Josh Greene vs. The Generals
Committee: Benny Sudakov (chair), Zoltan Szabo, Natasa Pavlovic
Special Topics: Algebraic Combinatorics and Algebraic Topology
April 22, 2005, 1:30-3:40 PM
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ALGEBRAIC TOPOLOGY (Zoltan)
What books did you study? (I didn't want to be responsible for too much, so I
just said Hatcher and Bott + Tu.)
Suppose that you have a space that fibers over a sphere with spherical fiber.
What can you say about its homology? (My first thought was to use spectral
sequences, but I was surprised that would arise in the first question on my
exam. So I mumbled on about the LES in homotopy of a fibration and how you
could use Hurewicz to recover the first non-zero homology group from that.
Zoltan sat non-plussed, so I did end up using the Serre spectral sequence,
which might have been overkill, but it turned out to be useful in thinking
about the follow-up questions. I also finally realized that the Hopf fibration
would be useful to mention here and did so, hence...)
Now let's specialize to the case of base S^2 and fiber S^1. Write down the Hopf
fibration. (OK.)
Can you give an example (of a space as in the first question) with non-trivial
homology in dimension 1? With homology Z/p in dimension 1? (I figured out that
a suitable lens space will do.)
How do you classify S^1 bundles over a given space? (I declared that I all I
knew was line bundles were classified by homotopy classes of maps into
RP^infty. He mentioned that CP^infty was the relevant space here, but what he
was looking for was for me to say "the second cohomology group." Which makes
sense -- that's the same as homotopy classes of maps into CP^infty = K(Z,2).
And now I understand why lens spaces give all the examples in response to the
previous question.)
What is CP^infty? Discuss its homotopy and co/homology. (I mentioned it was a
K(Z,2), and that Hurewicz gave the first non-vanishing co/homology groups. I
thought I was going to have to use spectral sequences again to deduce its
cohomology since I was in the K(G,n) frame of mind, but suddenly the right
neurons fired and I remembered the story is much simpler in this case.)
How about RP^infty? (Similar. I computed the boundary map in homology
explicitly, talking about degrees of maps. Therefore...)
What is the degree of a mapping? (First from a sphere to itself, then to
arbitrary manifolds. I had to state all the typical assumptions -- smooth,
oriented, closed...)
What is Poincare duality? (OK.)
What if the space is not closed or orientable? (You can manage.)
What is the Euler characteristic of a 3-manifold? (The alternating sum of ranks
was my first response. Then Zoltan asked it again, I realized what he wanted
me to say, and I enthusiastically and inexplicably announced "Three!" before
sheepisly correcting myself.)
What is the classification of surfaces? (I started fumbling here for no real
reason...)
How does the Euler number enter in? (Yeah, yeah...)
How does it behave under the connect-sums? (I completely blanked. Definitely
the easiest question so far, so it was embarassing to require all the nudging
that I did.)
How do you know that RP^2 # RP^2 is not the same as RP^2 # some orientable
surface? (Well, I'll use the formula I finally managed to scratch out just
before...)
What is the relationship between #^3 RP^2 and RP^2 # T^2? (He asked this after
I had already written that they were homeomorphic. I said you use some polygon
cutting-and-pasting, and that was enough.)
ALGEBRAIC COMBINATORICS (Benny)
Can you state the oddtown theorem? (I decided to start botching things in this
section right off the bat. So it took me a minute to formulate it correctly.)
What if all sets have even size and even intersections? How many can sets can
there be? (I gave the obvious construction of pairing people up and taking all
subsets of pairs. I didn't recall how to show that gives you the maximum
number, but eventually remembered the subspace generated by the incidence
vectors mod 2 is contained in its orthogonal complement, and that gave it.)
What if all the intersection sizes are the same? (Fisher's inequality.
Couldn't quite kill the proof at the end, but it was just some trick I couldn't
recall.)
Discuss more general results along these lines; i.e. (non-)uniform,
(non-)modular Ray-Chadhuri - Frankl - Wilson theorems. Prove the modular
non-uniform version. (OK.) What is the importance of the hypothesis that set
sizes are precluded from belonging to the list of intersection sizes? (The
eventown construction.)
Can you describe some applications? (Explicit constructions in Ramsey theory,
the Borsuk conjecture, both of which I screwed up immensely. At one of the
many points I got stuck I pondered aloud, "What would Noga do?" to no one's
amusement.)
What is Hilbert's Nullstellensatz? (Stated it.)
What is the version used in combinatorics? (Noga Alon's combinatorial
nullstellensatz.)
What is the corollary you use? (If a certain coefficient is non-vanishing then
the polynomial doesn't vanish identically on a certain product set.)
Give an application of your choosing. (I started to talk about list colorings
of graphs, but Benny told me that was the wrong choice, so I talked about
Cauchy-Davenport instead.)
