Will Schneeberger Set Theory and Logic Quadratic Forms Nelson (chair) Conway Trotter Real Analysis Suppose you have a function f, and you define a function g as g(x)=f(t-x) What can you say about the integral of f times g? {It doesn't converge unless f is measurable; it converges if f is L^2.} Is it continuous wrt t? Suppose you have a set of continuous functions [0,1]->R converging pointwise to 0. Do they converge uniformly? How about if the functions are decreasing? {Yes; then I tried to prove it. That took a while.} Complex Analysis Say you have a function which is analytic on the complex plane except at points 1 and 2. Now take the Taylor expansion around the origin. Where does it converge? {In a disk of radius 1, 2, or infinity around the origin, according as what it does at these two points} OK, suppose it has [nonremovable] singularities at these points. How would you expand about the origin if you wanted it to converge at some point of absolute value 3/2? State Cauchy's Integral Formula. {Forgot the n! term... OOPS} What kind of singularities can exist in analytic functions? {Removable, polar, essential, limit} What about the square root function? {Right... branch points too} What happens in the MacLaurin series for these singularities? Let's talk about doubly periodic functions. {I know nothing.} Can they be entire? {No; proved it, assuming function was nonconstant... oops} So assume they have a pole of order 1 in a fundamental region. Can it be the only singularity? Algebra What can you say about subgroups of a free abelian group? {Free} How many generators does it have? Can you say anything else? What can you say about subgroups of a free group? {Free, but can't prove it} What can you say about the number of generators? {At most countably many more} Suppose you have a degree 5 polynomial over a field. What are necessary and sufficient conditions for its Galois group to be of order a multiple of three? Can you give an example of an irreducible polynomial in which this is not the case? {x^5 - 2 over Q} What are the conditions for the Galois group of a polynomial to be alternating over the roots? If an irreducible polynomial of degree 3 does not have this property, is its Galois group necessarily the symmetric group? What about degree 4? Talk about the isomorphism classes of subgroups of Q. How many are there? {0 and one for any subset P of primes corresponding to the set of rationals whose denominators are divisible by no primes outside of P} Are these distinct? So that gives you the cardinality. Are these all of them? {I conjectured that they were; they are not} Quadratic Forms Talk about the quadratic forms over the rationals. {I talked about reduction to Q_p and R, and the determinant conditions. I then mentioned the epsilon, and that it should be 1 for all but finitely many p} Given all of this, what is the probability of success? {1/2, after a lot of trouble understanding the question, hoping they would let me define epsilon so that I could answer without completely understanding} What about forms over Z? What do you know about definite unimodular forms? {Talked about even and odd forms, and gave a few examples} What about indefinite forms? Can you produce each of the indefinite unimodular forms? Set Theory and Logic Write down the axioms of ZFC Set Theory. Can you reproduce these finitely? {No, because...} What assumption have you made? {I didn't know} Let's talk about regularity. How much mathematics could you get away with without it? {Almost all} Suppose that you had a result which actually did require regularity. How could you modify this result to get an interesting statement out of ZFC w/o regularity? What can you tell us about the cardinality of R? {Its cofinality} Anything else? Name an interesting result from logic, not from set theory. {I noted m-categoricity} What is an interesting consequence of this statement? {Algebraically closed fields} How would you prove that the theory of algebraically closed fields was consistent and complete? Comments My exam took about 2:10 in total, plus a 10-minute break after algebra, and plus the lateness of two of my professors (icy day). In addition Conway had to leave early (in the middle of the last paragraph above). The exam seemed to have little to do with anything that I had studied -- I didn't even get the so-called canonical Trotter question. Maybe that's a good thing -- I really didn't understand anything I had studied. I also found the mathematics being used much more informal than anything I'm used to. At times this was difficult to deal with.