Kollar (C), Pandharipande, Rodnianski.
May 2nd, 2003.
Time: 1:00 pm.
Algebraic Geometry:
K: Calculate the genus of y^3=x^6-1
i used blowing up to calculate the number of points above
the singular point, and used the map $x$ to \P^1 to calculate the genus
using Huruwitz theorem ( in the case when char(k) is not 2 or 3). Then i
mentioned that in char=3, the curve was a conic(non-reduced). For char=2,
fortunately, ramification was tame, so i could again calculate the genus
using the Huruwitz theorem.
P: What do you know about vector bundles on \P^1
nowhere vanishing section. Pandharipande pointed out to me that
it was not correct. (actually this is true if the vector bundle is
generated by global sections, but i couldn't think of this at that time).
Then i said that every vector bundle on \P^1 is direct sum of line
bundles (its a theorem due to grothendieck, which i didn't have to
prove)
P: What are all line bundles on \P^2? ( i had to prove that any line
bundle on \P^2 is $\sO(n)$ for some $n$.
P: Is the tangent bundle of \P^2 a direct sum of line bundles?
I thought for a few minutes but could not answer.
P: Do you know what are chern classes? calculate the chern classes of
tangent bundle of \P^2.
I calculated the total chern class : 1+3H^2+3H, where H is the class of
hyperplane). Everybody started laughing. I thought i had made some
calculation mistake. Pandharipande said "that's a bizzare way of writing
a polynomial!". I erased it and wrote 1+3H+3H^2 instead of 1+3H^2+3H .
P: What can you conclude from this?
I realised that the polynomial is irreducible over \Z and hence the
tangent bundle of \P^2 is not a direct sum of line bundles.
P: Is it possible to have a family, parametrized by \C, of rank 2 vector
bundles on \P^1 which is trivial at every point of \C except the origin?
I thought for some time but didn't know what to say.
P: What do you know of global sections of these vector bundles?
I stated the semicontinuity theorem. Pandharipande then asked me to
consider the specific example where the vector bundle was \sO(2)+\sO(-2)
at origin and trivial everywhere. It turned out that an
application of semicontinuity theorem shows that this is not possible.
Algebraic Topology:
K: What do you know about homotopy groups?
I gave the definition and explained why it is a group.
K: What do you know about homotopy groups of riemann surfaces?
For Riemann Surfaces of genus greater equal 1, I remarked that the higher
homotopy groups are zero, and found out the fundamental group using an
explicit cell structure. Then for the sphere, i showed that
second and the third fundamental group is \Z (by Hurewicz theorem and Hopf
Fibration respectively). Then i said that all other higher homotopy
groups are finite.
K: Why are they finite?
I gave a brief sketch of the proof using Serre spectral sequence.
Algebra:
P: Why does every matrix satisfy its characteristic polynomial?
I gave an argument, which works over \C, and which uses the fact that
diagonalisable matrices are dense in the space of all matrices and the
statement holds trivially for diagonalisable matrices.
K: Now what can you say about other fields?
I used the same argument, by base
extension of the field to its algebraic closure and using the fact that
diagonalisable matrices are Zariski dense in the space of all matrices.
Analysis:
R: What can you say about analytic maps from \C to upper half disk?
(louvilles thm)
R: What can you say if the map is harmonic?
I had to prove that every real harmonic function (on \C) is real part
of an analytic function (for which i required a lot of help),
which shows that it has to be constant.
R: If you have a function on a measure space of total measure 1 and whose
L^1 norm is 1, what can you say about the measure of the set where it is
greater than 10,000. (the measure is less than 10,000^{-1})
R: What do you know about Fourier transform of an L^1 function?
I defined Fourier transform, and proved the Riemann-Lebesgue lemma.
The interview lasted for about 1hr 30 min.