Category Archives: Opinion

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

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: