it – was one of the best moments in my life. So keep calm and hello world!

 

Why studying random dynamical systems matters

If I said I was studying multistability in random dynamical systems driven by Brownian motion, why should you care?

Martingales, filtrations, ergodicity, synchronisation, Monte Carlo methods: mathematics is full of words which invoke colorful ephemeral imagery for heavy, abstract concepts. Even if we explained the concepts and brought them to life, it wouldn’t be obvious how they were related to your world. Mathematics is powerful exactly because it exists a few steps away from reality, in a place where anything which obeys its structured rules is possible. But this means that to answer the question of why we study what we study (apart from that it’s beautiful and fun) we have to step out of mathematics and back firmly into the physical.

When we think about the Planet Earth mathematically we often think about understanding and capturing each of its parts, studying technical topics like turbulence, boundary currents, atmospheric waves, sub-gridscale processes or atmospheric chemistry. These are the building blocks of the Earth and investigating them shows us patterns and connections which let us project their behaviour forward a few days into the future by running them artificially faster inside our super-computers, predicting the weather or the dissipation of volcanic ash.

There is another powerful way to think about our Earth, though. For the past decade or so, biologists, for example, have stepped away from the reductionist  approach of looking at the components of their system (protein and DNA) to start looking at the high-level systems behavior. The pieces slot together intricately into such a complex system that some of its behavior (consciousness, for an extreme example, or an ant colony) is nearly impossible to discern by looking at the parts. They find that sometimes it’s better to drop the details and start at the top and work down, characterising as much of the emergent behaviour as possible since it’s part of understanding the animal, even if we can’t quite see where it comes from.

In the study of Planet Earth we’re lucky that we’ve been so driven early on by prediction even while we’ve been studying the pieces. We’ve always had an eye on the whole system, trying to simulate it in weather models since predictability is equivalent to safety and success as we try to flourish as a species. These models do an amazing job of mirroring the massive weather beast, especially as we tweak them year on year, checking them against reality and improving them. But what we don’t often do is explore theoretically the range of behaviour the complicated highly non-linear and multi-dimensional system we live within is capable of. We don’t often look at it from the top without the baggage of the details of the pieces. This is the truly “systems” approach.

Perhaps the simplest example of what you might see if you took a systems perspective is multistability: the idea that the Earth might have different states (or rhythms) it could comfortably fall into which would self-perpetuate themselves, even if in the same conditions another state would have been possible. The mathematical example is always a ball rolling down into a valley with another valley next door; the ball could stop at the bottom of either valley, both are stable states, but it can’t get from one to another easily. Ecosystems exhibit multi-stability – the same patch of land could be a birch forest or a grazed field and stay that way indefinitely. Even bodies exhibit multistability if you look at them from the right perspective: something alive generally keeps itself alive, but dead also stays dead.

A schematic view of bistability, in which the “ball” can be in either “valley” happily, but finds it difficult to transition between the two. Source: Wikimedia Commons.

Whether or not our climate has multiple stable states is vitally important because if it does and we start to push it, we could very possibly push it out of one and into another and it might be very hard to return. How would we know if it did? One way (the reductionist way) might be to carefully build a replica of it, to the very best of our ability, then start pushing it around and see if it switches. Not a bad idea, but with obvious drawbacks too: our model might not be quite perfect and it takes a long time to run these pushing experiments on a computer, so it’d be hard to draw solid conclusions. Another way is to think theoretically about what is required of a system to be a jumping system then study whether the Earth satisfies the conditions, looking back at the record we have about our real Earth system to see if we can see signatures of jumping. To do that, however, we need to know what jumping looks like.

That’s why we’re interested in a class of mathematical constructs called random dynamical systems. They’re an abstract construct and could be interpreted into the real world in lots of different ways so, like most everything in mathematics, they’re probably useful in a lot of fields. What they can do for us, though, is to represent the Earth like a point moving between states according to some update rule, complicated or simple, which summarises the evolution of our climate in time. What makes them even more complicated than the dynamical systems people like Ed Lorenz have been studying for decades is that we now allow the update rule to be a little bit random thanks to a Brownian motion term: to send the Earth up the valley when it should have gone down, occasionally. Suddenly, a whole new class of behaviour is exposed: with bistabilitity in the update scheme and additional noise comes stochastic resonance, the seemingly regular jumping of our proverbial ball between valleys depending on which way the wind (which wouldn’t usually be enough to move a ball between valleys!) blows. They also might simplify dynamical systems, with noise-induced synchronisation meaning that even if we don’t know where exactly the point started, it’ll end up in a similar place if it undergoes the same noise.

