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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.

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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.

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.

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.

Had a great time attending the Shape FRG meeting in London.
Interesting trivia fact. The idea of using Teichmuller spaces as the way to parametrize curves
(as in paper “2D-Shape Analysis Using Conformal Mapping”, pdf) was suggested to David by Curt McMullen. Although David worked on moduli spaces he was not aware of the definition of Teichmuller spaces. After Curt McMullen explained it to him, David decided to parametrize shapes this way.

A few interesting quotes.
Comparing the role of Catholic Church in Europe and Chinese bureaucracy in the development of sciences David noted: “As well known, computing the solar and lunar eclipses is a bitch!”

David is always careful in using proper dimensions of quantities and proper scales. It is surprising to see such care in a mathematician of an amazing caliber like David. It appeared in the metrics he developed and in his remarks at the conference.
David: “I would suggest you to state explicitly which kernel you used and the size of it in all your publications. This way you leave a paper trace. This is called the scientific method.”

And David was really happy when a student told him that the axes in his 3D image were millimeters.
David: “In Nature and Science they always show the scale! Always!”
Laurent: “But this is just an MRI image with 1-by-1-by-1mm voxel size. So in this case it coincides with millimeters. But in general nobody cares.”
David: “Shit!”

In my last Differential Geometry class I have introduced my students to the Calculus of Variations. This is an important topic when one wants to minimize anything, especially when one wants to find geodesics. There is a very neat fact that I came across again and I wanted just to write it down.

In general for the reader interested in Differential Geometry I would like to refer you to the nicely written Differential Geometry notes by my friend Mario Micheli.
He is currently at UCLA. The notes are available here and here. The lectures with easy introduction to Calculus of Variations are 26, 27, 28.

I will be very brief in describing the nifty little thing. So the question is “find function $q(t)\in C^1[a,b]$ with fixed points, i.e. $q(a)=y_0$ and $q(b)=y_1$, such that this $q(t)$ has the graph of minimal length”. Everything boils down to minimizing the functional
$J(q) = \int_a^b \sqrt{1+\dot{q}(t)}dt$
with respect to function $q(t)$.

So the interesting bit comes when one computes the gradient of this function. One gets (again, for details refer to the links above, or do the computation yourself. “It is an easy and helpful exercise, muahaha”):
$\nabla J(q) = -\frac{\ddot{q}}{(1+\dot{q}^2)^{3/2}}$.

If one imagines the simple gradient descent algorithm on any chose path between points $(a,y_0)$ and $(b,y_1)$, then one can see that the steepest descent will have to follow in the direction $-\nabla J(q)$. That means that where function $q(t)$ is concave, then $\ddot{q}<0$. Therefore the steepest descent will be flattening the concave part. Similarly, it would be flattening the convex part of the function, where the second derivative is positive. Also, the more concave (convex) the function is, the bigger the gradient, thus the function gets pushed to the straight line very quickly.
(As you have guessed straight line is the solution to this problem).

It is very good exercise to implement steepest descent for this problem. And then follow it with conjugate gradient and the some quasi-newton method. That’s how I learned these optimization techniques. Cheers.

In the class I am teaching I tried to count number of independent components of the Riemann curvature tensor $R_{ijkl}$ accounting for all the symmetries. We’ll call it RCT in this note.
It turned out to be not so straightforward, so I decided to write it down here.
First of all, here are the three symmetries/skew-symmetries of $R_{ijkl}$:
$R_{ijkl}=-R_{jikl}$,
$R_{ijkl}=-R_{ijlk}$,
$R_{ijkl}=R_{klij}$.
The first two expressions tell us that $R_{ijkl}$ is skew-symmetric in the first two and the last two indices. And form the last we can deduce that $R$ is symmetric in the pairs $ij$ and $kl$.
Read the rest of this entry »

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

In this blog I’ll be posting about mathematics, my thoughts regarding my research and some math related trivia. Everybody is welcome to comment and contribute.

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