Alison Miller's Generals
May 13, 2010
Committee: Manjul Bhargava (chair), Andrei Okounkov, John Mather.
Special Topics: Algebraic Number Theory, Representation Theory.
The following is a fictionalized account of my generals (written 3-4
months after it happened, so my memory of some parts is rather
blurry).
I show up in Manjul's office; he says that he has stuff on his
blackboard that he doesn't want to erase, so proposes moving to 1201.
Eventually the other two members of the committee arrive, and we move
to 1201.
People ask me what I want to start with. I suggest complex analysis.
M: State the cauchy integral formula.
I write down f(a) = integral of
f(z)/(z-a) dz.
M: Aren't you missing something there?
Oh, right,factor of 1/(2pi i) (It's possible I forgot the i here but
if so nobody mentioned it).
M: Prove it.
I sputter something about how one can show it when f is constant, and
then subtract off a constant term to reduce to the case where f(z) =
0, so then f(z)/(z-a) is holomorphic and one can apply Cauchy's
theorem.
Silence.
I ask them if they want me to prove Cauchy's theorem.
M: Sure, go ahead.
I ask if I can assume that the function is continuously
differentiable.
M: It was good enough for Cauchy, it's good enough for me.
I fumble for words.
People prompt me to say something about the differential form f(z) dz.
I start writing out real and imaginary parts.
M: Is it closed?
Me: Yes, so we're happy.
We go on.
somebody: analytically continue some function from number theory.
I write down the Riemann zeta function zeta(s)= sum 1/n^s. I ask if I
have to analytically continue it everywhere that it can be
analytically continued, or if a partial analytic continuation is
adequate. People say that the latter is fine.
I attempt to explain the argument from Lang analytically continuing to
Re(s)>0 by relating zeta to the function sum (-1)^n/n^s. People don't
seem to find this very convincing.
somebody: Why don't you try gamma instead?
Me: should I define gamma as the integral?
somebody: yes.
I write down the integral for gamma. I explain how one can apply
integration by parts to increase the region of convergence. people
are happy.
I think we moved to real analysis at this point?
Mather asks me if I know what some term means (I think it must have
been "residual set"). I say no. He tells me that it means countable
intersection of open dense subsets.
M: is the empty set residual?
Me: No, that's the Baire category theorem.
somebody: in what more general setting does the Baire category theorem
hold?
Me: it holds in any Baire space, which I believe includes complete
metric spaces and compact Hausdorff spaces.
M: Does there exist a residual set of measure 0?
Me: Yes, take the intersection of a family of open dense sets which
are each unions of a set of open balls, one centered at each rational,
and such that the sum of the radii of the balls becomes arbitrarily
small.
Mather: When you're doing the fundamental theorem of calculus in the
general setting, you need to require some technical condition on your
function. State the fundamental theorem of calculus and tell me about
the condition you need.
I figure out that he means the part where the integral of the
derivative is your original function.
Me: So f has to have bounded variation. I then assert that if f has
bounded variation, integral of f'(x) dx = f.
Mather: do you know about the Cantor-Lebesgue function?
Me: Yes, but that doesn't have bounded var-- darn, it does. I suppose
I really meant for f to be absolutely continuous.
Mather then asks me to state FTC again correctly. He still isn't
quite happy with it, because I neglect to mention that f' is
integrable (I do at some point put in the fact that f' is defined
almost everywhere, but somehow miss this). Eventually he gets me to
fix this.
He then makes some comment about something that I don't quite hear and
that the non-analysts didn't know about.
O: Here's a problem I gave to another grad student in his generals
last week, he had some trouble with it. Let's see how you do with it.
Suppose that f is a continuous function on (0, 1) and f is
differentiable at 0. Show that the Fourier series for f converges at
0.
Me (demonstrating that I didn't study any Fourier analysis when
preparing): but don't Fourier series for continuous functions always
converge?
O: It depends upon what you mean by convergence.
Me: Okay, so the nth Fourier approximation to f is given by
convolution with some kernel. I write down the kernel and sum the
geometric series.
I then attempt to give handwavy arguments that don't actually work.
I then try to figure out how I'm going to use the condition that f is
differentiable.
O: why don't you do something like the argument you gave for Cauchy
integral?
Me: Oh! this is clearly true for constants, so I can subtract off a
constant and assume f(0) = 0. Now I need to show that the limit of
this is zero. I realize that I probably want to combine the f term
with the denominator in the kernel and use differentiability, but
still sputter around for a while.
