Only showing posts tagged with 'maths' Show all blog posts

Selfish species: game theory and the ecosystem


I am studying for a PhD in Statistical Physics and Complex Systems at The University of Manchester. My research studies a system of many interacting species where the population of one species can facilitate or hinder the growth of another species. This relationship is determined by a specific interaction coefficient between the species. The interaction coefficients for the relationship between every pair of species are drawn randomly from a two-dimensional Gaussian distribution, and we use the parameters of this distribution to predict how the ecosystem behaves. We can then simulate these interacting species using a computer programme to check our predictions.

In Depth…

I studied Mathematics and Physics for my undergraduate degree at The University of Manchester. I chose this degree because I enjoy understanding how the world works, and appreciate how bizarre and counter-intuitive our reality is. I had a fascination for quantum mechanics and relativity, higher dimensions, and sub-atomic particles. I really enjoyed learning about these concepts as well as being introduced to many other fascinating ideas. I enjoyed the lecture style of teaching but I also developed my ability for independent learning, I became really good at managing my own time, and absorbing information at my own pace from reading textbooks and lecture notes. The most useful skill I learned during my degree was how to computer programme, I learned how use Matlab, C++, and Python, and I learned how to write codes for simulations, data analysis, solving complicated equations, and optimization algorithms. I decided to do a PhD after my undergraduate degree because I really enjoy self-study and programming, and I am further developing these skills with new challenges every day.

I became interested in population dynamics after reading "The Selfish Gene" by Richard Dawkins, where he described behavioural evolution using ideas from Game Theory. He described how an animal’s behaviour, and the behaviours of the other animals it interacts with, would determine how successful the animal would be at surviving and passing on it genes. These successful behavioural strategies would dictate how the behaviour of the population as a whole would change over time, and evolve to an Evolutionary Stable Strategy which could be understood as stable Nash equilibria. During my degree I took the opportunity to study Game Theory further by writing my second year vacation essay on the topic. I researched many areas of Game Theory and went through a short online course. I discovered how it can be applied to statistical physics, in the Ising model for ferromagnets, and really enjoyed learning about how ideas from quantum mechanics could produce Quantum Game Theory, where a player could play multiple strategies at the same time. In my fourth year I undertook a project with my current PhD supervisor on a population of individuals who had the choice of two behavioural strategies to interact with. The population evolved by the number of individuals playing the more successful strategy increasing, but this model also considered the effect of time delay, such as a gestation period in nature. I really enjoyed my project with my supervisor and through this I continued onto a PhD with him.

Going Further…

Here is a link to my supervisor’s webpage, if you are interested in my research you could look at his publications:

Here are links to the undergraduate Mathematics and Physics courses webpages:

If you are interested in game theory, here is a brief course:

If you are interested in “The Selfish Gene” here is a brief summary of the book, chapter 12 discusses game theory:

and the full text can be downloaded here:


Fractals: What are they?

by YPU Admin on May 4, 2017, Comments. Tags: Fractals, Geometry, maths, PhD, Pure Mathematics, Research, STEM, and UoM


Hi my name is Catherine Bruce and I’m a first year PhD student in Pure Mathematics at the University of Manchester. I study the geometry of fractals, which are objects deemed too irregular for traditional geometry such as straight lines, area, etc. (They have an infinite perimeter!) During the final year of a 4 year undergraduate degree I realised that I really enjoyed research and was lucky enough to be offered the opportunity to do it full time.

How I got here

After leaving secondary school I did A-levels in Maths, Further Maths and French. I was also interested in Politics and took it as an extra AS-level in my second year at college. I then applied to do a four year undergraduate degree in Mathematics at the University of Manchester. This meant I graduated with an MMATH degree which is called an integrated Masters. I enjoyed my whole degree but realised only while undertaking my final year project that research was for me and applied for a PhD. I took a year out to go travelling after graduating and returned to Manchester in September 2016 to start my research.

In Depth

Fractals are too irregular for their size and structure to be measured using classical geometry. The main tool of fractal geometry is dimension, which has many forms. A lot of research is done into finding the dimension of different fractals and the image of these fractals under different functions. Fractals are often very beautiful – they have detail at all scales which means no matter how much you zoom in to one part of the fractal it will always have an interesting structure, which is not true for classical geometrical objects like 2D and 3D shapes.

An important property of fractals is self-similarity. This vaguely means that an object looks like it’s made up of lots of smaller versions of itself. Notice that each branch of this fern looks like a smaller fern, and each smaller branch looks like an even smaller fern. This is a simple example of self-similarity.

