Saturday, May 22, 2010

A presentation and some introduction.....

Yesterday we had our MS project presentations, thats the last requirement for one to officially be a 'Master of something ' at IISc, I am still trying to find that out! Thought I would write about something that I said...
I studied some algebraic number theory as part of my Masters project, and proved a series of exercises that lead to the proof of what is know as the 'Kronecker-Weber theorem'.
It was a wonderful morning, in fact, it hardly dawned on me that it was morning when I went to sleep!!! So I got myself ready, and cycled to the department. It was 6:45 am, this was the first time that I reached the department so early (I have gone back to my room at this time from the department many a times though :P).
And then did the last minute preparations- check my slides, read my notes and blah blah blah.
My talk was scheduled at 10:10 am. The department had played it safe to ensure that there was a sizable audience, in that it had promised a delicious lunch to all those who attended the talks.
I made my last minute calls back home(with the number of exams that I have to answer reducing drastically, I do not miss an opportunity to let folks at home wish me 'best of luck').
So finally the moment arrived. I motioned to the screen, and began the jargon...
I mentioned before, I spoke on algebraic number theory. Here is some part of the my introduction -

The subject developed due to a rather silly comment of mathematician Pierre Fermat in the margin of Arithmetica by Diophantus as follows -I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.
 Fermat claimed that he had an elegant proof to prove the above fact. Unfortunately he was dead before one could publish books with fatter margins!!! The search for this coveted proof has lasted lifetimes of some of the best mathematicians in the past. This was notoriously labeled as the 'Fermats last theorem'(FLT). It was proved in 1994 by Prof Andrew Wiles at Princeton university. The proof is hardly elementary, it is about 290 pages long and uses some of the most sophisticated modern algebraic number theory. It definitely needs a lot more than a margin of a book.

However the alarming simplicity of the statement generated the most number of 'wrong proofs' for this problem in the history of mathematics( my attempt included-my undergraduate professor almost fainted when I told him that I was trying to prove the 'Fermats last theorem'!!!). In fact it was this theorem that in a way got me exited about mathematics. Very often it is difficult to prove some of the most simple questions in our lives. By simple I mean, ones with really simple statements, so simple that even a five year kid could understand them. Mathematics is flooded with such questions, and the FLT is a prime example.
The basic reason why many of the earlier proofs collapsed for due to the assumption of unique  factorization in certain domains. Mathematics in general, and number theory in particular has a predominant inductive reasoning, in that I mean that, one proves something for something small and finite, and then extrapolates it  to something much larger. A similar reasoning was applied to the set of integers. Integers have this marvelous property called the 'Fundamental theorem of arithmetic', which asserts that every integer can be written as a unique product of prime powers. One should note that this theorem(like many other useful theorems) has a dual nature, that is it not only talks about an existence of factorization but also guarantees its uniqueness. Uniqueness is a very important property, if something can be done uniquely, it means that it is independent of who does it. It also helps one to come up with important formulas(by computing something in two different ways and then equating them). This general principle was assumed for domains containing the integers, and this assumption was wrong. The moral of the story is that whilst proving something, check that the assumptions are proved to be correct-otherwise you might end up with absolute non-sense.

Algebraic number theory was developed to answer questions which generalize properties of integers to bigger and general domains... from here on the talk becomes a little technical and hence cannot write it up here...
I have still got to write out the report though, so will pushoff for now, till then all the best...may be you could start writing out conjectures in your textbooks, who know we might soon have a 'last theorem' after your name!!!

Monday, May 10, 2010

The God particle...and blah blah blah...

Couple of days back I attended a talk by Prof B V Sreekantan on 'The LHC and the God Particle'. For those who do not know- LHC stands for 'Large Hadron Collider', the biggest particle accelerator ever built. A logical question to ask is 'Why the hell does one need a particle accelerator? leave alone the biggest. I hope to share my understanding of the subject through this post.

The basic question that every human being wants an answer for is sometimes as follows: Why am I here?
just four words long yet this questions has made the scientific community slog for centuries. It has made theists out of atheists and vice versa. It has had profound implications on philosophy, the way we think. In fact it has made us question the very reason of our existence.
The biological front has thrown light onto this through the theory of evolution perceived by Charles Darwin, which besides  being a theory does have a lot of physical evidence.
The physical front however just has a theory, and the onus is on the LHC to provide the necessary evidence!!!

Getting to business, let me clarify that I am not an expert in this field( not that I am an expert in anything else!), so do not rely solely on  what I say. If I have aroused enough interest do find time to enlighten yourself better.

The LHC is a technological marvel. The International space station is the only other experiment that matches the magnitude of this project. Its cost so far is about $5 billion(US), and still growing, there are about 10 thousand scientist all over the world working in coherence at any given time. In terms of physical dimensions, it is 27km long, uses 1232 dipole magnets, 1600 superconducting magnets and 27 tonnes of liquid helium, and is about 175 meters beneath the surface of the earth. It generates terabytes of data in seconds, leading to the development of the high speed network called The Grid. The enormity of the entire project transcends all human imagination.