O: do you know how to prove the limit is zero if you didn't have the
denominator?
Me: I have no idea.
O: That's the Riemann-Lebesgue lemma: the Fourier coefficients of an
L^1 function go to 0.
(oh, so that's what Riemann-Lebesgue is! I was aware there was a
theorem with that name, but not what it was)
Me: Okay, then when we put the denominator back in I can use the fact
that f is differentiable at 0 to conclude that f/denominator is
continuous at 0, so that works!
M to others: Any more analysis questions?
People decide to go back and give me more complex analysis.
B or O: sketch the proof of the Riemann mapping theorem.
I do. I think they then ask me about Schwartz's Lemma.
They then ask me what the automorphisms of the unit disk are. I say
that an automorphism f that takes 0 to 0 and has f'(0) =1 must be the
identity by Schwartz's lemma, and it then follows that all
automorphisms are linear fractional transformations.
They ask me specifically which linear fractional transformations
actually induce automorphisms of the disk?
Me: I don't know offhand, but the automorphisms of the upper half
plane are SL_2(R)
someone: how do you transform from the upper half plane to the disk?
I say that something like z+i/z-i should work. They say that's almost
right.
They then ask me something about the relationship between holomorphic
and conformal mappings. I think maybe I say that all holomorphic maps
are conformal (oops).
They then ask me if holomorphic is the same thing as conformal. I say
no, z->zbar is conformal but not holomorphic (though actually that
seems to not be true under the standard definition of conformal).
They then ask about the other direction; oh, right, i realize: z->z^2
is not conformal at the origin.
It's been about an hour so people decide it's time to move on to
algebra.
B: (something like) Let f be a degree 4 polynomial with integer
coefficients; what's the smallest field in which f has four roots? (or
something like this)
Me: starts saying something about splitting fields.
B: no, that's not what I meant. "smallest field" includes finite
fields.
Me: Oh, then f has roots in F_16 if it's irreducible or F_8 if it
factors as cubic*linear, or...
I think maybe B gets me to say that f always splits in F_{2^6}.
B: how many irreducible polynomials are there of degree 4 over F_2?
Me: There are 12 elements of F_16 that aren't in F_4. Any irreducible
degree 4 polynomial has 3 of these as roots, 12/4 = 3. (O makes some
approving comment at the last step about how that's a calculation he
can get behind.)
I don't remember whether there are any more finite fields questions
here.
B: I have a present for you. (He pulls out a foam MAA icosahedron and
tosses it to me.) Stress reliever.
Me: Thank you. (I squeeze it a couple times for stress relief.)
B: Find the symmetry group of that object. (I thought it was supposed
to be a stress reliever! Fortunately this isn't very stressful.)
Me: Including or not including reflections?
B: Either way. (This is what one gets for asking for clarifications.)
So we first do without reflections (it doesn't make much difference
anyhow, I forget if we actually ever get to adding the reflections
in). People decide to make me compute the order with
orbit-stabilizer.
someone: what's the stabilizer of an edge?
Me: order 2, I demonstrate.
How many edges are there?
Me: well, I know what icosahedron means in Greek, so I know it has 20
faces, each face has 3 edges and each edge is on 2 faces, so 20*3/2
edges = 30 edges.
so then they prompt me to conclude the order is 60.
Me: Well, if it were a dodecahedron instead of an icosahedron I would
show that it's A5 using the group action on the five cubes inside the
dodecahedron.
B explains how you can also see the five cubes with the icosahedron
(using groups of 6 edges that are perpendicular to each other).
O: how do you know the symmetry group of the icosahedron is the same
as the dodecahedron?
me: they're dual to each other.
O: do you know the classification of higher-dimensional polyhedra?
me: Yes, but I'm not sure I remember them all in dimension 4.
O: what about dimension >4?
me: Simplex, hypercube, and dual of the hypercube.
O: in dimension 4, the most interesting one is the one that has to do
with some root system... (he might have said which one, I don't
remember)
me: yes, the 24-cell. (yay!)
now on to Galois theory/algebraic number theory
B: consider the field extension given by adjoining a root of x^3 - x -
1 = 0. Which primes ramify in this extension? (I think this was the
question; but he might have been asking me about the galois group of
the splitting field instead).
me: okay, so I need to check the discriminant of the cubic.
somebody: define the discriminant of a polynomial
me: writes down formula discriminant = prod_{i