Well, it’s been a while. I can imagine it is nice for students to do relevant and interesting HW. An idea for the Taylor Series HW for the Calculus class would be William Shanks’ computation of $\pi$, which is the most accurate approximation done by hand (527 digits of $\pi$). It’s a good exercise for Taylor series and has a nice historical background.

Inspired by the Quora post.

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.

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 🙂
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 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.

I have stumbled upon a talk that David gave at the International Congress of Mathematics in 2001 titled
“Trends in the Profession of Mathematics”. The first quote reflects David’s style a lot, as I remember him giving lectures and talks:

“I think mathematicians have a special problem in making new ideas accessible to their colleagues, a problem that is tough but not unsolvable if they only recognize it more honestly. It is our obsession with seeking to express each new result in its greatest generality! […] But do we want to live in the house that Bourbaki built? I want to express a radical alternative that I learned from Sir Michael Atiyah. His view was that the most significant aspects of a new idea are often not contained in the deepest or most general theorem which they lead to. Instead they are often embodied in the simplest examples, the simplest definitions and their first consequences. […] The most important message is often contained in a few simple, but profound observations which underlie the whole rest of the theory.”

This is something that I aspire to do and hopefully accomplish some day. As my friend told me once after the first encounter with David at the conference “his comments are simple, but deep at the same time.”

And the second quote tells a bit about applied mathematics:

…there is a teeming cauldron of phenomena present in the world asking for clarification and analysis. One tries to snatch out of this cauldron some specific things which lend themselves to a precise analysis. This can only be done by radical simplification but it _must_ preserve the essence of some aspect of the complexity of the full rich situation. I think mathematics can benefit by acknowledging that the creation of good models is just is significant as proving deep theorems. Of course, for a model to be good, you must show it leads somewhere: this may be done by mathematical ‘experiment’, i.e. by computations or by the first steps in its analysis. PhD’s, lectures and _jobs_ should be awarded for finding a good model as well as proving a difficult theorem.

I suggest to all of you to read the full talk.

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.