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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A Stroll Through Dynamics (and some Random Walks too)

Dynamical Systems is a vast area of mathematics, and one in which I have some small experience with a small bit of the research community. I won’t go into the overall field and all of its various subfields. Rather, I will concentrate on providing the interested reader some idea of what my Masters work was about. Modern research in many fields contains elements from a variety of areas of mathematics, so I will try and link to relevant concepts where possible. The reader is assumed to have some familiarity with basic mathematical analysis and elementary differential equations.

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Measurable Counterexamples

There are many important perspectives to take when trying to understand something. A very common theme in really understanding mathematical ideas is to conceptualize them as Theorems, and understand intuitively why the assumptions and criteria become useful conditions for the statement of a theorem.

“Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?” (Paul Halmos, “I want to be a mathematician”)

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