Ralph Thomas' Generals
Examiners:Browder (Chair), Stein, Hewitt
Topics: Algebraic Topology, Singular Integrals
Algebra (Hewitt)
Tell me about group rings (I defined them) What do you know about them?
(Nothing)
Ok, Let's try Commutative Algebra (Uh-oh) Do you know what the Radical is? (No) Well, its the intersection of all the prime ideals, and also equals the set of nilpotent elements. Use this fact to prove that F[x], F a field, has an infinite number of prime ideals.
Galois Theory: What groups can be realized as Galois groups (Sn, An, ..) Why Sn? Compute the Galois group of x^4-2.
What 3x3 matrices over the rationals (up to similarity) satisfy f(A)=0, where
f(x)=(x^2+2)(x-1)^3 . List all possible rational forms.
What do you know about homology of groups (I used Hi(G)=Hi(K(G,1)) where K(G,1) is the Eilenberg-MacLane space. This wasn't what he was looking for, so he stopped)
Real Analysis (Stein)
What is Absolute Continuity? (I defined it) What functions are Absolutely Continuous? (He was looking for functions of bounded variation) Are there functions which aren't differentiable anywhere? (I gave the standard infinitly peaked function. He asked me to describe the function, but I had no idea what he was getting at. I later realized he wanted me to say that the function is nowhere monotonic.) What functions are differentiable almost everywhere? (monotonically increasing functions)
State and prove Holder's inequality.
Why is L^2 complete?
What is the dual space to L^p, and why? (I was kind of vague, but he didn't seem to mind to much)
Complex Analysis (Stein)
What is the Gamma function (I gave the infinite product definition) What is the integral definition? Prove the Functional equation.
Riemann Zeta function. Do you know the functional equation? Talk about the meromorphic extension to the whole plane.
What is a pole. What does a function do near a pole? (It goes to infinity) How? (It becomes arbitrarily large. I have no idea why he asked me that)
How many zeros must an entire function of order 1/2 have?
Singular Integrals (Stein)
Tell me about bounded operators on L^2. (If they commute with translations, then (Tf)^(x)=m(x)f^(x)) Why? (I mentioned the fact that he omitted this proof from is book, which resulted in an eruption of laughter from Browder and Hewitt. I'm not sure how Stein reacted)
What do you know about the Hilbert Transform (I started listing facts, and he stopped me when I said it was bounded on L^2) Why? (I didn't know exactly. This was not well received. After fifteen minutes of me failing to prove why H is bounded on L^2, we moved on)
What general operators are bounded on L^2?
Algebraic Topology (Hewitt, Browder)
Compute Homology, Cohomology of RP^2. Why is it non-orientable? (Poincare Duality)
What is the Jordan Separation Condition? (I didn't know) Do you know what Alexander Duality is? (Yes) Use that to find the Separation Condition (It turned out to be pretty easy)
Why can't RP^2 be embedded in R^3? (Alexander Duality)
Then Hewitt had to go, so I was sent out of the room while they discussed the exam. I was supposed to go back in for more questioning, but they decided to pass me right then. The exam lasted two hours. Browder and Hewitt were helpful when I got a little stuck, but Stein will push you on a question he thinks you should know, whether you know it or not, without giving you any real assistance. If Stein is on your committee, DO NOT use Rudin's "Real Analysis". His view of analysis is very different from Stein's. I would recommend Royden's book instead.