# Marianne Johnson

## Lecturer in Algebra and Tropical Mathematics

Marianne Johnson (on the right)

I enjoy listening to students explain their ideas to each other and helping them to develop their ideas further. It’s great to see the 'light-bulb moment'.

How would you summarise your research?

My current research is in an area called 'tropical mathematics'. The basic idea is to consider a set of real numbers with an unusual algebraic structure. Instead of the usual operations of addition and multiplication, 'tropical addition' of two numbers, x and y, returns the maximum of the two, whilst 'tropical multiplication' of x and y returns their sum in usual arithmetic.

Why is this object interesting? One reason is that this structure is useful if you are interested in certain kinds of scheduling problems. The operations 'max' and 'plus' turn out to be exactly what is required to describe the timings of certain systems. For instance, imagine you are trying to create a train timetable. What time should a train depart? The earliest time it can leave is found by taking the maximum of the arrival times of any connecting trains and adding on a decent amount of time for the passengers to change platforms. Tropical mathematics turns up naturally in a number of other application areas.

As a pure mathematician, what is my interest in tropical maths? Although this new algebraic structure is quite different to the real numbers with their usual operations, there are some interesting similarities in the roles that they both play in a 'linear algebra' setting. Part of my research focuses on what these two structures (and others) really have in common and where the differences arise.

What do you think makes the School distinctive?

The School of Mathematics is one of the largest in the UK. As a consequence, we have researchers working on an extremely broad range of mathematical topics. This is fantastic for students and academic staff alike, who have access to a wealth of expertise.

What do you enjoy most about teaching?

Listening to students explain their ideas to each other and helping them to develop their ideas further. It’s great to see the 'light-bulb moment'.

What do you enjoy most about research?

Firstly, it's exciting when you spot some kind of pattern in mathematics. It gets you thinking, 'I wonder if this always happens. Or is there something special going on in this particular situation.' Making guesses or conjectures about how things work is part of the fun of mathematical research. Of course, you don't always guess right, and even if you do, how do you prove that you're right to someone else? This leads to the second enjoyable aspect; coming up with 'a good idea' to explain why your pattern exists. This might involve a lot of hard work and consideration of many 'bad ideas', but each new idea, good or bad, leads to a greater understanding of mathematics.

How long have you been at the School? What keeps you there?

I've been part of the School off and on for over a decade; first as an undergraduate student, then as a PhD student, later as a postdoctoral researcher and now I'm a lecturer here.