Maybe I should cancel brunch plans

One of the things I enjoy about Windows Vista is the slideshow gadget that comes with the OS. I have pointed the slideshow at my external hdd folders where most of my photos live. Hence this keeps playing back randomly selected thumbnails from my rather large photography set. This brings back rather nice memories at times.

Bloomington is rather cold today (its reached -14 degrees Celsius as of now) and I was thinking about warmer places I have lived in. As chance would have it, the slideshow brought up a picture of the Charminar in Hyderabad. I remember squinting to take this picture in the bright afternoon sun.

The Charminar, Renovations - March, 2005.

From http://en.wikipedia.org/wiki/Charminar

The monument was built by Muhammad Quli Qutb Shah in 1591 to commemorate the eradication of plague, shortly after he had shifted his capital from Golkonda to what now is known as Hyderabad[1]. Legends has it that the emperor Quli Qutb Shah prayed for the end of plague and took the vow to build a masjid on that very place. He ordered the construction of the masjid which became popular as Charminar because of its four characteristic minarets (possibly depicting the first four khalifs of Islam). The top floor of the four-storeyed structure has a masjid which has 45 covered prayer spaces and some open space to accommodate more people in Friday prayers. Madame Blavatsky reports that each of the floors was meant for a separate branch of learning - before the structure was transformed by the imperial British administration into a warehouse for opium and liqueurs.[2]

True to the legend, the city blossomed into a synthesis of two cultures. In 1591 while laying the foundation of Charminar, Quli prayed: Oh God, bestow unto this city peace and prosperity. Let millions of men of all castes, creeds and religions make it their abode. Like fishes in the water.

I just walked back home from Soma, the local coffee shop. Today is a cold day in Bloomington. I miss Cochin.

Spent most of the day brooding over over the fact that I have a lot of work to do which I am not doing because I am wasting my time brooding over it. Good ideas behave a lot like fear - in the right environment the fear is real and impossible to ignore. Category theory turns out to be an idea almost that good.

My category theory is acting up: It strikes me that many paradoxes can be explained away by the observation that there is no such thing as equality, the best you can have is isomorphism.

There are many profound discoveries that come your way as a phd student. I realized that I recently invented something of great strategic importance. The Junk Food box. Over time I realized that every now and then when I visit a less frequently explored corner of my house I discover junk food that I hadn't eaten. Most of the time, it would be very nearly expiring. Had I known it was there, I would have liked to snack on it. 

The junk food box is a great invention - its this box, a large cardboard box - that sits in my living room that has all my junk food. You should try it sometime - despite how simple it sounds, it works great. Now not only do I know where all my junk food and save me the trouble of searching, I have come to appreciate the fact that I don't waste as much.Also there is a quiet a bit more of variety immediately available. Plus you come up with all sorts of ideas. I recently mixed moong dal, cashews, cheetos and diced some onions in. Ah!

These are beginner level references for Category theory, ones that can be read by students of Computer Science. I haven't had a chance to look at all of this material myself. Ones that I have looked at have been marked with a star. Things that I think are better beginner material have more stars. At the time of this writing I am an absolute beginner in category theory and to most of abstract mathematics (with possible exception of Set theory), hence take my opinions here as just that - beginner level opinions. Finally, material that has been recommended to me by some of my professors has been marked with a +.

Categories
by T S Blyth (amazon) **

An Introduction to Categories in Computing
by Barry Jay (PS file)

Category Theory for Computing Science
by Michael Barr and Charles Wells (homepage of Charles Wells) (PS file of lecture notes) ++

Category Theory for Beginners
by Steve Easterbrook (PDF slides)

Elements of Basic Category Theory
by Alfio Martini (citeseer)

Computational Category Theory
by D.E. Rydeheard and R.M. Burstall (PDF)

A Categorical Manifesto
by Joseph A. Goguen (citeseer)

Basic Category Theory for Computer Scientists
by Benjamin Pierce (amazon) *+

Today evening, over pizza, Dr Hofstadter combined his Poetry class and his Group theory class and explained to the combined audience how Al-Khwarizmi visualized the solution to quadratic equations and how Tartaglia visualized the solutions to cubic equations. Using these simple visual models, he derived the solutions to the general form of the equations. He then went on to explain Tartaglia's 1534 AD poem that details his solution to cubic equations written in terza rima, the style of Dante Alighieri's La Divina Comedia. Needless to say, it was a fun class.

But what caused me to burst out laughing was this conversation I overheard where two students were discussing Group Theory and its application to Physics. Apparently there is a textbook by approximately that title and the university bookstore mislabeled it as "Group Therapy for Physics".

As I burst out laughing, something from my own ignorance struck me. Being somewhat culturally limited, I tend to think of some Chinese foods as "strange". Today morning I was trying to prepare some packaged Chinese food that I had purchased, when I noticed that the instructions on the cover said "Mix in the Soap Powder with constant stirring", and my instinctive reaction was "Oh, that's another strange thing that they'd eat". It was only several moments later that the awkwardness of of thing struck me. I went back and looked carefully at the instructions again. I had misread it, it really said, "Mix in the Soup Powder with constant stirring".

I am very moved by "Uncle Petros and the Goldbach Conjecture" by Apostolos Doxiadis. It's a work of fiction that makes several references to real people and real mathematics. It seems the author has done a fair bit of research to write this book. The book is about a mathematician "Uncle Petros" who spends his entire life trying to prove the Goldbach Conjecture.

In the course of the book, Doxiadis touches on the lives of G H Hardy, S. Ramanujan, Cantor, Kurt Godel and much of the pain, isolation, failure and achievement that is involved in doing research. The book is a fascinating read and is in many ways very true to life though the main characters are fictious.

The Goldbach Conjecture simply states that every even number greater than 2 can be expressed as the sum of two primes. Wikipedia has some more detail. The conjecture has been known for about 250 years now. There is still no proof, though the fact has been verified for very large numbers. It is a conjecture and not a theorem because it has no proof. Work still continues on this. Dare to try solve it? You may also enjoy looking at the Goldbach weave.

In the early 90's a long standing famous problem, Fermat's Last Theorem was proved by Andrew Wiles. It took him 7 years of exclusive work to prove the theorem. The theorem has been know since 1637! And has escaped all these centuries without proof despite many many mathematicians working on it. One of the things that made Fermat's last theorem famous is that Fermat has scribbled in the margin of his notebook a comment to the effect that he knew a proof but that the margin is not wide enough to note it. (Wikipedia) Wiles's proof is 150 pages and uses mathematics discovered in the 20th century - this is most likely not the proof that Fermat had in mind, if indeed he had a correct proof.

The interesting things about proofs like these are not just that they confirm the truth of statements that we always suspected to be true, but that they advance the state of mathematics. In the course of chasing hard problems, often new theories and new approaches to proof theory are discovered.