Mehdi Yazdi's generals
General Committee: Zoltan Szabo(chair), Sucharit Sarkar, Robert Gunning
Special Topics: Algebraic Topology, Differential Topology
05/08/2013 1:00 pm
First they asked me which topic do I like to start with and I said Algebraic Topology:
Algebraic Topology
[S] What are the homotopy groups of RP^2?
[Sz] What is the action of Pi_1 on Pi_n?
[Sz] Compute the action of Pi_1 on Pi_2 for RP^2?
I struggled here a lot and they gave me lot of hints but I wasn't able to solve the
problem which was quite simple, I was embarrassed, the answer is that the action of a generator of Pi_1 on
Pi_2 is the map x ---> -x from Z to itself.
[S] this may not be an easy question:
prove that RP^2 can not be made into a topological group.
I wasn't sure the difference between a topological group and a Lie group
so they asked me why it's not a Lie group, this was simple because the tangent bundle of a
Lie group is parallelizable but it's not true for RP^2(for example because the total stiefel-whitney class is nonzero).
Then Sucharit said it's not a topological group because if it was, the action of
PI_1 on Pi_2 should be trivial which wasn't(previous question!)
[Sz]How do you compute homology groups?
I said probably the easiest way is using cellular homology and zoltan asked me to explain it
more(chain complex, boundary formula and so on)
[Sz]compute homology groups of RP^2 * RP^2 with Z2 coefficient.
I said that it can be computed by giving it a cell structure or using kunneth formula for cohomology
and universal coefficient theorem, I wrote down the cohomology ring of the product and they said that it's
enough.
[S]what are Eilenberg-McLane spaces? what is K(Z,2)? How do you construct K(Z,3)?
[Sz]Prove that a simply connected 3-manifold is a homotopy 3-sphere.
I said H^1 is trivial because it's abelianization of Pi_1, then H_2 is trivial by Poincare duality
and H_3 is Z because it's a 3-manifold then we can use the Whitehead theorem. I was looking for a map
M ---> S^3 s.t. it induces isomorphism on H_3, which they led me through such a map,
during the argument they asked me to define degree for a map between two compact manifolds of same dimension.
Then they asked Gunning if he would like to ask some general questions, he started with complex analysis:
[G]Suppose that you have a function that is holomorphic on punctured plane, what can you say about its
behavior around that point? this led to discussion of three types of singularities, their characteristic,
Laurent series and giving an example of essential singularity.
[G]What is a conformal map? relation between conformal maps and holomorphic maps?
the most famous theorem in this subject? I stated Riemann mapping theorem but I forgot to say
simply connectedness hypothesis and they corrected me.
then Prof. Gunning asked about boundary behavior in Riemann mapping theorem. I discussed the sufficient
condition for being able to extend the map bijectively to the boundary(all the boundary points be simple[Rudin])
[G]What is your favorite non elementary function in complex analysis?
I answered gamma function, defined it using integral formula for half plane, writing functional equation and explaining its
extension to complex plane as a meromorphic function. then he asked me to draw graph of gamma function for real numbers!
[G]Do you know any intrinsic definition of gamma function?
I said it has poles at nonpositive integers and 1/f has exponential growth 1(I forgot to say 1/f but he allowed me to continue),
but he said that there are lots of functions with these properties, the he asked me do I know about convex functions and
I said no and he let it go.
Then he asked some real analysis questions:
[G]different notions of convergence and relation between them?
I stated dominated convergence theorem and the fact that L^1 convergence implies a.e. convergence for
a subsequence with a sketch of proof.
[G]What is the fundamental theorem of calculus in Lebesgue context?
define the absolute continuity, relation between absolute continuity and bounded variation,
a simple way of representing a function of bounded variation(difference of two continuous increasing functions).
Prof. Gunning was satisfied with analysis so they started to ask some differential topology questions,
I said that I've read characteristic classes but I didn't have time to read
h-cobordism completely(and they didn't ask any questions from h-cobordism!).
[Sz]Talk about cobordism groups.
I stated theorem of Thom about rank of oriented cobordism groups and the generators, also mentioned
stiefel-whitney numbers and pontryagin numbers as complete cobordism invariants of an oriented manifold(stiefel-whitney
numbers in unoriented case). Then he asked me to write unoriented cobordism groups and generators for low dimensions.
I knew it for oriented case but I wasn't sure about unoriented.
[Sz]Talk about Chern classes(inductive definition, its relation with cohomology of complex Grassmann manifold).
[Sz]Cell structure for complex Grassmann manifold?
I explained the cell structure then he asked me is it easy to compute homology groups using cell structure,
I said the number of cells is growing fast and it depends on the attaching maps, then he asked me to consider it
for special case of G_1(C^k) which is projective space, in this case its easy because there is no cells in
odd dimensions and the same is true in general case!
then he asked sucharit if he wants to ask any questions:
[S]prove that a closed surface with infinite fundamental group is a k(pi_1,1).(after a second they said to prove
it for orientable ones but the proof works for both cases)
I proved it using the classification theorem for closed surfaces and the fact that we know their universal covers(
in case of infinite fundamental group we see it's euclidean plane or hyperbolic plane which are contractible)
[S]prove that the tangent bundle of an orientable closed 3-manifold is parallelizable(Stiefel's theorem).
I proved that stiefel-whitney classes are trivial using Wu's formula and then just stated that the obstruction for
having two linearly independent sections is zero, if we consider their cross product as the third section(because it's orientable)
then we have three and we are done.
then they asked Prof.Gunning if he wants to ask some algebra questions:
[G]Classification of linear maps from a vector space to another one(probably with different dimension)?
I was confused because the standard forms(Jordan and rational) doesn't work here, finally it turned he
was looking for rank of the linear map. then he asked me about classification of linear maps from a
vector space to itself and I explained Jordan form and rational canonical form. then he asked me about
a theorem(with analytic nature) about deciding whether a matrix is positive definite or not. I wasn't
sure what he was looking for and finally I stated Sylvester's criterion(my problem was that I didn't know the name of this theorem!)
[G]classify groups of order 12.
I was half the way that he became sure that I can finish the argument and let it go.
[G]Fundamental theorem of Galois theory, definition of Galois extension, an example of a non Galois extension,
Galois group of x^3-2.
general comments: the committee were really nice and helpful, they led me through some of the arguments and I think they
were willing to ask questions that I knew not questions that I didn't.
The exam took about 100 minutes.