From Hydon’s Little Red Book to Parameter-State Symmetries

During my PhD, I came across a small red book that ended up shaping much of my later research.

It was Peter Hydon’s Symmetry Methods for Differential Equations, and I remember being completely fascinated by the idea that symmetries could be used to understand and analyse mathematical models. Coming from mathematical biology, I was struck by a simple question: why are Lie symmetries — so central in physics and connected to some of its most celebrated discoveries — almost unheard of in my own field?

That question stayed with me.

What began as curiosity gradually turned into a research direction of its own. It eventually led me to apply for a research fellowship with the Wenner-Gren Foundations with the goal of introducing Lie symmetry methods into mathematical biology. That application, in turn, took me to the Wolfson Centre for Mathematical Biology in Oxford, where I had the opportunity to develop these ideas further in a new and inspiring environment.

In hindsight, that was one thread of the story.

Another thread began much earlier, at the very start of my PhD, when I encountered a concept that I simply did not understand at all: structural identifiability.

I remember reading papers involving Lie derivatives and feeling completely lost. To be honest, that made it hard for me to appreciate the topic. Since I did not understand the concepts, I could not really see their beauty or importance. At the time, identifiability felt technical, opaque, and far removed from the parts of mathematical biology that I found most exciting.

Fast forward more than a decade, and the irony is hard to miss.

I now find myself writing a manuscript where Lie symmetries are used to study local structural identifiability and observability. In fact, while working on this recent paper, I ended up citing some of the very work that once confused me. That led to a slightly amusing exchange with one of my collaborators, who asked whether I knew Mats Jirstrand in Gothenburg. In a way, the answer is yes — if not directly through joint work, then certainly through the intellectual landscape that shaped my early years as a PhD student.

One of my main goals with this recent work has been to make these ideas more accessible: to provide a conceptually clearer way of thinking about identifiability and observability, and to show how parameter-state symmetries can be used to analyse them. For me, that has been part of the satisfaction of working on the project. It is not only about proving results, but also about finding a language and a framework that make difficult ideas easier to understand.

That, perhaps, is one of the things I find most rewarding in research in general. Often, the topics that seem most inaccessible at first are the very ones that, with time and persistence, become the most meaningful. What once feels alien can later become familiar. What once feels frustrating can later become fertile ground for new ideas.

Looking back, this journey has a shape that only becomes obvious in hindsight. A small red book, a new mathematical language, a fellowship application, a move to Oxford, and eventually a return to a concept that once seemed impossible to grasp.

Research is not always a straight line.

Sometimes, it is a loop.