• Quadratic reciprocity: $\bigl(\frac{p}{q}\bigr)=\bigl(\frac{q}{p}\bigr)$, up to sign.
• With massive loss of generality, let $n=5$.
• A comathematician is a device for turning cotheorems into ffee.
• How do you prove a cotheorem? Using rollaries.
• $0\to A\to B \to C \to 0$. Exactly.
• Let $\varepsilon\to0$. There goes the neighborhood!
• Take a positive integer $N$. No, wait, $N$ is too big; take a positive integer $k$.
• All metric spaces are Hausdorff - he's the only one.
• Let $X$ be a set. Call it $Y$.
• Calculus has its limits.
• There is a fine line between numerator and denominator.
• Without geometry, life is pointless.
• Think globally, act locally.
• Natural numbers are better for your health.
• There's a marked difference between a ruler and a straightedge.
• Statistics means never having to say you're certain.
• Suppose there were no empty set. Then consider the set of all empty sets.
• Q: What do you get when you cross a chicken with an elephant?
A: The trivial elephant bundle on a chicken.
• Q: What's green and really far away?
A: The lime at infinity.
• Q: What's an anagram of "Banach-Tarski"?
A: Banach-Tarski Banach-Tarski
• Q: Why is it an insult to call someone "abelian"?
A: It means they only have a 1-dimensional character, and are self-centered.
• Q: What's a polar bear?
A: A rectangular bear after a coordinate transform.
• Q: Why did the vector cross the road?
A: It wanted to be normal.
• Old Macdonald had a form, $e_i \wedge e_i = 0$
• Prof: Give me an example of a vector space.
Student: $V$
• Prof: Are $\mathbb{Z}/4\mathbb{Z}$ and $(\mathbb{Z}/2\mathbb{Z})^2$ isomorphic?
Student: The first one is, but the second one isn't.
• A poet, a priest, and a mathematician are discussing whether it's better to have a spouse or a lover.
The poet argues that it's better to have a lover because love should be free and spontaneous.
The priest argues that it's better to have a spouse because love should be sanctified by God.
The mathematician says, "I think it's better to have both. That way, when each of them thinks you're with the other, you can sit down and do some mathematics."
• $\aleph_0$ bottles of beer on the wall,
$\aleph_0$ bottles of beer,
You take one down, and pass it around,
$\aleph_0$ bottles of beer on the wall.
• A logician rides an elevator. The door opens and someone asks:
"Are you going up or down?"
"Yes."
• If the IRS had discovered the quadratic formula..., by Daniel Velleman
• Mathematical Proficiency Exam, by Isaac Solomon
• Finite Simple Group of Order Two
• I'm afraid we need to use... MATH
• Some excellent, but less well-known, Tom Lehrer songs
• Look Around You - Maths
• Bolzano-Weierstrass Rap
• Fry and Laurie - Open University blooper
• The proof is trivial! Just...
• Three logicians walk into a bar
• Dear Committee Members:
Thanks for your letter of March 30. After careful consideration, I regret to inform you that I am unable to accept your rejection at this time.

This year I have been particularly fortunate in receiving an unusually large number of rejection letters. With such a varied and promising field of schools, it is impossible for me to accept all refusals. Despite your outstanding record and previous experience in rejecting applicants, I find that your rejection does not meet my current career needs. Consequently, I will begin taking classes as a graduate student in your department this August. I look forward to seeing you then.
Best of luck in rejecting future applicants.

Sincerely Yours,
[Name Withheld]