Many things in nature have a fractal-like structure: clouds, mountain skylines and forked lightening. Fractal geometry can be applied to many different things in the real world. Examples that I know of are lasers, cancer treatments and fracking. However, I do not deal with any of these applications as I (along with all other pure mathematicians) study the theoretical side of mathematics.

In the first few months of my PhD I have been reading a lot to learn what everyone else in my field of research already knows. When I have done this I will be able to start answering questions that no one has answered before, and coming up with brand new research. This is the whole point of a PhD and is an exciting if not scary prospect!

Going Further

Learn more about what fractals are:

Have a go at constructing your own fractal:,0090,1,1,0,0,1

Have a look at the exciting research that’s going on in dynamical systems at the University of Manchester:



Developing environmentally friendly fuel

by YPU Admin on June 25, 2015, Comments. Tags: biofuel, biotechnology, computing, electricity, enzymes, hydrogen, maths, oxygen, Physics, redox, Research, and water


My name is Nick and I am a first year PhD student at the Manchester Institute of Biotechnology. At school I studied physics, maths and computing at A-level and then went on to study physics at the University of Manchester (BSc and MSc). My PhD research involves trying to find out how the structure of redox enzymes affects their redox potential. The redox potential is an important factor that needs to be considered in the design of biofuel cells. Biofuel cells use enzymes to help produce electricity from hydrogen and oxygen, with water as a waste product.

In depth

The redox potential (E0) tells you how likely a chemical species will accept an electron. When a chemical species accepts and electron, we say it has been reduced. The more positive the redox potential, the more likely it is that a chemical species will accept an electron and be reduced. The below reaction has a positive redox potential, so a copper ion will tend to accept an electron to become a copper atom.  

Cu+  +  e-  ↔  Cu  (E0 = +0.52V)

A chemical species may have a negative redox potential. This means it is more likely to lose an electron. When a chemical species loses an electron we say is has been oxidised. The more negative the redox potential, the more likely it is that a chemical species will lose an electron and be oxidised. The below reaction has a negative redox potential, so a sodium atom will tend to lose an electron to become a sodium ion.

Na+  +  e-   ↔  Na    (E0 = -2.71V)

Enzymes are a type of biological molecule which catalyse (increase the rate of) the chemical reactions that sustain life. Redox enzymes contain a metal ion which can either be reduced or oxidised. They help control the rate of many different reactions which involve the transfer of electrons. The structure of the enzyme around the metal ion influences the redox potential of the metal ion. Below is an image of an enzyme called Azurin, which has a Cu2+ ion in its active site.  The way in which the Azurin is bound to the copper ion affects how easily it can accept an electron.

You might be familiar with the idea that electricity is the flow of charge particles. For example, electrons flow in the wires that make up the electrical devices we use. Electricity can be made in many different ways, some more environmentally friendly than others. Biofuel cells utilise enzymes to help make electricity using hydrogen and oxygen and producing water as the only waste product. The redox enzymes help transfer the electrons through the cell which generates electricity. One enzyme takes electrons from hydrogen and passes them through the cell. The other enzyme collects the electrons and then uses them to make water.

My research involves working out how the structure of the enzymes changes their redox potential. The idea is to produce a computer program that will be able to adapt the structure of an enzyme so its redox potential is perfectly tuned for use in biofuel cells. I also plan to make the enzymes and experimentally measure their redox potentials, to prove the computer program works.

Going further

Manchester Institute of Biotechnology:

What are enzymes?

What are redox reactions?

Fuel cells:

Biofuel cells:

Could biofuel cells be developed for use in our bodies?


Researching the Squishyness of Foam

by YPU Admin on December 25, 2014, Comments. Tags: algorithms, applied, business, computer, cycling, finance, foam, helmets, industry, maths, model, patentlaw, pressure, and syntactic


My name is Maria Thorpe and it's now only 10 months until I have to submit my thesis for a PhD in applied maths.

My route to a PhD

I moved up to Manchester 7 years ago really excited to be going to university and studying for an undergrad masters in maths for the next 4 years. I loved every minute of my undergrad, but by the beginning on the fourth year still didn't really know what type of job I wanted when I finished. I was still enjoying my subject and I'd really enjoyed a research project I'd been sponsored to complete over the summer between third and fourth year, so I decided to apply for a PhD on a similar topic in applied maths. 

