PIMS Chats with L Mahadevan

Language and Reason

L Mahadevan began his education in engineering. However his interest in mathematics was piqued when he met his soon-to-be PhD advisor, Joe Keller, who was then a professor of mathematics and mechanical engineering at Stanford University.  As a graduate student, Mahadevan started reading Keller’s work, “I thought it would be amazing to do some small fraction of what he did, which was to look around and explain things. That’s what ignited my interest in using mathematics as a language.”

Now, Mahadevan considers himself a denizen of the borders between disciplines. “I am sort of a mongrel. I am not a real mathematician, not a real engineer, not a real physicist, but some poor mixture of all! And recently, leaning towards biological questions, although not exclusively.”

“Though many people have said this, it is worth repeating that mathematics is a language. I think it is the language of science so that interdisciplinarity comes naturally.” He explains that just like you can study and utilize language in many different ways, you can do likewise with mathematics and use it to describe all manner of phenomena – like a linguist, a writer or a poet use language in different ways.

At Harvard, he has appointments in Applied Mathematics, Organismic and Evolutionary Biology and Physics, reflecting various facets of his research and teaching. He teaches similar concepts in many of his courses, but the different disciplines name them differently and continue to rediscover old ideas in new forms.  For example, “historically, a large part of mathematics has been founded in observations of nature. However, in the last few years that language has been evolving to try and describe the artificial and social world as we have created it – in technology, society and beyond.”

In his PIMS/Niven Lecture Mahadevan discussed his research on the biological origins of shapes. “Mathematics is the study of patterns” says Mahadevan. “If you think of it as a language, it is used to describe ideas, and in this case, patterns in number, connectivity and shape. The initial stages of my study were focused on physical problems: how fluids flow, how solids deform, the behaviour of different phases of matter. More recently I have started asking the same questions in a biological context, i.e. living matter: why is the leaf shaped the way it is; how does a flower bloom; what defines the shape of a limb on a tree; why does skin wrinkle?”

“I find that I get most excited by things that you can use your own senses to observe directly. We looked recently at how birds sing; how snakes move, how worms crawl, how a Venus flytrap catches flies.” He explains that this type of research makes it easy to do experiments and immediately test theories, and the everyday scale that generates these questions is incredibly rich, meaning that you don’t have to look far to find interesting phenomena. “The questions are very simple” he says, “but while the answers are often not, they are very often applicable to a much larger set of examples than the original question.”

So, how does a bird sing? “A bird sings, from a biological perspective, because it wants to attract mates (usually males sing to attract females) or communicate information with another. If you were a neurobiologist you might ask how the brain controls the syrinx (a bird’s equivalent of the larynx), learns songs and introduces variations. If you are a mathematician or physicist, you can ask exactly the same questions from a quantitative perspective. For example: What is a song and how is it created? The answer: a coded reproducible sequence of sounds using a timed series of pressure pulses as a function of time, which impinges on your ear. How is that pressure generated? By vibrations of a membrane driven by a fluid. We are currently conducting experiments to explore how those vibrations are generated, controlled and varied. These are all mathematical problems.”

“I always work on specific questions for two reasons: they allow for bite size portions that are hopefully, soluble, and also because by looking at enough questions you can start to see the general picture.  An analogy to climbing may be useful here: you could go climb Mount Everest, which is very hard and you may never get there. Or you could start climbing small hills, slowly getting better and better and starting to see a broader view of the world around you. This latter is like the kind of science I do. Gradually, by climbing enough hills of increasing height and variety, you start to see the lay of the land and get a view of the bigger landscape.

So, as he explains, there haven’t been any big “aha” moments in his career, but “there were lots of little ones – indeed, although there is only a small likelihood of having many big ones, there is much larger likelihood of having many little moments of discovery – and this keeps one going!”

A major factor in beginning this research was reading a book written almost 100 years ago by D’arcy Thompson, On Growth and Form. “Thompson had a very inspiring view; he wrote from the perspective of a mathematician, but was using analogy… For example, in thinking about cell shape, he said that if you blow glass you can make something very beautiful. You must heat the glass and then use pressure and heat different parts differently. Certain cells, as they are being formed, use a very similar mechanism.” As Mahadevan explains, “That analogy is a century old, but Thompson didn’t push it, beyond pointing out that similarity. What I have been trying to do is take that to the next level by being quantitative and using experiments to flesh out the reasons behind it all.”

On May 24, 2012, L Mahadevan delivered the 2012 Niven Lecture, “On growth and form: geometry, physics and biology” at the UBC site.

L Mahadevan is a Professor of Applied Mathematics, Organismic and Evolutionary Biology and Physics at Harvard University whose work centers around using mathematics to understand the organization of matter in space and time, particularly at the scale observable by the unaided senses.