Committee: Tian(Chair), Neves, Andre, Salehi Golsefidy, Alireza

Date: April 21, 2009

Topic: Complex Geometry, Atiyah-Singer index theorem

 

 

Tian asked me to choose the order of generals, so I chose to do algebra first.

 

Algebra:

Alireza: Do you know anything about the character table of finite groups?

Me: It is defined to be the table of the trace functions of irreducible 
representations of that group.

Alireza: Why is the character table important?

Me: Because all the irreducible representations are the generators of all finite 
dimensional representations of that group.

Alireza: Why?

Me: Because we can define an inner product on the space of finite dimensional 
representations.

And I wrote down that inner product.

Alireza: What is that vector space?

I realized that the space should be the group ring CG.

 

Alireza: Now suppose we have an element g in G, such that \Chi(g) is real for 
every \Chi, what can you say about g?

I had no idea at the beginning, then he told me to consider \Chi(g^{-1}), and 
to see if I can diagonalize it. I understood that g and g^{-1} can be diagonalized 
simultaneously and \Chi(g)=\Chi(g^{-1}), so g and g^{-1} should lie in the same 
conjugacy class.

 

Alireza: Now for linear algebra. Can you show that A and A^t are similar?

Me: A^t is the matrix for the some linear transformation in the dual space....

Alireza: How about 2 by 2 matrix?

I classified all 2 by 2 matrix up to similarity.

Alireza: Then the general case?

I mentioned the characteristic polynomials, elementary factors, etc, but they 
were not what I need. He then asked me to consider the Jordan canonical form. I 
said that we can reduce it to the case when there is only 1 Jordan block in 
the matrix.

Andre: Why can we reduce it to this case?

I explained why. Then I understood I just needed to change the order of the basis.

Alireza: What is a Noetherian ring?

Me: A commutative ring with identity which satisfies the ascending chain condition.

Alireza: Can you give me an example?

Me: The polymonial ring.

Alireza: In C[x, y], given an example of a prime ideal and prove that.

Me: <y-x^2>. And I prove that.

Alireza: What property of the ring do you use in this proof?

Me: UFD.

 

Real analysis:

Andre: What is L^p space?

I told him the definition.

Alireza: Suppose you have a irrational number \alpha, why the fractional parts of 
n\alpha are equi-distributed in [0,1]?

I don't know the exact definition of equi-distribution, and I guessed a definition. 
Then Alireza told me that it should be in the weak sense, i.e. for a continuous 
function f,

 

(1/M)\sum_{n=1}^M f({n\alpha}) tends to \int_0^1 f(x) dx.

 

I didn't know how to show it. Then Andre told me to consider the equivalent problem 
on the unit circle, and consider the case f=e^{nx}. I realized that the left hand 
side tends to zero and integral of e^{nx} is zero. And I explained that triangular 
polynomials are dense in L^2, because they are an orthonormal basis for this Hilbert 
space.

 

Andre: So why continuous functions are dense in the space of L^2 functions?

Me: Because continuous functions can be approximated uniformly by triangular 
polynomials.

Andre: What about the space L^p?

I didn't know how to do. Then Andre asked me to consider the convolution. Then I 
understood that the convolution with a bumped function gives a sequence of smooth 
functions converging to the given function.

Tian: Why this sequence converges to the given one?

Me: Because the translation is a uniformly continuos family of operators in L^p, and 
I guess that convolution is an bounded operator.

Andre: What about in Sobolev spaces?

Me: We can transform the derivatives on the given function to the derivatives on the 
bumped function.

 

(There could be more questions but I don't remember.)

 

Complex analysis

 

Andre asked Tian if he could ask questions about conformal mappings, Tian said OK.

Andre: Is there a conformal mapping from the whole plane to the unit disk?

Me: I state the Riemann mapping theorem.

Tian: Why this theorem doesn't apply to the whole plane?

Me: Because then we have a bounded entire function.

Andre: Can you write down a conformal mapping from the upper half plane to the unit 
disk?

I gave the standard one.

Andre: Can you map a square to the unit disk?

I wrote down an elliptic integral. Then Alireza asked me to explain how the square 
roots are defined. He actually which rays whould be deleted from the whole plane.

Then Tian and Andre asked me to show why this integral maps the upper half plane to a 
rectangle. I showed them the path of real axis by the map.

Andre: Can you map the plane minus the origin to a cylinder?

I thought for a while and showed him that we can map an annulus to a finite cylinder 
and glue all the annulus together.

Andre: Do you know the conformal mappings from S^2 to S^2.

Me: They are the fractional linear transformations.

And I wrote down the general form.

Andre: Why they are all the conformal mappings?

Me: Because firstly the group of fractional linear transformations acts transitively, 
so we only have to consider all maps which fix the origin...

Then I got stucked on this case. I showed that the function should have a simple zero 
at the origin. Then Tian told me to consider the same problem at the infinity. Then I 
got confused on the change of coordinates. But finally I saw that all these maps are 
z\mapsto az.

Tian: Why a bounded holomorphic function reduces to a constant?

I wrote down the Cauchy integral formula, then realized that I need the derivative 
formula.

 

Atiyah-Singer index theorem

Tian: What is a Fredholm operator?

I gave the definition.

Tian: Why is the Laplacian a Fredholm operator?

I said that the Laplacian is bounded from the W^{2, 2} to L^2 and we can construct a 
parametrix of the Laplacian. But Tian was not satisfied.

Tian: Do you know any usual estimate for Laplacian?

I wrote down the Garding's inequality, but forgot what norms should be put on each 
side. Then Andre gave me some hints to guess the right norms.

Tian: Then can you use this estimate to prove that the Laplacian is Fredholm?
I realized that by Sobolev embedding, the kernel should be a subspace of L^2 whose 
unit ball is compact.

Tian: What is the simplest case of Atiyah-Singer index theorem?

Me: The Gauss-Bonnet.

Andre: In the surface case?

Me: The integrand is the Gauss curvature.

Tian: Why is it an index theorem?

I explained that the Euler characteristic is equal to the index of the Dirac operator 
(d+d^*), by Hodge theory.

Tian then asked me to give a brief proof of Gauss-Bonnet using heat kernels. It 
turned out that the proof is so long that he asked me to reduce the argument many 
times.

Tian: What is the transgression form?

First I explained in general dimensions. Then Tian asked to explain just in dimension 
2. Actually I didn't write down an explicit angular form on the unit sphere bundle 
but I briefly explained the Thom form, the Euler form and the augular form. And I 
showed how to prove Gauss-Bonnet by transgression and Poincare-Hopf.

 

Complex geometry

 

Tian: Compute the first Chern class of a degree d hypersurface in CP^n.

I did the computation.

Tian: Why do we have the adjunction formula?

I gave the standard local section of the dual of the normal bundle and showed that 
the induced trivialization has the same transition function as -V.

Tian: If the first Betti number of a compact complex manifold is r, is it possible 
that the manifold is Kahler?

Me: When r is odd, it is impossible.

Tian: Can you give an example of a compact complex manifold with first Betti number r?

I was considering the tori and Hopf surfaces. Then Tian asked me to do the case r=1.

I showed them the Hopf surface is an example.

 

I wasn't so nervous during the exam, but I still felt upset because I got stucked so 
many times. They gave a lot of hints when I didn't know how to proceed.

 

It took us about 2 and a half hours.