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.