It’s not that the Earth is a random dynamical system (it’s a planet, not an abstract mathematical concept), it’s that we hope that by trying to represent it as one, we’ll learn more about it. Climate certainly has some random elements: weather provides unpredictable and sometimes large change in local climate, not to mention volcanoes. We also have stochastic resonance-looking jump-like events recorded in the history: the Glacial-Interglacials and maybe Dansgaard-Oeschger events in the past million years. If we find that to some extent our reality fits well into this concept, we will be able to draw conclusions from the concept back out to the real world. It probably won’t be as revealing as complex wave functions were for quantum mechanics, but having a new way to account for the world around us is almost surely a good thing.

When it comes to complex systems that we count on for our survival, no one approach can be relied on. We don’t have a reason to believe that there’s a single bit of mathematics that nature obeys on this scale, but there will be patterns and properties at all scales which fit into mathematical frameworks. Random dynamical systems let us look at the pattern of stability and large changes in the whole system and is, as both a mathematical field and a climate perspective, relatively new and so very exciting.

When the mathematicians meet…

What’s the difference between an introverted mathematician and an extroverted mathematician? The extrovert looks at the other person’s shoes. You probably have heard this joke (very funny) many times and might believe that mathematicians work stuck in their offices (and their own heads). It was probably true a couple of centuries ago. However, things have changed after the rapid development of maths, especially applied maths. Why?

Nowadays we have to specialise, at least a little bit. We don’t have Da Vinci’s any more; it’s just impossible to fully understand more than your very narrow area. But while mathematics research narrows more and more, it tries to tackle more and more complex and multidisciplinary problems. What do we do now?!

We, as mathematicians, must get out of our comfort zones and collaborate. We need to accept our lack of understanding of certain aspects of each maths problem while, at the same time, being aware of how we can contribute to the solution. We have to identify what kind of experts we need to ask for help to make some progress. This is how it all begins.

When I started my adventure in maths, I certainly didn’t anticipate this. I was prepared for working alone and talking to colleagues only in my free time. While it could possibly work in pure maths, I would totally fail to succeed in applied fields if I tried to do so.

Examples? Mathematics of Planet Earth Centre for Doctoral Training! Yes, we pursue our individual degrees and work on our own projects. However, we operate as a cohort too. Sharing experience, tips and asking for help are essential for this programme to exist. I can’t even count how many times computer science experts saved my life (or at least my precious laptop) by preventing me from running a code that would destroy the system1 . In exchange I could give them a hand when they got lost in abstract multidimensional spaces (although I don’t claim I can visualise anything in more than three dimensions, though it disappoints my first year lecturer!). We all learn from one another.

Recently I realised that it’s not just a fake academic set-up, this is how the “real world mathematics“ works. I spent 5 days at the 116th European Study Group with Industry in Durham, UK (http://www.esgi.org.uk/). This event brought together about hundred mathematicians, physicists and industrial partners. The latter proposed eight problems they wanted to solve in fields as diverse as agriculture, banking and sepsis diagnosis. We divided ourselves into groups according to our interests — I chose the problem proposed by a digital bank. They needed help with marketing their product to the best target audience (of course the ultimate goal was to spend less and earn more). We sat down in a room and…well, and started thinking, talking, brainstorming and arguing. Within 3 days we managed to produce whole models and get some useful results for the industrial partners. Something infeasible for one genius became a reality for a group of people with different backgrounds.

Yes, you might still meet a mathematician staring at her/his own shoes while talking (or avoiding any contact) to you. But this is not a norm anymore. And definitely not the only way to succeed. We can tackle real world problems together because together impossible becomes possible!

  1. Note, not everyone is as lucky as me, you can read what happens when you’re not careful while programming here.

Not “too silly”, not “too girlish” for maths

– “What do you study?”
– “I’m doing PhD in maths.”
– “Wow, you must be so smart! And you’re a girl!”

