A Mathematical Voyage

By Rhiannon Kirby, Monash Univeristy

”The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful.”        

Henri Poincaré

Science et méthode (1908), translated Francis Maitland (1914)

Mathematics is a beautiful manifestation of man’s desire to understand the physical world he is bound by, an attempt to pave pathways between human perception and the objective exterior. These pathways weave through abstractions, exploring rational thought and logic within the mind, and reconnect with empirical knowledge via experimentation. Through mathematics we discover a universal communicative tool, capable of capturing notions that prove difficult to describe using everyday terminology, and by overcoming the barriers of unique discourse found in different fields of research, we enable interdisciplinary collaboration. Mathematics also embodies the human imposition of structure on all that confronts, to catalogue and control, to seek meaning and definition in the blank, but, few beautiful human phenomena exist without such dichotomy.

Throughout primary school, and a large part of my secondary schooling, mathematics presented itself merely as a weekly schedule of classes to attend, and a symbolic challenge. Endless questions waited numbly in black text, on their white stage, labelled 1a, 3f, 7g. They came with rules of engagement, and a warranty of being useful ‘later’, but without context, without images. The practice that ‘made perfect’ was the practice of conceptual amnesia, and a growing herd of spherical cows went unnoticed as crucial assumptions were disregarded. Beyond comparing the relative size of chocolate pieces, scheduling routines, or halving a cake recipe, mathematics was but a small curiosity in my formative years. Maths, for me, needed motivation.

To the teacher that showed me maths was ubiquitous in application, and, dare I say, enjoyable, I thank you for opening a portal. I was sceptical, but my mathematical voyage had begun.

During my three years of undergraduate studies at university, I have been privileged with exploring ideas in classical and quantum physics, thermodynamics, and fluid flows, paralleled by the development of an applied mathematical toolkit. It is this thrilling simultaneity in learning and immediate application that has allowed me sufficient freedom to appreciate the profound relevance of mathematics itself, and to see the sheer charm in math-for-math’s-sake. Through reconciliation of abstraction with physical theory, I have discovered underlying beauty in the formation and analysis of mathematical models. I am delighting in using maths to model natural processes, in learning how to implement routines computationally, and gaining further insight into the world around me.

During the summer of 2015/16 I was offered the opportunity to play with new mathematical ideas through an AMSI Vacation Research Scholarship. In a short several weeks, I visited realms of dynamical systems and elementary bifurcation theory, in the context of population ecology, with diversions off to interacting chemical systems, patterns on leopard tails, stripes on fish, and spots on (rectangular) cows. It is precisely this diversity in mathematical modelling that I find truly captivating.

My research focused on a simple continuous model of two competing populations of mosquitoes. In a model of competition, it is particularly important to determine if the system may support a coexistent or lone population, and to consider the long-term fate of the system, which may be sensitive to the initial population conditions. Natural systems also fluctuate in space and time, and so it is vital to consider whether long-term fates are stable to small changes. In this project, we observe that instability can give rise to a plethora of complex and intricate patterning behaviour for the mosquito populations, and this behaviour is heavily dependent on the physical parameters chosen to describe the system.

As a newcomer to the incredible world of mathematics, I look forward to another year of discovery and connection ahead.

 

Rhiannon Kirby was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.

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