# Charles Walkden

## Director of Teaching and Learning and Senior Lecturer in Pure Mathematics

Charles Walkden

At some point a PhD student starts having their own insights into their research problem, which are unforeseen by the supervisor.  This moment – when the relationship changes to two equals working on a research problem -  is probably one of the highlights of being an academic.

I work in an area of pure mathematics called Ergodic Theory.  This is closely related to dynamical systems, more popularly known as ‘chaos theory’.  The idea in dynamical systems is that one has a simple mathematical rule and one wants to understand what happens when one repeatedly applies (or ‘iterates’) it.   Often the behaviour is surprisingly complicated; Ergodic Theory is about studying the long-term behaviour of iterating dynamical systems.  For example, suppose you start with the number 1 and repeatedly multiply by 2 to obtain the sequence 1,2,4,8,16,32,64,128,… .  Now look at the sequence of left-most (or leading) digits of this sequence: 1,2,4,8,1,3,2,1,… .  It appears that in this infinite sequence the digit 1 occurs more frequently than digit 2, which occurs more frequently than digit 3, etc.  One can use Ergodic Theory to explicitly calculate the frequency of each digit in this sequence (the answer is that digit r occurs with frequency log (1+1/r)).

I work in an area of pure mathematics called Ergodic Theory which is closely related to dynamical systems.   One has a dynamical system (for example, a map T : X -> X) and one wants to study what happens to a point x in X when one repeatedly applies the map T to it.  Instead of trying to understand the behaviour of every orbit, ergodic theory takes a more qualitative approach: can one understand the long-term behaviour of typical points under iteration?  To make ‘typical’ precise one needs to use ideas from measure theory.   I’m interested in the ergodic theory of a class of dynamical systems called ‘skew-products’.  These are systems that allow one to model a wide range of dynamics, for example random dynamical systems, forced systems, iterated function systems.   I’m particularly interested in the fractal structure and probabilistic properties of such systems and use techniques from thermodynamic formalism to study them.

What do you think makes the School distinctive?

We’re one of the largest schools of mathematics in the country.  This means that we have a very large number of academic staff with a very wide range of research interests.  This means that we can put on a huge range of courses that comprise our undergraduate and taught masters programmes.  Many of our 3rd year and 4th year undergraduate courses and taught masters courses are taught by leading experts in the area.  Being part of one of the largest universities in the country, and situated in the heart of Manchester, also means that our students have access to excellent facilities.

What do you enjoy most about teaching?

There are many different aspects of teaching at a University and I enjoy them all.   Giving a lecture to large class of 1st year undergraduates is a very different experience to teaching an advanced 3rd year course or to supervising undergraduate projects.  When giving a lecture with potentially several hundred people in the audience, one had to think carefully about how to make material come alive and how to make sure that audience is actively, rather than passively, engaged in the lecture.   Supervising undergraduate projects can be a hugely rewarding experience for both myself and the student: working together on a piece of mathematics that develops in unexpected directions is always fun.

How do you make your teaching up-to-date, innovative and inspirational?

A lot of mathematics teaching is still the traditional ‘chalk-and-talk’ – although often blackboards have been replaced by visualisers and data projectors!    In my own lectures I try to point out wherever I can the connections between the current topic and other areas of mathematics – mathematics is not just a series of unrelated course units!   I also link the material to current Oresearch or important unsolved problems wherever I can – it’s good to show students that there are still a lot of things we don’t know the answer to.   I also try to make my lectures and support classes as interactive as possible (‘mathematics is not a spectactor sport’), either by carefully designing student-led worksheets, or by encouraging the use of suitable iPad apps or websites. More widely in the School, I’ve recently been involved in helping to run demonstration classes for academic staff on preparing short instructional videos.  I and a few colleagues have trialled these over the last few years and they’ve proved very popular with students.

How long have you been at the School and what keeps you there?

I came to Manchester in 1997, initially as a postdoc, and then as a lecturer.  I set-up and ran the MSc in Pure Mathematics for many years, before taking over as Director of Postgraduate Studies in 2012.  I then became Director of Teaching and Learning – with lead responsibility for all undergraduate and taught masters programmes in the School – in 2015.   I also teach two courses and do research in ergodic theory, as well as run several schools outreach and public engagement activities.  It’s this variety in the job that keeps me at Manchester – although it also keeps me extremely busy!

What kind of balance do you strike between teaching facts and developing skills?

Most courses have a mixture of theory and application. This enables students to acquire knowledge of important mathematical results and methods, and to develop problem solving and analytical skills.

How does research feed into the syllabus?

Our students are taught by leading researchers in their fields.  As well as finding out about current areas of mathematical research, students also develop the study skills that prepare them for a range of postgraduate programmes at both Masters and Doctoral level.  Other students develop these research skills and go on to use them in careers in industry.

How do you make sure the course meets the needs of industry and business?

We are in close contact with representatives from many of the companies that employ our graduates. We run a series of talks, workshops and careers fairs (such as the annual ‘Calculating Careers’ Fair run within the School) to make sure that staff and students are aware of the knowledge required by graduates in industry.   We also embed careers and employability within the curriculum, for example we have a course on Mathematics Education aimed at undergraduates who are considering a career in teaching.

What are the course highlights for you?

As one of the largest departments in the UK, with a staff of exceptional academics, we are able to offer a broad range of high quality undergraduate courses covering pure, applied and financial mathematics, probability and statistics.

What are you doing to improve NSS scores?

We work closely with our student representatives to identify ways to improve our teaching and the student experience.

Why do graduates from your course stand out in the job market?

As well as strong subject knowledge, our graduates have the ability to apply mathematics to real world problems.

What kind of industry relations do you have and how do students benefit from them?

Through our maths focused careers activities we have developed strong links with graduate employers. We have developed several initiatives such as the ‘Managing My Future’ activity, mock internship interviews and the Calculating Careers event to help students find out more about careers and employability.

What distinguishes this course from similar ones in other institutions?

The range and quality of course units available to undergraduates means that students may choose to study any area of mathematics they wish to specialise in. Graduates from The University of Manchester are highly regarded by employers. Our students go into a wide range of sectors where analytical and problem solving skills are important. Many of our undergraduate students go on to study at postgraduate level at Manchester and at other leading universities in the UK and around the world.