I hear it so often. I’ve done a small amount of research and my results are sad: my friends studying linguistics, architecture,medicine and so on don’t get such a routine reaction. Why is it the case? Where does the assumption that a mathematician must be smarter than the rest of the society come from? And why are we still surprised that women are capable of pursuing this career path?

I had thought the same before I decided to study mathematics. It shouldn’t be surprising. As a ten-year-old I fell in love with John Nash (or rather Russell Crowe starring in A Beautiful Mind). Than I laughed at how nerdy and out of touch with life the characters of The Big Bang Theory were. Media portrayal of mathematicians didn’t make my decision to study maths easy. Would I become like them? Would I spend my adulthood bending over equations, unable to engage in social life and relationships?

Moreover, I was afraid that I wasn’t smart enough, that one needs a brain of Gauss or Newton to be a mathematician. But I took up the challenge and… I’m still here! Even though my IQ isn’t high enough to join Mensa. Even though I’m a GIRL!

Einstein nailed it: Genius is 1% talent and 99% percent hard work.During my undergraduate studies I’ve seen apparent geniuses being expelled from the university because talent and intelligence aren’t enough. Nobody is born with maths knowledge, it takes years of hard work to gain enough experience  experience to earn a diploma.

I believe that talent is helpful but what counts most is your interest in whatever you’re doing and determination to work hard. Although if you truly enjoy maths, the work might be turn in to fun, as crazy as that sounds. I’m not claiming that I loved every evening spent going through some complicated proofs (especially the ones beginning with the words “It’s obvious that…” – maybe it’s obvious for you, author, but it isn’t for me!). But some problems and ideas were really my thing. I even kept reading about them after the exam!

Ok, but what about the girl part? Is mathematician really a job just for men? Does the gender matter at all? Personally, I get very annoyed when someone admires me for studying maths despite being a female. There’s no correlation between the excellence in mathematical subjects and gender. Neither positive nor negative – I don’t agree with the common statement that girls are more hard-working so they get better results than “smart but lazy” boys.

Unfortunately not everyone agrees. Last year I went to my first mathematical conference. The organiser (male) came to me the day before my talk just to say something along the lines of: You don’t need to worry about your talk. You’re a women, you can’t be as good in mathematics as your male colleagues so nobody expects you to give a good talk. I was shocked! I did well because my research was of good quality, not because of or despite the fact that I’m a woman.

To sum up, if you feel that you like maths (or some parts of it) but are afraid of pursuing the degree because you’re not a genius or (even worse!) you’re a girl, don’t hesitate to give it a try! You have every chance of success and you don’t want to regret not having done something you really wanted. It’s far better to regret something you’ve done!

If you’re interested in articles about the need (or lack of need) for extraordinary mind to do mathematics, take a look at:

 

How Climate Model Uncertainty Should Influence Climate Policy

“All models are wrong; some models are useful” –  George Box

“I don’t believe anything, but I have many suspicions” – Robert Anton Wilson

Climate models are our primary method for predicting the future state of the Earth, and so are a crucial influence on climate policy.  Politicians often demand firm evidence that climate change is real. Scientific evidence of climate change has been around for decades, however, skeptics still manage to blow any uncertainty in scientists’ models out of proportion. In the following post I will discuss a short article by J. Norman et al. in which the authors argue for action to mitigate climate regardless of whether we have perfect models.

In the article they pose a powerful question which helps us think critically about what factors come into our decisions about climate policy. This question can be phrased as: “what would the correct policy be if we had no reliable models?” Thinking carefully about this question allows them to dismantle the assumption that it is only worth acting to combat climate change if you believe in climate scientists predictions. This is important to consider since many climate skeptics arguments rely on pointing out the flaws in scientists models. Furthermore, this question represents an interesting limit case to think about since it encourages us to think about how we should behave given that there are uncertainties in our models.

Their argument is based on the so called precautionary principle which, from a risk management perspective, posits that if something is potentially harmful to the public, the burden of proof on the people who want to carry it out rests on establishing that it isn’t, not the other way round. In short: we shouldn’t dabble with things when we don’t know what the unintended consequences might be. This is especially true when there is even a small risk of a complete catastrophe. Ignore this principle at your own risk in everyday life, but when we alter things that might affect whole ecosystems or planets it is definitely worth being extremely cautious about the consequences of our actions, which may not be reversible.

