Thoughts on Uncertainty and an Unpredictable Future

I have recently been reading about when certain methodologies or paradigms fail. My current research has taken the direction of when classical models of certain kinds of physics are not appropriate. Assuming the results of my proposed alternatives look good, I will share some of them here. I have also been reading things very much outside of my area, and it has made me think more about the nature of academic, government/industrial, and layperson thought. The rest of this post will be a very brief response to this article by Nassim Nicholas Taleb, as well as a few other pieces such as a book on Street-Fighting Mathematics, which is an excellent read for anyone interested in approaching mathematics from a very different perspective.
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A Note on Multiple Scales: Homogenization Theory

As I have mentioned before, many problems in science can be profitably attacked from a large variety of different angles. Today I want to discuss one particular aspect of theoretical work in a technical way, in order to provide a bit more motivation to why a mathematical method of analysing a scientific problem provides insight that complements computational or experimental work. This is the technique of multiple-scales analysis, and specifically Homogenization. Most of the following was written as part of a broadening course I took in a Perturbation Methods lecture that is taught at Oxford for fourth-year undergraduates.

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The Spectrum of Rigour: PDE Theory, Shocks, and the Devil’s Invention

An overly simplistic (and hence incorrect) view of science is one in which we think all scientists have the same level of certainty in their descriptions of natural phenomena. One might suggest that different scientific fields could be arranged by their level of rigour, but even this misrepresents the idea of our certainty in a scientific result. The reality is that rigour and careful methodology don’t depend linearly on the complexity of the field or even the problem at hand. Nevertheless, we can make some progress by trying to understand the rigour of theoretical results on some linear spectrum.

With this idea in mind, I want to briefly touch on a few areas where this spectrum can be observed. I will not give anything close to a full account of these areas, nor can I list the countless textbooks and online sources that discuss them, but I will litter the discussion below with some links to help make sense of things. Ideally, the reader will have some mathematical maturity, but I hope the discussion is illustrative even if the details are too abstract to follow. It should also be noted that this is perhaps a very subjective opinion, and that my experiences are primarily as a student, and less as a researcher.

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Percolating Ideas

Today I want to write briefly about some of what I have been learning about Percolation Theory. I will try and make this interesting to the layperson, but some of the details require a bit of mathematical maturity. As always, questions are welcome. I won’t spend too much time on the history or significance of the subject, but the Wikipedia article above is a good place to start for that, and I can always provide some good references if you ask. Continue reading

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Fluid Dynamics and Mechanobiology

Today I want to discuss some of the mathematics I’ve been doing, and mention where I hope my research goes. My doctoral research hasn’t exactly narrowed down on a particular question yet, as we are asking around to find experimental collaborators, but I have been learning quite a lot in the way of solid and fluid mechanics, mechanobiology, and a host of related ideas.

First I want to discuss a classical problem in fluid mechanics (flow through a cylindrical pipe), and then a slight modification of it that makes it a bit more interesting, and possibly related to biological capillary flow. The notes are in part for my own review, but any comments or questions would be very welcome. After this I will briefly touch on the kinds of problems my research interests are becoming, although everything at the moment is so interesting that this might change. Continue reading

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A New Beginning

I have recently started as a DPhil student at the University of Oxford in the UK. The trip was a bit longer than anticipated, but the experience so far here has been incredible. My wife’s blog details many of the day-to-day experiences we have had in moving, so I will speak more to the research and academic culture here. Although I have to say that walking about five miles a day to commute has been a wonderful experience so far!

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Toward an Understanding of Mathematics

“Young man, you never understand mathematics. You just get used to it.”

-Von Neumann

I remember as an undergraduate, and even still to this day, trying to optimize any time I put toward learning new material. It is clear from the psychology of education that some methods of learning are better than others, and this is quite dependent on the individual. But questions always arise: when do I say I understand something? Is the kind of understanding I have of this object good enough for this purpose?

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