Quotes about Mathematical Modelling

Below is a list of quotes I have come across related to modelling, widely interpreted. I will try and update this as I come across interesting statements. Please do send me things I have missed here (errors, quotes to consider, your own comments about the place of theory and modelling etc).

Also see: The Unreasonable Effectiveness of Mathematics in the Natural Sciences

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Dynamical Phase Portraits

So this is a short tale about how stumbling upon two curious ideas led me to a cathartic afternoon of making some pretty videos. Skip to the bottom for some pretty animations unless you’re keen to see where the ideas and tools came from.

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Theodicy

“Do not be daunted by the enormity of the world’s grief.

Walk humbly now. Do justly now. Love mercy now.

You are not expected to complete the work, but neither are you free to abandon it.”

-various religious sources including the Pirkei Avot and Micah

Theodicy is an attempt to resolve the problem of evil: how could a good God permit so much evil in creation?

Theodicy is hard. It is also important, irrespective of one’s religious beliefs. Theodicy helps us think about good and evil in general and concrete terms, as well as suffering and our obligations to alleviating it.

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An Interesting Year

This year has been an exceptionally productive one, partly due to many things from the previous few years accumulating, and partly because it was a long year of hard work. I formally completed my DPhil at Oxford (though my Thesis was submitted last year). I had 13 papers accepted for publication, and several more have been submitted. We made several trips to Edinburgh, Cambridge, and Cardiff to meet with collaborators, as well as St Andrews for a conference. Finally, we have spent the last part of the year back in New Mexico, and will return to Oxford in just under two weeks time after a long break (as we haven’t visited home in three years). Many of these things were the culmination of last years’ work, though some new projects and collaborations occurred within this year.

I am incredibly grateful to many people for having made this year so successful and productive, and I look forward to another year of doing science, as well as teaching and mentoring students. This has also been a good year for the latter, as I’ve co-supervised a number of projects in various areas, and will continue to do so for the next few years. I’ve also had the opportunity to do a small bit of public outreach in the form of two “Research Case Studies” discussing work from this year: Constraining Nonequilibrium Physics and Following up Turing. Blogging, on the other hand, has fallen by the wayside as I’ve only posted once this year before today. I did manage to draft about eight other posts, but it may be some time before I complete any of these.

Scientifically I’ve been very fortunate to now have several collaborations and many questions to pursue over the next few years. These collaborations range from projects with experimentalists working on mice models at the Roslin Institute (known for cloning Dolly the sheep) and recent work on E. Coli with Microsoft Research, to several different theoretical projects. Most of my research is concerned with patterns in time and space, and fundamental models of how such things emerge from simple dynamical rules (which can ideally be related to biological or physical processes). Going forward I hope to learn as much as I can from my experimental colleagues in order to really get a sense of the biological and physical processes, and to use this insight to think of new ways of understanding complex phenomena. I have been increasingly interested in statistical physics, and non-equilibrium processes more generally (both in terms of transients of dynamical systems, and ways of understanding systems driven away from thermodynamic equilibrium). There is a huge amount of work I have yet to absorb in these fields, but I find the questions exhilarating.

Research has and will dominate most of my energy for the next few years, though I do want to do some blogging or other “outreach” kinds of things as I move forward. Finding the most meaningful medium is a bit difficult, but I do think such things are valuable. I plan to have a post about pedagogy, as well as a statement of teaching philosophy, posted sometime next year (which may be in the spirit of the Lament, though with perhaps a more positive outlook). Later on I plan to properly teach myself some areas of mathematical physics and dynamical systems, and may post notes here as a way of absorbing things for myself. I have been very lucky in my endeavours so far, and I hope to maintain a good commitment to returning that investment.

 

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Some Surprising Things

Below are two surprising mathematical results which I found interesting recently. They kind of demonstrate some really fascinating ‘arbitrarily complex’ behaviours of very simple objects, and are primarily useful for philosophical grounding in terms of understanding tools used in science, such as data fitting and mathematical modelling. Continue reading

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Some Projects from 2017

This has been an incredibly busy and hectic year, but overall an extraordinarily rewarding one. I will catalogue some of what I did here in case anyone is interested, and point out some of the things that will hopefully happen early next year. I do not know how often I plan to blog going forward, but I think having at the very least one reminiscence a year about things that I have been involved in is useful.

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Numerically Approximating Conformal Maps with the Zipper Algorithm

Update: A commenter points out some errors in the terminology used. Additionally, there are now much better codes available for zipper-like algorithms in Matlab, as well as Julia implementations.  Additionally there are codes for the circle packing algorithm which someone may find useful.

As with the previous post, I want to mention some other work I did related to my DPhil course. This project was about implementing an algorithm to numerically compute conformal maps between simply connected domains and the unit disk. By the Riemann Mapping Theorem we know that for every such domain, a map must exist, but finding it analytically is rarely easy if possible at all. A good discussion of these ideas can be found, for instance, in Terence Tao’s lecture notes on the subject. Given a domain, there are several ways to numerically construct a map which approximates the conformal map guaranteed by this Theorem, such as by approximating the domain by polygons and using Schwarz-Christoffel maps to map it to the unit disk.

I implemented the Geodesic or Zipper algorithm due to Don Marshall. The full code (in Matlab) can be found here, along with a few graphical tools to explore it. It has some precision problems for domains with “sharp” boundaries for the inverse map (that is, mapping the unit disk to a given domain) but overall I’m pretty happy with the results. I did not include all of the approximations one could to make the algorithm both more accurate and more efficient, but I may come back to this later and do that, as well as demonstrate some of the applications of these maps. If you’re interested, I would also check out this Thesis on the topic which I found to be useful. It also contains Python code for this algorithm as well as another approach using sphere packing. Below I will include a few examples of what this code can generate, but I encourage you to download it and play with it yourself. As always, comments and questions are appreciated! Continue reading

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