Ryan Bissell-Siders' Generals.  John H Conway Chair, Hale Trotter
Algebra & Number Theory, Almut Burchard Analysis.

Set theory:
What implies "any set can be wellordered"? (Ax. Choice)
What did Frankel add to Zermelo's set theory? (Functions.)
What does this allow us to construct? (Aleph_omega)
What might 2^{Aleph_0} be?  (Aleph_1,2,... lotsa stuff... the only
restriction on 2^x is: cof(2^x) > x)
Define cof(x). 
What is the first set that can't be 2^{Aleph_0}? (2^{Aleph_omega})
Find two sets with the same Aleph_0 power ([Aleph_omega]^Aleph_0 and
Aleph_omega, because Aleph_0^2 = Aleph_0).  
What else?  2^x = x^cof(x).  Forcing doesn't give us much freedom to
determine a small cardinal exponent of a large cardinal.
Tell us something about Godel's proof of the consistency of Axiom of
Choice (talk about constructible sets). 
Whose names go on the theorem: if we can inject A into B and B into A, we
can isomorph A to B?.

Number Theory:  
What are the extensions of a p-adic field? (there are a finite number of
extensions of each degree).  What are the quadratic extensions? (there are
3 unless p is 2).
State Units theorem.  What are the units in a quadratic field (no
new (other than -1 or 1) units unless we have adjoined sqrt 2 or 3).  
State the correspondence between polynomials factoring and primes
splitting (a prime splits in a number field extension just in case the
polynomial defining it factors.)

Algebra:
The eigenvalues of a hermitian matrix are hermitian (real); those
of a unitary matrix are unitary. 
When do the powers of a matrix tend to zero? (eigenvalues < 1).
State and use Sylow's theorems (there is 1 group of order 15).
Classify finitely generated modules over Z, pid, Dedekind ring. (Direct
sum of cyclic modules.)
Subgroup of finitely generated free abelian group is? (free and has fewer
generators).  
Subgroup of f.g. free group is? (Free with unbounded generators).  Prove
(covering spaces of a wedge product of loops). 
f.g. torsion free abelian implies free abelian.
Tell an infinite group which is not product of cycles (2-adic rationals
mod 1; they are divisible, but a product of cycles is not.)
Galois theory: state main theorem, show the trees of subgroups and
intermediate fields for Q adjoin roots of X^3 - 2.  

Complex:
Prove that an analytic function has an expression as a power series.
(in Cauchy formula, expand 1/x-z into a power series, approximate the tail
with $\int_circle (x/z)^n z f(z) /(x-z).$ 
State Arzela-Ascoli (if a sequence of equicontinuous functions
converges pointwise, it converges uniformly). 
State Morera.

Real:
State dominated convergence, fubini inequality...