Any predictive model will have uncertainty in its outcome. In addition to this uncertainty, we must also strive to remember that the model isn’t reality, no matter how hard we try, meaning that there will always be events which are out of the range of our predictive powers. The importance of events which are out of our predictive range was popularized by Nassim Nicholas Taleb in the 2007 book The Black Swan. “Black Swans” are extreme events that lie out of our predictive range (no matter how good our supercomputer is). In the management of risk, the impact of these events may also be referred to as Knightian Uncertainity, this is a risk that is essentially uncomputable.

Just as an unpredicted market crash will render forecasts produced by economists for the future obsolete, there may also be unexpected events which will have a dramatic effect on current climate predictions. For example, a huge volcanic eruption might occur or some new carbon capturing technology might conceivably allow us to remove a large amount of CO2 from the atmosphere in a matter of months. In fact, we even have examples of relatively sudden changes in atmospheric composition happening in the past, as is the case with the “Paleocene-Eocene Thermal Maximum”.  Taleb’s message in the Black Swan is not that we should try and predict these events, but that we should instead be aware that, good or bad, unknown unknowns are out there in the future, whether we like it or not.

Figure 1. [3] Forecasting Skill plotted against year for different forecasts. This demonstrates the improvement of our ability to predict weather patterns over relatively short time scales. We shouldn’t naturally assume that this transfers over to the quality of our climate predictions. However, these types of observations provide evidence of the plausibility of physical models in the prediction of future states of the climate system.
That being said, we shouldn’t be totally pessimistic about our ability to predict. Weather and climate models can and have been statistically tested to perform fairly well [2]. In addition, the accuracy of weather forecasts and our ability to “see” longer into the future has indeed improved over the years (see figure 1). Given this, we should be careful not to misinterpret Norman et al.’s article as saying “Policy makers should never use climate models because we simply can’t predict things.” The main point is summarized in the final two sentences of the article:

“The popular belief that uncertainty undermines the case for taking seriously the ’climate crisis’ that scientists tell us we face is the opposite of the truth. Properly understood, as driving the case for precaution, uncertainty radically underscores that case, and may even constitute it.”

There will be plenty more to come on the blog about this difficult situation! For now, the main thing to take away is that we need to keep an open mind about the different possible outcomes which might arise in the future, predicted or not.

In summary:

  • We shouldn’t blame scientists for failing to include the presence of certain rare events in their models. After all, with limited computing power there is only a certain amount one can include in a model of such a massive system.
  • However, we can blame scientists or policy makers if they attempt to implement policies which ignore the possibility of anything that the model doesn’t explicitly predict.
  • Uncertainty is reason to act, not a reason to not act.

“The ancients knew very well that the only way to understand events was to cause them.”  – Nassim Nicholas Taleb

Matt

References and Further Reading:

[1] Taleb, Nassim Nicholas. The black swan: The impact of the highly improbable. Random House, 2007. Highly recommended for anyone who wants to read more about the impact of rare events on the course of history. Taleb goes into detail about the reasons why we should be very careful when transferring predictive models inspired by those in the physical sciences into domains such as the social sciences and economics.

[2] For more on how good climate models and their predictions have been for us so far: https://www.skepticalscience.com/climate-models.htm

[3] Image Reference: http://www.nature.com/nature/journal/v525/n7567/images/nature14956-f1.jpg

[4] Cover Image: http://www.billfrymire.com/gallery/weblarge/global-network-earth-space-night-sky.jpg

 

What on Earth is Mathematics of Planet Earth?

Until recently I had no problem explaining what I was studying; I was just an average maths student. I could reliably predict the reaction of a person informed of this fact: “How can you do that? I’ve always been hopeless at maths. And anyway, what are you going to end up doing with that degree”? Things got much more complicated when I started a PhD at Mathematics of Planet Earth. The first reaction is now usually the question used as a title to this article: what on earth is mathematics of planet earth?! After a brief explanation that I am learning how to use and develop maths for climate and weather predictions, I just get a reassuring statement: “I know that this whole climate change thing is very dangerous/rubbish” (choose the option that applies to you) and a question: “But why did you resign from doing proper maths?”

