The Wonderful Thing About Tensors

(This one was pure inspiration, one day in algebra class.  It actually helped my on my candidacy exam, when I had to remember whether tensors were left or right exact!)

The wonderful thing about tensors
Is tensors are wonderful things.
They're made out of maps middle-linear
On modules made out of rings;
Right exactness, adjoint Hom, fun fun fun fun fun!
But the wonderful thing about tensors is:
Up to isomorphism there's only one!


Somewhere Into a Spectrum

(This song was written, with some help from my officemate Rebecca, as an introduction to a graduate student seminar talk I gave on cohomology theories and Eilenberg-MacLane spaces.  No disrespect intended to either Eilenberg or MacLane:))

Somewhere into a spectrum
Saunders lives.
Samuel waits with omega
For your map in from X.
And when the cohomology
Is found with coefficients G
You will find that map you see
When you take homotopy...


The Mathematicians Who Don't Prove Anything

(Okay, so this song shouldn't be taken as representative of our work ethic around here...  It just came to me while watching Veggie Tales.  I never did think up good verses for it, however.)

We are the mathematicians who don't prove anything -
We just sit at tea and talk around.
And if you ask us to prove anything,
We'll just tell you - we don't prove anything!


The Weiner Covering Lemma Song

(Rebecca gets the credit for this one.  She was inspired one day in real analysis class.)

Oh I wish I were a covering of Weiner
To bound the size of measurable E
With a subset that is disjoint times a constant
Depending on dimensionality.


The Cat in the Hat  (The Abstract Algebra Version)

(This spoof of the children's book was written at the end of the 1997-98 school year, after my friend Emily and I completed a two-semester abstract algebra course together.  We were also on a retreat together with a theme of The Cat in the Hat, which inspired me to "rewrite" it for her as a graduation present.)

