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With the help of my friend I have discovered the whole submanifold of people, writing on the stuff I am currently interested. The topic being geodesic PCA (or PGA). I was going through all the references from our groups (Johns Hopkins, Utah, Paris etc groups) and digging. Then a friend forwarded me a paper, which uncovered the whole trove of papers that I haven’t discovered yet. There is a lot of mutual linking in that new submanifold, but very few links cross over to the subset of papers I was reading initially. Hence the difficulty in finding.

Of course, a google scholar search of “PGA” would have revealed them eventually. And in retrospect one can say “aha! here is the reference! how did you miss it?” But I guess before the AI is invented one has to carefully parse through all possible links on the topic. We are lucky to have google. It is difficult to imagine how one would search for an article before all the digitization.

It would be interesting to set up a Mathematical Picture of the Day website.

Interesting and troublesome, it would require a lot of work to come up with exciting and varied content.

I am inspired by the Astronomical Picture of the Day, APOD.

The images there are magnificent, descriptions are interesting and content is stunning in many ways.

I feel, the MPOD version has to be user-generated, with a few editors approving the postings. Maybe it has to be MPOW, i.e. a weekly affair in the beginning. Mathematics could be as visually stunning and revealing as the astronomy.

Maybe this is a project for my another mathematical life đź™‚

Cheers.

PS I stumbled upon this MPOD site, but the author seems to have stopped updating it. But the beginning looked good.

Nice article “The Mighty Mathematician Youâ€™ve Never Heard Of” in the NYTimes on the occasion of the 130th birthday of Emmy Noether. I remember being stunned when I learned that Noether was a she. There are not that many female mathematicians.

There is an attempt at explaining the Noether’s theorem for a layman:

Wherever you find some sort of symmetry in nature, some predictability or homogeneity of parts, youâ€™ll find lurking in the background a corresponding conservation â€” of momentum, electric charge, energy or the like. If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noetherâ€™s theorem, youâ€™ll see itâ€™s angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move.

I have to say, I do not find this convincing. I wonder if there is a more intuitive and more direct way to explain this theorem for a layman? That is an interesting thing to ponder about.

P.S. On a more scientific note (and to get back to the research part of this blog :), Noether’s theorem is at the heart of the conservation of momenta on the groups of diffeomorphisms. I am using this conservation (and people have used it many times before, for example, Statistics on diffeomorphisms via tangent space representations) to study statistical variability of the hippocampus dataset. The paper writing is in the progress.

Lowering or raising indices or, in other words, canonical isomorphisms between a tangent bundle and a cotangent bundle are called musical isomorphisms.

I will post some reviews of books that I find useful in the course of my study. I am not sure how technical I want my reviews to be or how general and vague.

For the time being I’ll try to do both đź™‚

The first book I would like to review is the book “Shapes and Diffeomorphisms” by Laurent Younes, Johns Hopkins University.

This book summarizes a lot of work done by the author at the Center for Imaging Science, where Michael Miller’s group does a lot of interesting and exciting work on medical images.

This is a wonderful book for anyone who would want to learn about shape representation and matching. Very thoroughly written. It starts with curves, how they could be represented, then moves onto surfaces. Talks about Euler-Lagrange (Euler-Arnold) equations on the groups of diffeomorphisms which is a fundamental equation for all the Computational Anatomy, or pattern matching in general. A wonderful overview of methods and techniques that are in use today. Author also discusses numerical implementation and issues that arise in the field.

Sometimes I find notation a bit too heavy, with many sub- and super-scripts and names for variables that are not intuitive (to me, at least). In particular in the Diffeomorphic Matching chapter (Chapter 11) first a general construction is presented and then it is demonstrated on several examples. The general notation is reused in the examples which I find a bit cumbersome. One can be better off rewriting a specific construction in the landmarks case introducing his own notation. After moving on to the general case it seems to make more sense. At least that’s how it worked for me.

Later I am going to post a computation, which is just an expanded proof of a theorem from the book.

I recommend this book to anyone trying to learn about Pattern Theory, and in particular the emerging applied discipline, Computational Anatomy.

P.S. If your institution is subscribed to Springer publishing house, then you can view this book for free online.

My research is related to the TeichmĂĽller theory.

This is an interesting piece of trivia that I’ve found about Oswald TeichmĂĽller from wiki:

TeichmĂĽller was a passionate Nazi, joining the NSDAP in July 1931 and becoming a member of the Sturmabteilung in August 1931.

Upon personal authorisation from the FĂĽhrer, he joined the Wehrmacht in 1939 and was killed in fighting on the Eastern Front.