Actually, I am more involved in studying mathematics than ever. No science would exist without mathematics, in particular climatology or meteorology. Some people can predict the rain by feeling it in their bones; I can “predict” the rain more or less based on the fact that we are in UK. But do we really want to risk our life on someone’s body niggles? No, I do not exaggerate. Our life really can depend on it. Do not forget that a bad weather prediction not only can get you wet, but also farmers might not prepare for a drought (and the crops would get extremely expensive next year), local authorities might not decide to grit ice-covered roads (so you might get stuck in traffic or even have an accident) or a dangerous storm might hit citizens completely not ready for it. It is something worth looking at, is it not?

To get more reliable predictions about the state of atmosphere in the next couple of hours, days or even centuries, we need… mathematics. No, not that boring multiplication table, but nearly every field of very advanced mathematics. Let us take a look at a couple of examples.

Chaos Theory

You will have heard of the “butterfly effect” which allegedly can provoke a hurricane. This is all about the chaos. The intuitive definition, given by E. Lorenz, the creator of chaos theory, is [1] when the present determines the future, but the approximate present does not approximately determine the future. It means that if we were given infinitely precise initial conditions (i.e. full description of the state of the weather now), we could predict the weather at any time in the future. So why do meteorologists sometimes get it wrong? Because this is just wishful thinking. In reality we are not able to get perfect measures of the weather components, for example due to the limitations of measuring devices. Thus mathematicians need to choose the most important measurements with the available precision and try to get the best prediction they can. However, chaos theory states that, under some conditions, starting from almost the same state we can get completely different results. It complicates weather prediction so chaos theory is still something we need to study.

Numerical Analysis

There would be no weather forecast without very advanced computers we are using. Some of them are even supercomputers, such as the one used by Met Office. It costed a trifling £97 million. Why do governments invest such enormous sums into such equipment? Before we understand that, we have to see how the weather prediction works. As mentioned above, we cannot forecast it exactly. Hence mathematicians have to get rid of some parameters that seem to be less important (by the way, deciding which are those is far from obvious) and, using the ones that are left, build a model. This is a set of equations (sometimes thousands of them!) that describe the system. Do you remember solving systems of two equations at school? You might have struggled with it. So now imagine solving thousands of much more complicated ones. Yes, this is exactly why we need supercomputers; they make this job feasible. However, mathematicians still need to make sure that the result produced by a computer is sensible. They do it by carrying out a numerical analysis, checking the properties of the system.

I’ve mentioned only a tiny fraction of the whole range of mathematical tools used in the weather prediction. Next time when you listen to your favourite weather forecast, keep in mind that it would not make any sense without mathematics. And if you happen to have a child, encourage them to study maths. Just in case.

 

[1] Danforth, Christopher M. (April 2013). “Chaos in an Atmosphere Hanging on a Wall”. Mathematics of Planet Earth 2013. Retrieved 27 January 2016.

It’s OK When Weather Forecasts Are Wrong

Weather forecasts don’t have a great rep. Since I started studying the Mathematics of Planet Earth, I’ve lost count of the number of friends and family members that have asked me, “But weather forecasts aren’t any good are they?” Sure, weather forecasting isn’t an exact science. You only have to follow a forecast for a week in Autumn or Spring in the UK to notice that it rains when you were told it would be sunny and vice versa. Have a look at Figure 1, which shows the number of times the Met Office correctly forecasted rain throughout 2014 — they get it right about 70% of the time (this is very good — the Met Office claim to be second in the world for quality of their forecasts [1]). However, I don’t think this is a problem with the science. It’s a problem with how we interact with the forecasts.

Figure 1: The fraction of times rain was accurately forecast by rain symbols one day ahead of time by the Met Office throughout 2014. Source of data: http://www.metoffice.gov.uk/medi a/pdf/5/6/MOSAC20_2015_Anne x_III_forecast_accuracy.pdf
Figure 1:
The fraction of times rain was accurately forecast by rain symbols one day ahead of time by the Met Office throughout 2014.
Source of data:
http://www.metoffice.gov.uk/medi a/pdf/5/6/MOSAC20_2015_Anne x_III_forecast_accuracy.pdf

In fact, weather forecasts should not be correct all the time. Ewen McCallum, a former Chief Forecaster at the Met Office says: “If we got it [the weather forecast] right every time, we’d be God.” Trying to predict the weather is a bit like trying to predict dice rolls — we just don’t have enough information to be able to do so. In order to have a perfect prediction we would need to know the exact air conditions — temperature, pressure, wind velocity and humidity — at every single point in the Earth’s atmosphere with perfect accuracy. This clearly isn’t possible: even if our measuring instruments were perfect, it’d be impossible to know these properties everywhere. Even the flap of a butterfly’s wings or a baby’s breath will cause these quantities to change.