 Whats the deal, why do we even need it? A fundamental question in modern physics has been to find the most basic particles in the universe, basic means- smallest and indivisible. It so happens that the universe is not only about particles but also forces, and hence the real search is to find the most basic 'particles and forces', from these everything else evolves, and hence they are called fundamental particles.
Physicists have come with 'The standard model' for the universe and according to this model there are about a dozen fundamental particles(electrons, quarks, leptons,...) and four fundamental forces (strong nuclear, electromagnetic, weak nuclear and gravitational). The fermions are supposed to be the ultimate constituents of matter. This theory is pretty solid, and answers most of the questions, however the problem with this theory is that it does not answer why some particles are heavy while others have no mass at all(you see we need a theory that answers all the questions). The answer may be the so-called Higgs mechanism. According to the theory of the Higgs mechanism, the whole of space is filled with a ‘Higgs field’, and by interacting with this field particles acquire their masses. Particles that interact intensely with the Higgs field are heavy, while those that have feeble interactions are light. The Higgs field has at least one new particle associated with it, the Higgs boson(this was infamously called the 'God particle' by Prof Leon Lederman, director at Fermi labs, Illinois, one should read his book 'The God Particle' ). If such a particle exists, experiments at the LHC will be able to detect it. The existence of such a particle  will answer questions related to the Big bang theory and the creation of the universe.

 It is believed that the universe has a dual nature. For example corresponding to something hot there exists something cold, if there is light there exists darkness, then newtons third law says that every action has an equal and opposite reaction and so on. One could then ask, since there exists matter, does there exists something called anti-matter? The answer is believed to be in the affirmative, and again we rely on the LHC to get some answers.   

 However there are doubts whether the experiment will be a success. Many physicists feel that creating conditions similar to that at the time of Big bang is virtually impossible. Presently the accelerator is running at low energies, at higher energies, they claim that the accelerator will give rise to mini black holes(objects with extremely high densities) which would cause the earth to collapse onto itself, hence bring an end to this planet and in fact the entire solar system. The desire to know how it all began might just lead to a premature end!!!

Towards the end of the talk, we were told that what we see is just 3.5% of what is there, about 70% is what is called the Dark energy and another 25% of Dark matter( seems like we are almost totally in the dark!!!). The physics that we develop here, is subject to only 3.5% of the universe, most of which is beyond experimental reach(since what we see is light years away), and most of the theories, including the 'Standard model and insanely extrapolated to all of it.

The problem lands into philosophical and theological realms and deeply entwines with the question of the existence of God. Are we trying to grab more than what our hands can hold? It would be in Gods best interest that we know the universe that He has created. 'Ask and you shall be given, knock and the door shall be opened, search and you will find'-yes we shall search with faith that we will find the answers someday. Till then carry on with research...

Saturday, May 1, 2010

Run maadi run.......Science and technology go for a run!!!

Hi, its been a long time, I am sure you guys missed me :P, anyways had a lot of activity happening. I was sick for sometime and then we had our end-semester exams, which went on for almost a fortnight(more about it sometime later :) ).
On the 10th of May we had the 'Science and technology run'. This is a time when the outside world tests how fit people in IISc are, they are surprised not only by our fitness but also the enthusiasm that some guys show!!!
It also surprised me, in fact there was a substantial crowd from the math dept, which defied the opinion about math researchers that many have.
The run started at 7:50 on a beautiful Sunday morning, before that, one had to register and then collect the t-shirt(a complementary gift which nobody wants to forsake). The marathon is open to the local community around IISc and also to the corporates(most of whom had dressed well with 'just do it' from head to toe-just made one feel that life will be rosy once we leave IISc, soon).  This time the organizers had done a marvelous job, in that they had put up the route on display a week before the run, the publicity done well and they had live music playing.
The main marathon was for 10km, in fact they managed to keep the path within campus. In the process I got to explore some untrodden  roads in IISc. It was fun, I managed to complete the run, that was a pretty consolation :), in fact was devoid of most of the post workout blues. My logic in running was the fact that I run about 3km regularly, which I insanely extrapolated to 10, it worked though.
There was an event for everyone. Besides the 10K marathon, there was a 5K half marathon, a 5K relay, a run for the children and for the old...truly, it was a celebration of running. It is essential that we spend time exercising regularly. We often resort to bikes and cars even to travel very short distances, and to do simple errands.
People of all ages participated. There was a prof. who completed it on a wheel chair, really inspiring. The skinny, the obese, the athletic and the not so athletic, the young and the old, all had a moment of fun and gaiety...they went back with tired legs but a unfatigued lets run!!!
(All the photos are taken from