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A nice write-up by Leo Brewin of different quantities (metric tensor, geodesic, parallel transport, law of cosines) in terms of normal coordinates.


Here is a thought of the day: in general a principal geodesic component does not pass through the intrinsic mean on the manifold. Which is not the case in the usual Euclidean PCA. The reference is the article by Stephan Huckemann. I think it is worth to carefully go through it.

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.

Nice article “The Mighty Mathematician You’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, you’ll find lurking in the background a corresponding conservation — 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 Noether’s theorem, you’ll see it’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.

If you are in the job market for an academic job or do not have ideas to put down in
your teaching statement, here is a nice summary of misconceptions about student learning.
(this advise is actually hurting me, cause I will have more competition, haha)

To apply this knowledge is another matter. I don’t know how these facts would help in teaching
math classes (calculus, differential geometry etc). I guess one just has to try…

A great article, Don’t Lecture Me: Rethinking How College Students Learn, about peer-teaching. Resonates a lot with my thoughts.
I encouraged students to explain each other problem sets.

But here’s the irony. “Mary is more likely to convince John than professor Mazur in front of the class,” Mazur says.

“She’s only recently learned it and still has some feeling for the conceptual difficulties that she has whereas professor Mazur learned [the idea] such a long time ago that he can no longer understand why somebody has difficulty grasping it.”

I have encountered a very nice way of referencing in one paper. Along with the usual way of citing a paper in the body of the article there is a back reference. In the bibliography after each paper there is a page number where this paper has been mentioned. I find it very sexy.

The thing that annoys me with references is that one cannot see the reference right next to the place where the author is mentioning it. While reading on a computer this issue could be solved: you click on the references, jump to the bibliography, and then can get back through the “back” button. This approach is still buggy though. But when one reads a physical print-out there is no ‘back’ button, it is very inconvenient to jump back and forth. When somebody will come up with the solution to this, she/he will earn my most eternal gratefulness.

P.S. I keep telling myself I should post more often here. We’ll see.

Lowering or raising indices or, in other words, canonical isomorphisms between a tangent bundle TM and a cotangent bundle T^*M 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 🙂

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.

Here are the slides of the talk “What’s an infinite dimensional manifold and how can it be useful in hospitals?” that was given by my adviser, David Mumford, at the University of Coimbra, in September 2007.

It should give you a general idea of what the field is about. The field being Computational Anatomy, or Pattern Theory (a branch of it). Some familiarity with differential geometry concepts is expected, but one can just look for some nice pictures and graphs. My favorite is the multiple image of a galaxy due to the gravitational lensing. This is probably the most powerful and the most direct observation for the curvature of our space-time.