The sun did not shine;
It was too wet to roller blade,
So we sat in Herak,
With calculators we played.
I sat there with Bubbles,
We sat there, we two,
And I said, "How I wish
We had some math to do!"
Too wet to go out,
And no problems to solve,
So we sat in Herak.
We did nothing at all.
So all we could do was to
Sit! Sit! Sit! Sit!
And we did not like it,
Not one little bit.
And then
Something went BUMP!
How that bump made us jump!
We looked!
And we saw him!
The Cat in the Hat!
And he said to us,
"Why do you sit there like that?
I know it is wet,
And the sun is not sunny,
But we can have
Lots of prime fun that is funny!
Some abstract algebra I can find,"
Said the cat.
"It will mess up your mind,"
Said the Cat in the Hat.
"A lot of good proofs,
I will show them to you.
Your professor
Will not mind at all if I do."
Then Bubbles and I
Did not know what to day.
Our professor was out of town
For the day.
But our advisor said, "No!
Make that math go away!
Tell that Cat in the Hat
You value your G.P.A.!
He should not be here.
He should not be about.
He should not be here
When your professor is out!"
"Now! Now! Have no fear!
Have no fear," said the cat.
"My proofs are not bad,"
Said the Cat in the Hat.
"Why we can have
Lots of fun - here's the scoop -
With a game that I call
UP - UP - UP with a group!
They''ve identities and inverses,
They're associative and closed.
If you're lucky they're abelian."
And some problems he posed.
"But that's not all!
If you get this down pat,
There are subgroups, too,"
Said the Cat in the Hat.
"And you can go further!
If you want to be formal,
You can find subgroups
Which are said to be normal.
These subgroups are special,
As we shall see -
Of homomorphisms, they're kernels,
Take it from me!
Of course there are theorems
Written just for you
By Lagrange, Cauchy, Fermat,
Just to name a few.
And then we get to cosets,
When we look at a factor group -
Yes, this all maks up
A very fine math soup!
And there is a fine thing
To you I must show
By subgroups discussed
In theorems called Sylow.
And here's something else -
It's better than vacations!
We really must look at
Some permutations!"
He meant to go on,
Talking 'bout groups,
But he saw we were lost,
Going in loops.
And our advisor came in
And right there on the spot,
He said, "Do I like this?
Oh, no! I do not!
This is not a good course,"
Said our advisor as he sat.
"No, I do not like it -
I'll say that out flat!
Now look what you did!"
Said he to the cat.
"Now look at their minds!
Look at this! Look at that!
You fried their brain cells
So they cannot think!
You messed up their minds,
And their grades you did shrink.
You SHOULD NOT be here
When the professor is out.
You get out of this place!"
Our advisor did shout.
"But I like it here!
Oh, I like it a lot!"
Said the Cat in the Hat
To him on the spot.
"I will NOT go away.
I do NOT wish to go!
And so," said the Cat in the Hat,
"So so so I will show you
Another prime game that I know!"
And then he ran out,
And then, fast as a fox,
The Cat and the Hat
Came back with a box,
A big red wood box.
It was shut with a hook.
"Now look at this trick,"
Said the cat.
"Take a look!"
Then he got up on top
With a tip of his hat.
"I call this game FUN-IN-A-BOX,"
Said the cat.
"In this box are two rings,"
Said the cat with a bow.
"I will pick up the hook.
You will see something new.
Two rings.  And I call them
Ring One and Ring Two.
These rings will not bite you.
They want to have fun."
Then, out of the box,
Came Ring Two and Ring One!
We looked at the rings,
But our advisor said, "No!
These rings should not be
In this class! Make them go!
They should not be here
When your professor is out!
Make them leave! Make them go!"
He continued to shout.
"Have no fear, my dear sir,"
Said the Cat in the Hat.
"These rings are nice rings."
And he gave them a pat.
"They are tame! Oh, so tame!
They have come here to study.
They will give you prime fun
So you won't all go nutty.
Now a ring has some properties
That aren't so diminutive:
Under multiplications they're
Closed, associative, distributive.
And it gets even better,
Yes, even more fun -
A ring that has unity
Is a ring with one.
If there is an inverse
For a particular element,
We call it a unit,
And to one they are sent.
It gets even better,
So let's start to sing!
If all elements are units,
We have a division ring!
And here's a big secret,
So keep your lips sealed -
If a division ring's commutative,
We call it a field.
There's something else -
It's really quite plain.
A ring with no zero divisors
Is known as a domain.
And still more there is
To fill up your brain -
If a domain is commutative
It's an integral domain.
And if it has factorization
That is unique, you see,
It has a really cool name:
A UFD!
And just as from groups
The subgroup springs,
Inside of rings there are
Little subrings!
And there's a special one, too -
Oh, such a deal!
If it works as a kernel,
It's made an ideal.
And ideals are so special,
If you can get near 'em,
Because they helped solve
Fermat's Last Theorem!
And if you think prime numbers
Make for a good time,
You should try ideals,
Which, too, can be prime!
And some prime ideals
Are not very small -
Sometimes, you see,
They are maximal.
There are still more kinds
We can find of rings
When we put the elements
Into polynomial things.
For one more fact
That's just not refusable,
There are polynomials
We say are irreducible.
And if you have a polynomial
(Monic it must be),
You can use Eisenstein to test
Its irreducibility.
Oh, there's one more thing
I guess I should mention -
You can find fields
As an extension.
But before you do that,
I should mention in places
You will need to know linear
'Cause you'll use vector spaces!
You''ll use algebraic numbers
And polynomials minimal
But then after, you see,
You find you can pin 'em all,
You can learn a small theory
By Galois, it is probable,
To see a fifth-degree polynomial
Just isn't solvable!"
"That's prime!" was all that
Bubbles and I could say,
But who but that advisor
Should get in the way.
He pointed, said, "Look!"
And his voice shook with fear.
"Your prof is on his way back!
Do you hear!
Oh, what will he do to you?
What will he say?
Oh, he will not like
Your new G.P.A.!
So DO something! Fast!" said he.
"Do you hear?
He called me.  Your professor!
Your professor is near!
So as fast as you can,
Think of something to do!
You will have to get rid of
Ring One and Ring Two!"
"Oh, dear!" said the cat.
You did not like our game . . .
Oh dear.  What a shame!
    What a shame!
       What a shame!"
Then he shut up the rings
In the box with the hook.
And the cat went away
With a sad kind of look.
"That is good," said our advisor.
"He has gone away. Yes.
But your professor will come.
He will find a big mess!
And your brains are so muddled,
And your grade is so low,
We cannot change things.
There is no way to do so!"
And then!
Who was back in the class?
Why, the cat!
"Have no fear of this mess,"
Said the Cat in the Hat.
"I always pick up my lecture notes
And so . . .
I will show you another
Good proof that I know!"
So we looked at numbers
Constructable and not
And why we couldn't
Trisect an angle he taught.
And we learned about planes,
Over BIB's we laughed loads,
And we ended talking about
How to break secret codes!
The cat closed up his books.
Then he said, "That is that!"
And then he was gone
With a tip of his hat.
Then our professor came,
And he said to us two,
"Did you learn anything?
Tell me, what did you do?"
And Bubbles and I did not know
What to say.
Should we tell him
The things that went on there that day?
Should we tell him about it?
Now what SHOULD we do?
Well . . .
What would YOU do
If your professor asked you?