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Eaman Eftekhary's generals (Jan 17, 2001)
Topics: Algebraic Geometry, Algebraic Topology
Committe: J.Kollar (chair), W. Browder, R. Gunning
(lasted about 1:40)
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Gunning began with asking some questions in analysis:
G: geometric property of holomorphic functions? regions conformally equivalent
to the interior of unit disk (Riemann mapping)? talk about extension of a
hol. map from a triangle to disk to a neighborhood of triangle. different
types of singularities? (Big Picard?) [and some other questions that I've
forgotten]
G: different notions of convergence for functions over [0,1], L^p spaces? how
are L^p spaces related? prove it. does the convergence in L^1 imply
pointwise convergence ? what if the sequence is bounded ? [dominated
convergence]. how do you prove that L^p is complete? [again, I don't
remember the rest of it!]
Then Kollar started with algebraic geometry:
K: compute the genus of y^3 = x^6-1.
I started with char(k)=2,3 and said that I should use Frobenius map to
reduce to another equation, Kollar said what's the inverse of this
map,... then I moved into computations, blowed up the curve at its
singularity a few times till I got a nonsingular curve, counted the # of
pts mapped to the singular pt of the initial curve, composed this map with
(x;y;z) --> (y;z) to P^1, computed the ramification divisor and finally
got the genus of the normalization using Hurwitz formula, but I didn't
know that this is actually the geometric genus of the initial curve and
Kollar was not happy with it.
Then he asked me to define arithmetic genus, and I did. he asked me how
I prove the genus formula, I said through Hurwitz formula. He didn't like
that way, and asked whether I can do it differently. I wrote the sequence
0--> L(-C)--> O_X --> O_C -->0 (X=P^2) and computed the arithmetic genus
using [C]=[dH]. Then he asked me to state the explicit computation giving
h^i(P^n,O(m)) and Serre duality. He mentioned that I didn't use nonsingularity
of C and hence the genus formula is true for arbitrary curve, giving
arithmatic genus.
K: talk about the extension determined by the function fields of C and P^1
determined by the map you gave. I said it's finite of degree 6 and showed
that it's separable when char(k) is not 2 or 3, in case char(k)=3, I
mentioned that the function is reducable,....
Then they asked Browder if he is willing to ask some algebraic topology
questions!
B: what are Stiefel-Whitney classes good for?
(I showed that P^n does not immerse in R^{2n-2} for n a power of 2)
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comments:
as everybody will probably note, the exam was not as fast and smooth
as it appears here, I made some mistakes, but they kindly tried to let
me find them, by myself.
doing math, when you're doing with 6 expert eyes following you, is harder
than doing it for yourself.
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