How to blabber mathematics

This is a portion of my sermon on the “dais” in 1BSc Mathematics classroom – where I randomly walked in as a substitute for staff on leave.

Prologue: ‘‘Generalisation is a heartbeat of mathematics. If the teachers are unaware of its presence, and are not in the habit of getting students to work at expressing their own generalisations, then mathematical thinking is not taking place’’ John Mason (1996)

Here’s a disclaimer: I am here as a substitute for your ma’am, who is on leave today. I will be speaking subject, but not what you have to study this semester. That also means, if you are planning to disturb the class – I do not mean little chats in between – but the ones which may upset me – you must walk out of the class now.

I waited for a minute, to assure that I have silence, and their complete attention on me.

I don’t exactly remember if I came to your class before – I vaguely remember coming here. Anyway, hope you know me – I am Jesse, and I teach here.

I had their attention, and all eyes were on me, wondering what I am going to speak about. A bunch of students walked in, claiming that their French class finished late.

Have you ever heard about the exciting bit of mathematics? I am not talking about the kind that you do now – and that you did in your school. I believe you have to wait for a year more to get a picture of it – the exciting bit of mathematics, called “pure” or “abstract” mathematics.

To begin with, let’s do some addition. We all know 2+2 is 4. Let’s take a look at this so called “+”, which we have been studying since standard 1. Easily, we notice that a+b, where a and b are numbers, is another number.

I cannot go dry like this – I must interact with them more – I was happy to see those eyes curiously running all over the board, waiting to see where I am taking this to.

Now, what can you say about a+b and b+a? (shouts here and there that they are equal) You are right! Similarly, we can see that (a+b)+c = a+(b+c). Also, we know that if a+b = c, we can write b = c – a; and turns out there will be a unique number which satisfy this, that is, for two numbers “a” and “b”, only one number “c” will satisfy this condition.

Interesting! Did you ever notice that our little “+” sign had so many beautiful properties? Wait! There is one more familiar “operation” that has these nice properties – multiplication!

Clearly, a x b is a number again – and a x b = b x a! That’s not it – (a x b) x c = a x (b x c)! “x” also has the nice properties of “+” we have been talking about! Does it have all those properties?

Tell me, if a x b = c, what is b = ?

I waited for the automatic response – that b = c/a.

But turns out this fails for one magical number – the number we mathematicians are very scared about – 0! If 0 x b = 0; can you say b = 0/0? I must admit – we mathematicians may be fearless about quite a lot of things in this world – but we are very scared of 0/0 – I will tell you a little secret, if you will not tell anyone – it’s because we don’t know what it is!

There were little growls here and there – they knew 0/0 is troubled waters – but many of them confessed, with their eyes that they never knew that the matter was this deeply disturbing, having joined the league to be a mathematician.

Back to b = 0/0 – it is absurd! Notice that in this case, our “x” doesn’t hold the last magical property our “+” holds, for the special number “0” – but it holds all of the others. Are there any other operations that holds these beautiful properties? Turns out there are many!

That’s why we study them separately, in a subject called “Group theory”. We will define a world, where we will see how each operator that follows these nice properties of “+” fare, given a set of numbers. In a group, we associate one operation to a set, and study it’s properties.

In other words, we will be “generalising” the properties of “+” using the groups, for many operations.

When it comes to group theory, I am tempted to talk about one man, who made enormous contributions to this field – his name goes this way: I write “Galois” on board. Any guesses on how it is pronounced?

They were in the league – I could hear “Galoee” and “Galoees” et al in backdrop.

It’s pronounced “Gal-wa”! These weird French people, I tell you. (The French students who walked in late, now looked up). Our man, contrary to popular belief that we mathematicians are “nerds” and do not have a life – had an exciting, adventurous, short time here on Earth – while he wrote a tiny theory – which no one were able to comprehend when he had written it – may be because his handwriting was bad – just like mine (I do some scribbling on board which they tried to read and failed) turns out it was one of the greatest theorems ever written in mathematics!

Well, he only did tiny things – like walking into king’s palace, and threatening to kill the king – something like shouting “I’ve got a gun!” When Prime Minister is in same room as yours – and he was arrested for it! He was a revolutionary, who died at the age of 21 – in a “duel”. There are multiple reasons quoted for that duel – like cowboys where they point gun at each other and start shooting, but one of those reasons is interesting – and I would leave that to your further research online.

Talking about the little theorem he wrote – it was “decoded” only after he passed away – and in fact, today, it is one of the most celebrated theorems in maths world. You will, if you continue in mathematics, have the fortune to read and study it, by the time you are in MSc.

He was a “child prodigy” – just like Akshay Venkatesh, who won the Fields medal recently.

