July 16, 2018 Monday
Bedtime Story
Group Theory Slowly Advances post Galois
If Galois, very unintentionally, laid down
the foundations of group theory, he was followed by some very great
mathematical minds that not only grasped the significance of his complicated
theory but also developed it further and formalized it.
The men of mathematics who made significant
contributions to the group theory as a subspecialty of abstract algebra are the
Norwegian mathematicians Niels Abel and Sophus Lie (to name a few notable ones
whose names are littered all over mathematics) but sadly for now, I cannot go
in detail into the nature of their mathematical works and lives.
Suffice to say that Group theory, even
though primarily originated with respected to solution of polynomials and was
deeply embedded in abstract algebra, has bearing on all branches of mathematics
including the very simple numbers.
Besides that, Group theory has strong
connection with idea of symmetry, in aspects both of the general meaning and the
formal mathematical definition.
How the group theory is linked with
symmetry has to be assumed for now and you must take a giant leap of faith in
what I am saying.
A questioning and a skeptical person though
ought to make his own enquiry and find out for himself the veracity of my
statement.
In mathematical physics, symmetry is that
property or a feature of a system that remains unchanged under some
transformation.
Transformation again in mathematics is
quite different from the general understanding of this word.
In mathematics, a transformation is a
specific type of function that maps a set to itself.
In the language of function, f : X →
Transformation functions are of various
types such as linear, affine, rotations, reflections and translations.
Each function has some specific rules that have
to be followed and these functions can be performed either geometrically or
algebraically and can even be described accurately using matrices.
Perhaps the simplest transformation to understand
is that of translation where every point of a desired structure is shifted or
moved by a fixed distance in the same direction.
So in mathematics or in mathematical
physics, if a certain property of a system that is either observable or
intrinsic remains unchanged after transformation, then the system would be said
to be symmetric with respect to that specific kind of transformation.
In mathematical physics, symmetry is often
used interchangeably with the word invariance which literally means lack of
change under any kind of transformation.
It is incredible that something that
originated in pure abstract algebra related to solutions of abstract higher
order polynomials today (since some decades at least) finds resonance in
mathematical physics.
It is now well accepted by most theoretical
physicists that the concept of symmetry and thereby Group theory is extremely
powerful as evidence is pouring in from experimental physics which seems to
affirm the idea that virtually all the fundamental laws of nature seems to
arise from the mathematics of symmetry.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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