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, V = I R. 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 x \longmapsto x^2 + 5x -3 represents a curve in the 2-D space. Indeed, two curves of degree m and n intersect in atmost mn 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 mn 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,

\boxed{\text{Do not think; let the equation think for you!}}

Advertisements