Innumerable times, I have been ‘clean-bowled’ by the question — “What kind of maths do you do?” This has been asked by all kinds of people, my neighbours, my doctor, even my engineer friends. Some have gone so far as to ask, “Hasn’t all the mathematics that was to be known, known?” Following is an attempt to demystify the air of suspicion they might have on the way I (shall) earn my bread-and-butter.

The mathematics that I study has three pillars that form the core part of any undergraduate Maths program — Algebra, Analysis and Topology. There are other branches like number theory, differential geometry, representation theory, differential equations, combinatorics. Here I am elaborating on Algebra.

Algebra, in simple terms, is solving equations. Apart from its own intrinsic beauty, it is an indispensable tool in other parts of Mathematics; even Physics and engineering.

Given a problem, one method is to convert it into a problem in algebra, do the calculations and reinterpret the result in your subject. I present two illustrations.

Let us suppose an electrical engineer is to analyze the current-carrying capacity of a load in a circuit diagram consisting of (voltage / current) sources and (resistive) load. The problem is electrical; the equation is Ohm’s law, . S/he applies the Kirchoff Current and Voltage laws to get a system of equations. Thus the electrical problem is converted into a linear algebra problem. (Had the circuit had inductive or capacitive load, we would have had a system of ordinary differential equations; the basic *funda* remains the same.)

During our college days (and I really mean the 12th grade), it was algebra that was used in proving interesting geometrical properties. Any beam of light parallel to the axis of a parabolic mirror reflects into the focus. When a coin placed on one focus of a parabolic carrom-board is flicked, it rebounds and passes through the other focus.

A polynomial, say represents a curve in the 2-D space. Indeed, two curves of degree and intersect in atmost points. Two nonparallel straight lines (degree=1) intersect in a point; a line intersects a circle (degree=2) in atmost 2 points and a circle intersects a parabola or ellipse or hyperbola (degree=2) in atmost 4 points. The bound points is necessarily attained if we work over complex numbers instead of real numbers. (This property is called being algebraically closed). This beautiful result is called Bezout’s theorem in Algebraic Geometry.

This Algebra, or solving equations reminds me of my old line,

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April 3, 2010 at 02:43

pushkarThis is not a demystification of your work at all. You really cannot convince me that solving a polynomial equation is going to earn you bread and butter. You have to tell me the power of mathematics, what you can achieve with your work. May be, http://www.cs.toronto.edu/~mackay/conway.pdf

April 3, 2010 at 08:19

abhishekparabHahaha, very true indeed! I should have had a still broader view when writing this. I will try to add something here.

Surely solving a polynomial is not an easy job. Read the biographies of Galois and Abel. A lot of algebra was discovered to solve the Diophantine equation , the famous Fermat’s Last theorem. But Algebra

doesgo beyond solving polynomials. It is the abstraction of everything we see around us. Take a good idea or a nice problem, solve it and then generalize it so that the problem becomes just a trivial case of your broader theory. ( Indeed, there are two types of mathematicians – problem solvers and theorizers , cf. here. )So, the theory you develop to solve your Algebra problem can be widely used by other mathematicians, physicists and engineers. Finite field theory is used in Coding theory, e.g. the Chinese Remainder theorem is used in the RSA. Category theory, a strongly abstracted concept is used by Computer scientists, I hear. Homology and cohomology is used in so many branches of mathematics itself, most importantly, topology. All this is pure Algebra.

Here is a line due to Alexander Grothendieck —

“The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps…”

April 3, 2010 at 12:16

AnonymousYou should not have been so biased to Algebra!!

April 3, 2010 at 12:26

abhishekparab1) Read above:

This post is about Algebra2) It is most likely that I shall earn my living by doing Algebra 😀

3) I don’t encourage anonymous commenting but knowing your IP, I can guess who you are; don’t worry, I love Analysis too. In fact, aren’t we discussing real analysis since the past couple of days? 🙂

April 3, 2010 at 20:47

Amar MainkarThe illustrations were nice, I was suddenly reminded of good old high school math that I was terribly good at, and how I have fallen from grace. You will make a terrific algebra teacher, I sure. I never had a good Algebra teacher, and I blame that for turning into an Algebra-hater, though hate is too strong a word.

And people not only earn their bread by doing algebra, they also win million dollar prizes for doing math.

June 14, 2010 at 03:58

Shashank Sawant@ Pushkar

It might be alright if a layman questions the profitability of maths. But I guess at least engineers should understand the long term benefits maths has to offer.

I think it’s analogous to questioning the bread and butter of those working on the LHC. Even in that case there are neither immediate gains nor guaranteed ones. But those who claim to be a part of the scientific community should come forward to support the work being done in the areas which are fundamental – and upon which all science is based on.

June 28, 2010 at 10:00

pushkarI never questioned the importance of maths here. Nor the scientists at LHC ever describe the M Theory when asked what is it they do. They know how important their work is, that is why they are doing it. They have to make others realize the implications of their research and discoveries. That is why you will find Brain Cox’s talk convincing people and justifying funding at LHC.

For example, justification for funding research into robotics is to make robots that can explore other planets, automate manufacturing process. It just turns out providing such justification for LHC and indeed parab-like math is hard, but that is where the demystification lies… And I think Parab did a decent job in his first comment at that.