In Depth

Since then I've been trying to mathematically model the way in which a specific type of composite squashes under pressure. I work with a material similar to syntactic foam, similar to the  sort of foam cycling helmets are made from, however instead of creating small cavities within the material by injecting air into it, tiny hollow balls (called shells) are mixed into the foam before it sets, forming a composite. These micro shells are created from very stiff, glass-like materials and help stiffen the material under low pressures, but under high pressures they crumple like a coke can. I want to understand whether having shells close to each other changes the way the composite reacts to pressure: do the shells reinforce each other and allow the material to withstand higher pressures? Or do they have the opposite effect and cause the composite to squash more than if they were far apart?

The company sponsoring my research wants to understand how their material works so that they know how to improve it. It would take too long to try out all the different ways the shells could be mixed into the foam, and might involve buying new machinery, so it makes sense to model the material instead. Creating a very flexible model means that the same model can be used for many different applications, so I try to model the material theoretically, by extending the models previous generations of mathematicians have created. This means that most days are spent making very small steps forward with my research, but when a whole section comes together it can be really rewarding.

Aside from working on my thesis my PhD has enabled me to travel to some really great places: I spent a month in New Zealand with a company having a go at the more experimental side to my research; I've traveled to conferences all across Europe; and I've spent three months working in parliament to learn how science influences policy.

Moreover these last three years have allowed me to discover all the ways maths is used in industry and business, from patent law and government policy to computer algorithms and financial trading, so that this time round, when it comes to looking for post PhD careers, I have a much clearer idea of where I could go from here.

Going Further

If you'd like to read more about my research and that of the group I work with, the waves in complex continua group, check out our webpage:

There’s also an interesting article on the use of syntactic foams for deep sea exploration here:


Applying mathematics to real-world problems

by YPU Admin on November 13, 2014, Comments. Tags: aerodynamics, appliedmaths, manufacturing, mathematics, maths, and Research


I'm Phil, a student at the University of Manchester in the final year of a PhD in Applied Mathematics, which means I apply mathematical ideas to solve physical problems. I started out doing a Mathematics MMath (a four year undergraduate course) here in Manchester and my A-levels included Maths and Further Maths. You might think that all this studying would make me feel like I know a lot of mathematics, but in fact the more you learn, the more you realise there is left to learn.

In Depth

So, what do I actually mean when I say I “apply mathematical ideas to physical problems”? Well, imagine you’re a car manufacturer who wants to test how aerodynamic a certain part of a new car is going to be. You could build a prototype of this part, using your current knowledge of how to build something that’s really aerodynamic, and test this prototype in a wind tunnel. During this experiment you could measure all sorts of useful data such as the drag caused by the wind as it flows past the part.

Now, what if you want to slightly change the shape of the part and test it again? You could, of course, build a slightly different prototype and test this again in the wind tunnel. But this would cost both the time and money necessary to build the prototype all over again. If you want to test a lot of different prototypes, the investment of time and money (on just this single part) could really start to mount up.

This is where an applied mathematician could come in. With mathematics, you can build a model of the prototype in a wind tunnel using all of the physical laws that we know it would obey. You can then write down the equations that the object would satisfy and either use clever mathematical tricks to solve them on paper or put them all into a computer to solve them (essentially using a computer to perform a virtual experiment). This can gain you a crucial advantage; once you’ve built the model, you can (hopefully) solve it for many different versions of a part much faster than the time it would take to build a whole set of prototypes and test them in a real wind tunnel. And, best of all, solving the model doesn’t cost you anything at all!

This kind of thinking can, of course, be applied to an almost limitless array of physical problems. In my specific research, I investigate how flames propagate through gaseous mixtures of fuel and oxidiser. Since my work is all in the form of solving equations on paper or simulating the physical system on a computer, I never have to get burnt or accidentally set myself on fire (you might see this as a positive or a negative, depending on your viewpoint). I’m hoping that this will lead to a post-doctoral position at a university, where I can continue to both research mathematics and teach the mathematicians of tomorrow.

Going Further

The website for the School of Mathematics at the University of Manchester can be found at

Information on the applied mathematics research at the University of Manchester can be found at

It’s not specifically about applied mathematics, but if you’re interested in maths I’d strongly recommend reading a book called “Euclid in the Rainforest” by Joseph Mazur – it’s both interesting and very readable.

Finally, I recently wrote a blog on cryptography for the Young Persons University, which can be found at

Find me on twitter @pearce_maths