My name is David Marchant, and I am studying permutations patterns. Permutations are fairly simple combinatorial objects, but when we start asking questions relating to the way in which a “small” permutation can be contained in a larger permutation, the subject becomes complex very rapidly.

At school my favourite subject was mathematics, followed closely by computing. When I started work, it was developing software commercially. I started a BSc in Mathematics in my early forties, and enjoyed it so much that I went on to an MSc in Mathematics. I enjoyed this even more, and decided that I wanted to carry on doing mathematics, so applying for a PhD position was the obvious next step.

Doing research is very tough, but also very rewarding. Finding ways to split the problem you are working on into smaller, manageable, pieces is an interesting challenge, and finding a solution to one of those pieces is a real thrill. Even if you can’t find a solution, you learn something, and there’s always the possibility that you can come back to a problem later with a different approach.

It was a fortunate combination of several things. Firstly, the Open University is one of the universities in the UK that does research in the area I’m interested in. Secondly, while I was able to stop full-time work to concentrate on research, the OU’s support of off-campus (“part-time”) students meant that I didn’t have to attempt to persuade my wife to relocate. My experience with the OU prior to applying also meant that I knew the academic staff were friendly and approachable, which, I’m told, is not the case in some other universities.

My intention is to spend as much time as I can doing mathematical research. If I’m lucky, then this will be in a university context, so I may end up being paid to do something that I really enjoy – but if not, then I’ll carry on doing research anyway.

Read! Read papers and articles in the area you’re interested in. Try things out. You don’t have to understand every single part of a paper, but picking up something and digging deep broadens your knowledge, and may give you insights you can use in your own work. Early on, find a way to record what you’ve found in a way that means you can find it later. If you don’t do this from the start, then you’ll find yourself having to go back and do it later (says the voice of experience!) Pick up problems and try and solve them – or, at the very least, understand them.

My name is Margaret Stanier and I am studying areas of graph theory, group theory and hyperbolic geometry. I am using the properties of a certain kind of graph to find new results in apparently unconnected areas of mathematics such as number theory.

After retiring I greatly enjoyed doing an Open University M.Sc. in mathematics, and was encouraged to pursue a Ph.D. I find doing research really interesting, challenging and exciting. It is great to meet up with other researchers. The Open University well understands the needs of part-time students, and I find it straightforward to fit my studies flexibly around other activities.

I worked as a teacher and then in education administration, a busy and challenging career alongside bringing up two children. I decided to start taking OU courses after I retired, and concentrated on maths as it has always been my main interest and I had not been able to do any seriously for a long time. The OU distance learning model suited me as I wanted flexibility and do not live near a traditional university.

I spent some time taking undergraduate courses to achieve an MMath, then embarked on the MSc course as I was encouraged by the successes I had achieved and the excellence of the materials and support I had received. I was not disappointed - the different MSc modules which I took were all most interesting and challenging in their different ways. The course materials help to study the set texts in depth. The assessments with their deadlines can seem daunting at times, but they do provide a framework for regular study. I actually generally enjoyed getting the whole course together in my mind in preparation for exams. Being able to demonstrate my understanding whilst answering the papers was quite satisfying, though of course there were a few anxious moments.

Advice and support from tutors was always there when needed. I found engaging with other students on the course forums helpful and interesting. I was fortunate in being able to attend most of the residential revison courses and course reunions, meeting with other students and experiencing working with them, which was invaluable.

The dissertation was a daunting prospect, but again it became possible with the help and support I received. It gave me a real taste of doing some actual research and appreciating what is involved when searching for background papers and writing up.

After I had finished the MSc I spent a couple of years taking OU physics and astronomy courses which were very interesting. However I found that I really missed seriously studying mathematics. I had dismissed the idea of doing a PhD as I thought I was too old, but I contacted students I had become friendly with on the MSc forums that I knew were doing PhDs with the OU. They encouraged me to give it a go. I found a research project which I was really interested in on the maths school web site and plucked up the courage to apply, which I would never have been able to do without the confidence my M.Sc. experience had given me. I was accepted and am really enjoying my new challenge.

My advice to students would be to be bold, never underestimate what you are capable of doing. Don't hesitate to ask for help if you get stuck on something, other students and your tutors will help. You do need to be well organised and be realistic about the amount of time you need to free up to study.

My name is Rosie Cretney, and I am studying the history of mathematics. My thesis will be about some aspects of the contributions of the Swiss mathematician Leonhard Euler to the mathematical theory of ship-building.

At school my favourite subject was maths, but I was also quite interested in the humanities. During my undergraduate degree in maths, I took a third-year course in the history of maths, and enjoyed all the new perspectives it gave me on the maths I'd been studying. In particular, it gave maths a human side that is often lost or suppressed in modern presentations of the subject: how did past mathematicians come up with their ideas, and why did the maths develop in the way that it did?

