Ganesh Swami

One of the best things of an open platform is the community that develops around it. The huge community around Wordpress has built a number of plugins that extend and enhance the base installation in many ways. I’ve listed a few of my favorite plugins that I use on this site. Hope it’s useful.

Optimal Title

Wordpress by default prefixes the title of your blog posts by the name of your blog. Now, this isn’t very search engine friendly. Using Optimal Title reverses the order of your title to have the name of the post at the beginning followed by the name of the blog (this is customizable.)

Image headlines

No matter how much time I spend at the typography at this site, it simply doesn’t look as good as static pictures. Image Headlines is a plugin that converts your titles to beautiful, lick-able images. These are cached, so you don’t have to worry about needlessly hitting your web server.

LaTexRenderer

LaTex is the defacto inputting method for physics and math, so much so that even Microsoft is implementing something along these lines for the next generation of Office. The LaTexRenderer plugin implements a nice wrapper over mimeTex. I’ll save my discourse on the virtues of LaTex for later.

Markdown

I use emacs to write my posts, and do so in plain text. Markdown converts specially formatted text to valid XHTML. I think this plugin is included by default with Wordpress. This lets you spend more time with the content rather than bothering about how it’ll look when published.

Google Sitemap

A plugin to automatically update the Google compliant sitemaps for you. This makes the search engine crawl process for your site more efficient. One less thing to keep in mind.

Related posts

This plugin lists related posts at the end of each post. This is a great way to reduce the number of “exit points” on your site. In the future, I intend to write a small script to visualize how people jump from post to post as a graph (with normalized weights for the edges.) Highly recommended.

Footnotes

One feature that I missed from my previous blogging software was the use of footnotes. This plugin lets you do that.

Acronyms

This plugin will automatically put the expanded form of an acronym as a tooltip.

Time Since

This plugin will replace the date with the time that has elapsed since that date. I find this easier to read than just the date.

I was invited by the Computational Anatomy group at the Medical Imaging Analysis Lab to see a PBS documentary on Fermat’s Last theorem. Fermat, a lawyer by profession, was also a prolific mathematician. He left behind one of the hardest problems in mathematics.

Mathematically, his theorem stated that for the equation:

is unsolvable for the case n>2 (a, b, c and n are integers.) For the n=2 case, the equation is simply the Pythagoras theorem. He also left behind a note which said that he had a proof for statement, but the margin was too small to contain it.

The solution to the problem was done by Dr. Andrew Wiles from Princeton using 20th century mathematics. It’s highly unlikely that Fermat had thought of the same proof.

The key to his proof were in the connections between elliptical curves and modular forms. Elliptical curves are equations of the form:

The equations are non-singular (don’t intersect.) Cryptography based on elliptical curves have been proposed to replace RSA because the problem is fully exponential, unlike prime factoring which is sub-exponential.

Back to our story…

Right after the second world war, two Japanese mathematicians Shimura and Taniyama proposed a conjecture that elliptical curves are just modular forms in disguise (known as the Shimura-Taniyama theorem.) Think of modular forms as highly symmetric functions.

Decades passed without anyone making any breakthroughs in proving the Shimura-Taniyama conjecture, but that didn’t stop people from developing math atop it. Then, in the mid-80s, Dr. Kenneth Ribet from UC Berkeley made a connection between the conjecture and Fermat’s theorem.

His argument is as follows: If somebody were to find a set of numbers that satisfies Fermat’s equation, then this set of numbers could be used to construct an elliptical curve that is not modular, thus disproving the Shimura-Taniyama conjecture.

Working backwards from the graphic, if the Shimura-Taniyama conjecture is true, then all elliptical curves are modular, thus with no solution to Fermat’s equation, making it true.

The rest of the documentary was about how Dr. Wiles took on the challenge to prove the Shimura-Taniyama conjecture. Solving Fermat’s last theorem had been his life-long ambition. He worked for many years in isolation and when he finally proposed a solution, it was found to have a mistake. Disappointed, he re-traced his steps and proposed an alternative argument to the solution. He had solved the hardest problem in math and become a part of history.

I think this documentary was more about ambitions, dreams and aspirations of a person than the mathematics. Dr. Wiles took upon this monumental task of achieving his childhood dream. Fame, money and prestige were unimportant to him.

In our discussion that ensued, it was clear that major breakthroughs can no longer happen by one person alone. One of the scientists in the documentary read out a list of at least 30 people who directly contributed to the proof.

This was the first week in about four or five weeks that I didn’t have projects, labs, reports, or anything else related to school that I had to work on during the weekend. I borrowed a bunch of movies and watched them over the weekend:

The Da Vinci Code

I wanted to read the book first before watching this movie, but I just couldn’t get to it. Pros: I really liked this movie – it had all the elements that I was looking for. I probably would have enjoyed it more if I had a background in Christianity, but that didn’t stop me from doing some research on the internet after watching the movie. Cons: The subtitles on the DVD I had didn’t work for some reason, and it was hard to simply stare at the screen when those guys were talking in Latin or whatever it was. And, the movie was really long.

Mission: Impossible III

This movie left me disappointed. This seems to be the general trend amongst sequels – Final Destination, Pirates of the Caribbean and the Matrix just weren’t as the original was. Can somebody tell me what the significance of the rabbit’s foot was, apart from it being a nuclear hazard? I was expecting something to happen with the rabbit’s foot, when instead they started showing the credits.