Ganesh Swami

Some random math facts that I came across last week.

Poincaré group

The Special Euclidean group combines rotations and translations in Euclidean space. The SE group is an isometry, which means it preserves angles and distances.

In special relativity, the 4-vectors are three space variables and one time variable. Rotations in this space can be generalized to the Poincaré group. Instead of having sines and cosines in the rotation part as in SE, they have hyperbolic sines and cosines.

The norm of a vector is defined through its inner product:

which in special relativity is

A friend pointed out that the Lorentz transformation was only recently proved to be linear.

Geodesics and Metrics

One of the most beautiful things I learnt in the last few weeks is the connection between geodesics (shortest paths) and metrics in that space. If is a geodesic path and is the velocity of the path:

then the metric is defined by

Moreover, the path satisfies the Euler-Lagrange equation. This was first shown by Arnold in a hydrodynamical context, and subsequently borrowed by control systems and computer vision folks.

A Poor Joke

Two functions and were going in a car. looked ahead and said, “Oh shit! There comes a differential operator.” says with a smirk, “It can’t do nothing to me!” On approaching the differential operator, it says “Haha! I’m d/dy.”

As narrated to me by a random math major last week…

I just came back from four days of non-stop math action at the Canadian Undergraduate Math Conference. It was well attended for an undergraduate conference: we had about a hundred people, with only eight from SFU.

The opening keynote speaker was by Peter Borwein on the Riemann Hypothesis. The Riemann hypothesis is a conjecture about the connection between the zeros of the Riemann-Zeta function and the distribution of prime numbers. The first quarter (being generous) of the talk was interesting to me: large computations, experimental mathematics and the millennium problems. The latter part of the talk quickly jumped into complicated number theory and other things I’ve never seen before. An interesting note is that the Riemann hypothesis has been tested to about 1.5 billion numbers.

The second keynote was by Frank Morgan on Soap Bubble mathematics. Keeping volume constant, the shape with the minimal surface area is a sphere. With more than one bubble, figuring out the shape of the bubbles is non-trivial. Another interesting fact is that it’s more efficient if the highways meet at 120 degrees instead of 90 degrees as in crossroads. The net distance that you’d have to travel is reduced.

The third keynote was by Leah Edelstein-Keshet on Mathematical Biology. The math part of the talk was interesting because I was familiar with some of the pattern formation PDE models that you see in nature as in Zebras and Giraffes for example. The model is based on Reaction-Diffusion systems, which I had studied in the Ginzburg-Landau equation. The Ginzburg-Landau equation also forms patterns when carbon monooxide diffuses and reacts on Platinum surfaces.

I’ll be posting details about my talk on Fluid Image Registration soon.

My experiments with Python continues on. This time it’s playing with the numerical and plotting capabilities of Python.

The shortest path between two points on a plane is the straight line connecting the two points. There’s a way to derive the equation of the shortest path using variational calculus. This experiment derives such an equation by doing a high dimensional gradient descent on the space of equations that satisfy the Euler-Lagrange equation.

We are looking for a curve that minimizes the arc length between the two points. We define y(x) as the curve, y_x as the x differential of y.

The corresponding Euler-Lagrange equation is

The plot below was generated by matplotlib, a plotting library that is almost exactly similar to the Matlab plotting interface. I was surprised at how close Python code can be to Matlab code. Nowhere in my code do I have to loop over the dimensions of the vectors. The numerical calls are passed over to high performance math routines written in Fortran. The initial condition was a sine wave and you can see that it eventually converges to the true solution.

Get the code here.

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Jakob Nielsen has put it down quite nicely. This explains why posting has been sporadic lately — it’s significantly harder to write original content. I usually do not have the time or the energy to write in the evenings. Weekends are spent living my “real life.”

This is going to be one of those meta posts, with the only common link between the topics is that it has to do with me.

Firstly, I’m beginning to appreciate Vancouver more than ever. On Canada Day, my family and I went to the Capilano Suspension Bridge. There’s a fair amount of history behind the bridge. The tour also covered some of the First Nations history. This might be controversial, but I think that by accepting special rights and privileges from the Canadian government, they’ve distanced themselves from the mainstream. For someone like me who is genuinely interested in learning about different cultures, it makes it harder than usual. Contrast this with St. Patrick’s Day, Bisaki or the Chinese New Year celebrations which attracts a few hundred thousand people. If you live in Vancouver, you’ll know exactly what I’m talking about.

Secondly, I’m giving a talk at the Canadian Undergraduate Math Conference. As usual, I was only able to register a few minutes before the deadline, so I didn’t have much time to think about what I wanted to talk about. I registered with a very generic abstract on fluid image registration. There are many approaches I can take in the talk (as a physics guy with an engineering major presenting to math majors on applications to medical images.) More on this in a later post.

Lastly, if you’re looking to get a comprehensive understanding of math (or mathematical physics,) check out Penrose’s Road To Reality. It runs about 1100 pages, but it goes through all the math from the ground up for a general audience. It starts with all the way from the Pythgoras theorem, complex numbers, calculus, Fourier decomposition, group theory to more complicated stuff in theoretical physics. A good read.

That is all.