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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.

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.

My colleague and friend, Mario Micheli, has co-authored (with P.Michor, D.Mumford) a paper “Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks” (arxiv). This is a landmark (sic!) paper, that expresses the curvature of the manifold of landmarks in terms of the cometric, which is a much simpler formula (but by no means simple) than the expression in terms of the metric (cometric is the inverse of the metric).

Another interesting part is that authors decomposed the curvature into four terms, thus attempting to describe what exactly makes curvature positive/negative. The decomposition is given in physical terms (strain, force, compression). Armed with a strong physical intuition one might be able to decipher what exactly makes the curvature non-zero.

A more general paper on the expression of curvature in terms of cometric is coming out later, as I have been told by MM.

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.

A nice post by wonderful Terry Tao about the Arnold formalism.
Sums it up in a concise way, where everything is in one place.