Q: What did the constipated mathematician do? A: He worked it out with a pencil!
OK, not going to try to explain this one.
Q: What's purple and commutes? A: An Abelian grape.
A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian
Q: Why do you never hear the number 288 on television? A: It's two gross.
A "gross" is a dozen dozen, or 144. Not a very mathematical joke.
Q: What do you get when you cross a mosquito with a rock climber? A: Nothing. You can't cross a vector and a scalar.
The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.
Q. How many mathematicians does it take to change a lightbulb? A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.
When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.
Q: What's big, grey, and proves the uncountability of the reals? A: Cantor's diagonal elephant.
The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)
Q: What's yellow and equivalent to the Axiom of Choice? A: Zorn's Lemon.
Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.
Q: What's yellow, normed, and complete? A: A Bananach space.
A Banach Space is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.
Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic? A: An antique tractorisation domain.
A pun on an unique factorization domain.
Q: What is hallucinogenic and exists for every group with order divisible by p^k? A: A psilocybin p-subgroup.
A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).
Q: What is often used by Canadians to help solve certain differential equations? A: the Lacrosse transform.
The Laplace Transform is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.
Q: What is clear and used by trendy sophisticated engineers to solve other differential equations? A: The Perrier transform.
The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.
Q: Who knows everything there is to be known about vector analysis? A: The Oracle of del phi!
The Del operator is used to express various types of vector derivatives, and phi is just a greek letter that is often used to represent vectors.
"I am just a simple Pole in a complex plane"
Given a complex function, a pole is just a point where the function is not defined (usually because something goes to infinity).
So, they just had to rely on the method of steepest descents.
A way to find the nearest local minimum of a function - works whenever the function is smooth near that minimum.
Adders can't multiply without their log tables.
This is how slide rules work too, BTW - to multiply x and y, look up the logarithm of x, and the logarithm of y, then add those, and then take the exponential of the result.