COMPLEX AND REAL ANALYSIS (Natasa)
What does it mean for a function to be analytic at a point? (I said that meant
it had a power series at the point, and she asked how I would prove that.
Prove my definition? I knew what she wanted, and said it was equivalent to
being C-differentiable.)
What are the Cauchy-Riemann equations?
Why do they imply that the real part of a holomorphic function is harmonic?
Cauchy's integral formula and its generalization to higher derivatives.
Deduce Liouville's theorem as a corollary.
Apply this to prove the Fundamental Theorem of Algebra.
What is Cauchy's residue formula?
How can you use it to compute integrals of real functions on the real line?
What is the L^p norm?
What about p = infty? (I answered this and all previous questions, but nothing
noteworthy arose.)
Why do you need to consider equivalence classes of functions in order to define
L^p as a normed linear space? (I had forgotten. She told me to write down the
definition of a normed linear space, and just as I was about to write down the
first axiom for the norm I figured it out.)
How do you prove that it is complete? (This I severely screwed up. I could see
the page from Rudin, but couldn't write it down. She prodded me: show that
completeness is equivalent to every Cauchy sequence possessing a convergent
subsequence. This got the ball rolling a bit more, and we basically did the
proof.)
What are the basic facts about Fourier transform? (There was the standard
discussion about how to normalize properly.)
What is the transform of the convolution of two functions? (I stated what.) How
do you prove it? (I said it was straightforward, which almost satisfied her,
but finally I had to mention you need Fubini-Tonelli.)
Suppose that you had a calculus student who was stubborn and wasn't satisfied
with the Fourier transform on L^1. How could you extend it to L^2? (I didn't
ask, but I wanted to know what this person was doing in a calculus class. I
mentioned Plancherel, starting with L^1 \cap L^2 and going from there. She
mentioned that there is also an approach using Schwarz spaces; I asked her for
a reference on that, and she referred to the book (by Duoandikoetxea) she had
suggested I look at prior to the exam. Whoops.)
ALGEBRA (Benny)
What is Sylow's theorem? (I wrote it all out.)
Show that there is no simple group of order 132. (I was surprised that counting
up elements of different orders actually takes you all the way through here.)
What can you say about finite fields? (Just the standard stuff. I asked how
you know there are irreducible polynomials of every degree, Benny said you just
count them, and then I figured out a different response.)
What is the structure theorem for abelian groups? How many of order 2000? (OK.)
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REFLECTION AND ADVICE (me)
At first, I didn't want to write anything on the board, much to the chagrin of
my committee. I was also quite timid in getting into details about anything,
failed to mention important things that I thought were just obvious, and also
very sloppy in talking through proofs in logical order (especially in the
combinatorics section; my performance there was a major disappointment, which
Benny more or less confirmed afterwards). When it got to Natasa's turn she had
to keep insisting I write on the board for every question. So I would aim to
be a better communicator on my next generals exam.
Very little in the way of proofs were requested throughout. Basically anything
that could fit, in large handwriting, on the back of an index card was
sufficient. Understand the moral arguments to proofs. Much more important to
the committee was that I could apply what I knew in concrete situations.
If you are nervous about the exam, talking to your upperclassmen is a very
consoling experience. These written accounts are limited in how much they can
convey all the bumbling mistakes you make and prodding you receive throughout
your exam. Once you get going you start to calm down, the time flies, and you
just accept the fact that you're going to make mistakes when you think on your
feet. A mock exam would be a good idea in case you are very nervous. Take
your exam as soon as you feel there is nothing more you could do to prepare --
prolonging the wait will just strain your nerves.
These accounts usually don't speak to the extent to which the material you are
examined on is culled to you. Choose and talk to your examiners prior to the
exam so that there won't be any surprises. Zoltan told me early on that
knowing the basics from Hatcher would be sufficient, and I confessed knowing
really nothing about classifying spaces and characteristic classes. I prepared
for combinatorics by using a syllabus for a course Benny had taught in the
past, although "algebraic combinatorics" could mean lots of other things.
Also, a former student of his -- Peter Keevash -- wrote this wonderfully
condensed set of notes on the Babai-Frankl book and Alon's Combinatorial
Nullstellensatz that I studied from, and Benny even kept referring to his copy
throughout the exam! So if you elect to do this as one of your topics, ask
Benny or Peter or me for a copy.
There are some bizarre questions on the standard questions list (as of the time
I took the exam). Just be realistic and don't stress out over them. You're
not going to have to know about the Laplace transform if your topics are
combinatorics and topology, and your committee is there to provide lots of
hints. I think it's pretty easy to identify what is core material from the
available resources, and knowing that is sufficient for passing. There's no
need to get intimidated by more advanced material that you read in some of
these generals accounts, although it is nice to know something more than the
bare minimum.
Bringing a water bottle is a great idea.
Good luck!