I didn't want to stop after studying that third-year course, so between the third and fourth years of my degree, I did a summer research project (funded by an Undergraduate Research Bursary awarded by the Nuffield Foundation; this scheme is now administered and funded by the London Mathematical Society). After ten weeks of puzzling over a problem that hadn't been fully explored in the literature, I finally made a breakthrough, and the buzz of that got me hooked on research. I then did a fourth-year dissertation in the history of maths, and applied to do a PhD in the subject.

The OU was recommended to me by my dissertation supervisor, as it has one of the UK's strongest research groups in the history of maths.

One of my favourite aspects of research is the diversity of approaches and sources that I need to use in my work. One day I might be building an interactive visualisation to better understand a dataset; the next, I might be visiting an archive and trying to make sense of scribbled calculations in a 400-year-old notebook. I'm also very lucky in that thanks to OU funding and international contacts made through my supervisors, I've been able to travel to lots of conferences and meet new people.

As my field is interdisciplinary and draws upon a whole host of other fields, there are lots of skills that I didn't really need during my undergraduate degree, but which I've needed to pick up for my research. To cite a few examples, I've had to brush up on my foreign language skills, learn how to approach rare books and archive material, and learn some basic Python programming. A fellow OU PhD student and I also co-organised an international conference for PhD students and early-career researchers in history of maths. This taught us a lot, and I found it a valuable experience for what I hope will be a future career in research.

If you are thinking about applying to do a research degree in the history of mathematics, do some reading around the subject first. Find a specific problem, a mathematician, or time period that particularly interests you, and use that as a starting point for your research proposal. If you are looking a short, accessible introduction to what historians of maths spend their time thinking about, I can recommend Benjamin Wardhaugh's *How to read historical mathematics* (Princeton, 2010), and Jacqueline Stedall's *The history of mathematics: a very short introduction* (Oxford, 2012). You could also look at *A concise history of mathematics* by Dirk J. Struik (Dover Publications, fourth revised edition 1987) for a brief overview of the history of maths written in a more traditional style.

My name is David Martí Pete and I do research on transcendental complex dynamics, which means that I study the iteration of holomorphic functions that have some essential singularity. This is a particular kind of discrete dynamical system, and the main goal of this area is to understand the dynamics of models of evolution processes which may occur in nature.

As far as I can remember I have always enjoyed doing mathematics. I believe that this is because I encountered some very good maths teachers in my high school. So when it came to the decision of choosing an undergraduate degree it was clear to me that this was what I wanted to do, and as time went by it became more and more obvious that I liked doing this.

I like doing research because it is quite challenging. Every week you have a set of small problems to deal with and if you succeed then you are led to more questions! Also you often have the opportunity to go to conferences abroad and meet people from very different places and backgrounds.

The main reason why I came here is that there is a very strong team working on transcendental dynamics. I had met my current supervisors before and they have a very good relationship with my master's supervisor in Barcelona. Once I arrived here I discovered that the department has a very good atmosphere and everyone is very welcoming and friendly.

The Open University holds a series of doctoral training workshops to make sure that you get not only the subject knowledge but a wide range of skills which includes career development, communication methods, time management, etc. On the other hand, my supervisors always encourage me to do talks in conferences and support me if I want to do other kind of academic activities like joining a teaching organisation as the UKMT which I did recently.

For me it is sort of important to be in a research group that has more people working on the same area to which you can speak and discuss about what you are studying. In my department there are a couple of other PhD students, a post-doc, and my two supervisors working on transcendental dynamics and we have a weekly meeting plus some seminar talks.

My name is David Bevan, and I am studying enumerative combinatorics, particularly permutation classes.

For twenty or so years before embarking on my PhD, I worked in software development. After reading Proofs from THE BOOK by Aigner and Ziegler in about 2002 (and noticing that one of the proofs wasn't optimal), I continued to investigate the problem (which eventually resulted in a paper) and also read lots of maths papers in my spare time. A few years later when I was becoming increasingly dissatisfied with my job, the discovery that I could be paid for three years to explore what had become my main 'hobby' was sufficient catalyst to seek to embark on a new career.

Exploring the universe of mathematics, or "spending time in maths world", which is what I tell my family I'm doing when I'm lying on the sofa with my eyes shut, working hard! In my psyche, the experience is much like discovering strange new worlds and investigating unfamiliar landscapes (with apologies to Prof. Brian Cox). It is a challenging pursuit and can be very frustrating, with many dead ends and false summits, like trying to find a way to the top of a steep mountain, but I find that the satisfaction of making even a minor breakthrough is considerable.

With the Open University, the following coincided: The availability of a fully funded PhD studentship, the possibility of doing research in an area that interested me, and a location not too far from where I live. The OU also has lots of experience with 'more mature' students like myself. One thing that characterises the department is how friendly and approachable people are; there is not the big divide between lecturers and research students here that is present in some other places.

I would love to be able to spend the rest of my working life doing mathematical research in a university setting.

Read lots of mathematics papers. This is now very easy with the arXiv. Learn how to get the gist of an article without needing to take in all the detail. Try to solve (hard) problems, and keep trying. If you don't enjoy that, you won't enjoy research. Find out what sort of mathematics enthuses you and is likely to keep your interest for several years.