Surely these effects are too small to make a difference though? It turns out this isn’t true. The Earth’s weather is known as a complex system — the tiniest changes to its state cause the system to evolve in a different way. This phenomenon is known as chaos and is observed in many types of system. It means that no matter how close we get to measuring or guessing the current conditions, at some point in the future we won’t be able to determine the weather.

One of the ways that forecasters try to account for this is by trying their simulations many times with slightly different conditions at the start, to account for the unknowns in their measurements of the current state of the atmosphere. This technique is known as ensemble forecasting. Rather than telling forecasters what will definitely happen at 3pm on 19th of April, they might see that in 70% of their simulations it rains over London at this time. This gives them a probability for it raining, rather than a definite answer to whether it will or won’t.

However, we as the public don’t like this kind of uncertainty. We simply want to know if it will or if it won’t rain, and it almost looks weak of a forecaster to avoid giving a definite prediction. Unfortunately giving a definite prediction is poor science as it does not represent truth about the weather. Given that the nature of the weather is unpredictable, demanding a definite forecast will inevitably lead to failure in the long term.

So the problem is not so much with the quality of the research at the Met Office — it lies with our expectation for a definite prediction rather than one that contains uncertainties (such as one saying there is 70% chance it will rain today). These uncertainties are an inherent part of a chaos theory, which we see in the weather as we can’t have perfect knowledge of the whole system at once.

References:

  1. http://www.ecmwf.int/en/forecasts/charts/medium/monthly-wmo-scores-against-radiosondes?time=2016041200&parameter=vw850

Why bring maths into it?

To thrive on planet Earth, knowing what it’s going to throw at us is key. We need to know what crops can be grown where and what time of year to plant and harvest them. We need to know whether the mosquito that’s about to bite us is likely to be carrying malaria. We need to know how to manage flooding, and how much snow or wind our structures have to withstand.

So how do we know these things? Experience, first. The security that what will happen tomorrow will probably not be too much different from things that have happened before. But more and more, we are profiting from a delicate and precise understanding of the “why” and “how” of the system encoded in mathematical models to deduce more exactly what tomorrow will bring: how much light will shine on the solar panels or even when to shut the roof on Wimbledon to avoid the rain.

With climate change, the need for strong predictive structures become even more dire. Humanity is taking a complex, intricate system and dramatically altering a key component – greenhouse gases. We really are putting a cat amongst the pigeons, and, without doubt, things are going to change. The future is not going to be like the past: how it will change and how we can best avoid the worst–those are questions that require a mathematical and physical mastery of the system.

And of course, this is desperately important. While we in Britain might have enough resources and padding to quickly adapt to these changes, a lot of the people who share the earth with us are in a much more precarious position. Think of what famine does to a country with ethnic tensions, what water shortages in a country with a strong military imply for its neighbors and how a poor country deals with a new set of diseases traveling in with the weather.

Realizing the patterns, quantifying the interactions and building models is not a fix by itself. We still have the problem of wanting ever more growth and energy at as low a price as possible, of prioritizing today over generations from now.

But what we, as mathematical and physical thinkers, can contribute is a demystification: revealing the behavior of the complex Planet Earth so that as a society, a species, we can make the large ethical decisions facing us with more determination and confidence.

Mathematics, physical, geological, biological, ecological and chemical sciences don’t offer a pre-packaged answers: they’re ways of thinking to be drawn on depending on the question at hand. In this rest of this blog, we’ll be talking about some of the puzzles important in a responsible response to climate change and what tricks and elegant techniques from the mathematical world especially we can use. Expect calculus in the form of Partial Differential Equations used to describe a system changing over time, linear algebra to make those equations approximately solvable by a computer, dynamical systems thinking to try to simplify the complicated evolution and understand the butterfly effect and finally statistics and probability to express what we know and don’t know after all of that.