Well, I went on to gossip about Nobel, and why we needed a “mathematician’s Nobel” called Field’s Medal, then a little introduction to life of Euler, and how his evening walk lead to start of two new fields in mathematics, called graph theory and topology, how he said “Now I can focus more on mathematics” when he got completely blind. We touched up a bit on a few puzzles, and observed that we have mathematics in them too – then to a beautiful subject, called “real analysis”, where we generalise the concepts in puzzles we just did – another pure paper they will come across by 4th semester. We indulged ourselves in history of mathematics department in MCC and why it is in Arts block, instead of science, few great names like Gift Siromoney who saw mathematics in everything, including music, archeology, zoology, botany (and the list goes on), Rani Siromoney, a world famous Formal Language researcher, V. Rajkumar Dare who made many of us fall in love with mathematics etc. and their mathematical contributions, all of which I will not elaborate here – because that’s another few pages I am talking about. For now, Auf Weidersehen.

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About Teaching Topology

This is a random thought, exclusively for absolute math nerds:

The blame is on me. I taught him topology, he deformed my favourite coffee mug. What better gift can a topology lecturer expect?

Couldn’t help but post this. Sorry, muggles who are randomly reading this post!

ODE Cheat Sheet: Few Relevant Definitions

It’s been a while since I wrote here, and this is an attempt to resurrect the blog, and get inspiration to contribute something for Wikibooks as well.

This is a part of the book “Ordinary Differential Equations: Cheat Sheet” that I am compiling in Wikibooks, and it will be published in five instalments before/along with finalising the book in Wikibooks. And just like Examples and Counter Examples Series (Which again I am planning to compile in Wikibooks), this will also be edited as I learn.

Wronskian of Two Functions

Definition

Wronskian of two functions, y_1,y_2 is given by W_{y_1,y_2}(x)=\left|\begin{matrix} y_1 && y_2 \\ y_1' && y_2'\end{matrix}\right|

Useful Facts

If two functions y_1,y_2 are linearly dependent in an interval, then it’s Wronskian vanishes in that interval.

Laplace Transforms

Definition

\mathcal{L}\{f(t)\}=F(s)=\int_0^\infty e^{-st}f(t)dt

Properties

  1. \mathcal{L}\{af + bg\} = a \mathcal{L}\{f\} + b \mathcal{L}\{g\}\,,
  2. \mathcal{L}\{e^{at} f(t)\}(s) = F(s - a)\, for s > \alpha + a.
  3. If F(s) = \mathcal{L}\{f(t)\}, then \mathcal{L}\{f'(t)\} = sF(s) - f(0)
  4. Similarly, \mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f''(0)

Laplace Transform of Few Simple Functions

  1. \mathcal{L}\{1\} = {1 \over s}
  2. \mathcal{L}\{e^{at}\} = {1 \over s-a}
  3. \mathcal{L}\{\cos \omega t\} = {s \over s^2 + \omega^2}
  4. \mathcal{L}\{\sin \omega t\} = {\omega \over s^2 + \omega^2}
  5. \mathcal{L}\{1\} = {1 \over s}
  6. \mathcal{L}\{t^n\} = {n! \over s^{n+1}}

Convolution

Definition

f(t)*g(t)=\int_0^t f(u)g(t-u)dt

Properties

  1. Associative
  2. Commutative
  3. Distributive over addition

Examples and Counter Examples for Abelian Groups

Note: This is the first of “Examples and Counter Examples” series. I plan on expanding such posts in course of time, as I come across examples.

Interesting fact: These are the OEIS lists of number of distinct groups of order n: number of groups,number of abelian groups, number of non-abelian groups

Examples

  1. \Bbb Z
  2. \Bbb Q
  3. \Bbb R
  4. Integers modulo n \Bbb Z / n \Bbb Z
  5. S_n,n\le2, the symmetric group.
  6. A_n \subset S_n,\forall n\ge1
  7. Any group of order < 4

Counter-Examples (or Non-Abelian Groups)

  1. Symmetric groups S_n,n\ge3

New kid in the block

Welcome the new kid in the block.

Almost two and a half years ago I kicked off my first WordPress blog Far Far Away, an attempt to showcase and improve my writing skills, and thankfully it was decently received in its first year before I got lazy and stopped writing regularly.

Off late I have been fascinated by the idea of “maths blogging” which is expected to improve my maths writing skills, and help me improve my skills in communicating mathematics. Thankfully WordPress.com supports LaTeX (\LaTeX fails to render though).

For the record, this is my sixth or seventh blog, and second to hopefully survive my wrath and an attempt to learn writing maths. I am a maths post graduate.

I have also transferred my following tech posts from Far Far Away, an attempt to make that blog more focussed on non-maths, non-tech content:

I hope I will be able to be able to post more frequently after December 20th. Since then, I plan on at least one post a week.

The content will be mostly about (and not restricted to),

  • Interesting facts in/about mathematics
  • Interesting theorems I came across
  • Interesting math.SE questions that caught my attention
  • Geek jokes
  • Interesting mathematics contest questions
  • …and hopefully compilation of definitions and important theorems, arranged topic-wise, for my own future reference.

I am also an active member in math.SE. I am more inclined to abstract side of mathematics, hope you now know what to expect here.