I have a project to use some similar jokes as quizzes: a student who laughs, knows enough of the subject to understand a joke:-) (They say a similar method of testing what students know was actually used by Prof. I. M. Glazman in Kharkov).

1. Oral exam in Moscow University.
Professor: What is a root of multiplicity m ?
Student: Well, this is when we plug a number to a function, and obtain zero; then we plug it again, and obtain zero again... and this happens m times. But on the (m+1)-st time we do not obtain zero.

2. A Polish mathematician Mark Kac (who escaped to the US in 1939, just in time) was questioning a student (in the US):
Prof. Kac: What singularity does z+1/z have at infinity?
Student: Well... I forgot... Could you give me a hint?
Prof. Kac: OK, here is a hint: Who am I?
Student: ???
Prof. Kac: I mean a simple Pole!

The idea of Kac was used in many other jokes. The following one requires some prerequisite in linear differential equations (MA 366 would be enough:-)

3. A Polish airplane crashed, because an engineer was taught that for stability, ``all Poles have to be in the left half plane''.

(They were talking about the poles of the ``transfer function'', that is the inverse matrix of (sI-A). Here is a recent paper about these "poles")

4. In general, many jokes can be made with the word "pole". For example: a mathematician named his dog Cauchy. Because it leaves a residue at every simple pole.

5. Jokes about sine.

5.1 In a written exam in freshman calculus, a student solves the equation sin x=2. He writes: x=arcsin 2, and gets an "F". He comes to ask what was wrong, and his professor explains that arcsin 2 does not exist, and that the equation has no solutions.
Few years later the same student has an exam in complex analysis with the same professor. He is very glad to see at least one problem, whose solution he knows: to solve the equation sin z=2... Well, you can invent the end of this story yourself.

5.2 My teacher (A. A. Goldberg) used to say, that a teacher has to understand the soul of a student. And gave the following example.
Professor: why did you divide by (sin x-5), when solving this equation?
Student: because sin x never equals to 5, thus sin x-5 cannot be zero.
Professor: OK, very well...
This professor does not understand the soul of a student... Here is another one, who understands:
Second professor: OK, but WHY sin x never equals 5 ?
Student: Well, we know that in the first quadrant, sin x changes from 0 to 1. But the total number of quadrants is 4, so sin x cannot be more than 4.

6. Perron's Paradox. (This joke has a somewhat deeper meaning).

THEOREM. The greatest natural integer is 1.

Proof. Let N be the greatest natural integer. Assume, by contradiction, that N>1. By multiplying both sides by N, we obtain NN>N. So N is not the greatest. Contradiction. So N=1.

7. Classified research in former Soviet Union was an object of many jokes.

A colonel from a top secret military research institution comes to a math department, and asks to find a conformal map from an equilateral triangle onto the upper half-plane. They tell him. A week later he comes again and asks about a conformal map of a square onto the upper half-plane. They tell him. Next time he comes and asks about regular pentagon and hexagon (which is much harder). The mathematicians are starting to suspect something... They ask him: What is your ultimate goal? He replies: Well, I think I can tell you, though this is a secret research. We are trying to find a conformal map of a disc onto the upper half-plane, by approximating the disc by regular polygons with many sides!

It is a very nice research project for a math 525 or 530 student, to find explicitly a conformal map from the regular 5-pointed star (the one which is on the flags of many nations, including USA and USSR) onto the unit disc. Use the Symmetry Principle to reduce the problem to a mapping of a triangle, then write the Christoffel-Schwarz formula, and try to reduce the integral to a simple standard from.

They say, a paper with this formula was published in one Soviet journal. The paper was dedicated to the 50-th Anniversary of the Great October Socialist revolution.