Mar Gonzalez

Date: May 16, 2000
Special topics: Differential Geometry and Functional Analysis
Commitee: Alice Chang (chair), Edward Nelson and Hale Trotter.
Length: 2 hours and a half, more or less.

The exam started at 9am. When I arrived, Trotter was not there yet, so I
had to wait. Then they sent me outside to discuss what they wanted to
ask. Finally, we began. They decided the order of the topics: Algebra,
Complex, Real, Functional and Geometry.

Trotter asked all the questions in algebra. He started with: What can
you say about finite groups? ('a bit' vague question...). Then he
suggested me to talk about Sylow theorems, and some applications (groups
of order 15, p-groups...), simple groups... 

We changed to finite fields, and I had to list some properties. Trotter
gave me some specific examples. 

I had to talk about the structure theorem for abelian groups, and a bit
about field extensions. The last thing was to describe the ideals of the
ring of square matrices.

The algebra part was over. Nelson and Chang then asked me about complex:
Fundamental theorem of algebra, Liouville's formula, Maximum modulus
principle...

Nelson wanted a version of Phragmen-Lindelof (an analytic function on
the strip -1<Re(z)<1, continuous on the closure, |f(z)| less or equal
than 1 on the boundary, less than 20 in the interior of the strip).

Map a vertical strip to the unit disk. With this mapping, where does it
go a line through the origin? (Chang)

What about mapping the unit disk to a rectangle? (Riemann mapping
theorem and Schwarz-Christoffel formula - they didn't ask for the
proof!)

Harmonic functions: What's its main property? Talk about Harnack's
principle, first for a ball, then for a general domain. A bound for the
ratio between the max and the min of an harmonic function. (Alice Chang
again).

A function with zeros in the integer numbers. Weierstrass factors in for
the general case. Why these factors?

And now real: Again, just Nelson and Chang.

Suppose you have a sequence of continuos functions in [0,1], pointwise
convergent to f. Is f continuous? What happenes if the functions
decrease to zero? (Dini's theorem). What can you say about the
integrals?

Now a bit deeper question about the same: we don't know if the limit
function is continuous, but... can it be so in some region? Nelson
realized that the question was a bit vague and gave a specific example
to work with: the characteristic function of the rationals. (Baire
Category theorem).

This was supposed to be the real analysis part. However, Nelson changed
to functional (Alice Chang mentioned that she wanted to come back
later).

Take a linear functional defined on L2(0,1) into C[0,1], which we don't
know if it's bounded. Compose then with the delta function. Is the
composition a bounded operator? I mentioned closed graph theorem.

I had a lot of problems with this question, and I was getting tired. The
committee deciced to stop then for five minutes and I threw myself on the
sofa.... 

After the break we didn't go back to this question. So it was time for
more Functional Analysis. They asked what topics I had studied (!!!); it
took me a while till I could concentrate again and say anything...

Talk about the operator defined in L2(0,1) by Tf(x)=int{K(x,y)f(y)dy}, K
square integrable. Is it self adjoint? How does it look like? -another
vague question, but Nelson made it more precise...-.

Talk about the laplacian. How do you define it? Fourier transform and
some properties (regularity, inversion theorem, Plancherel...)

How does it look like a self-adjoint unitary operator? Along the way,
spectral theorem.

Finally the functional analysis part finished; I screwed it up a bit....

Alice Chang moved to Geometry, and she asked everything (we never went
back to real).

Talk about an helix: parametrization, curvatures... She wanted also a
geometrical interpretation, Same with the torus. 

Why is Gauss curvature invariant under isometries? Christoffel
symbols... I didn't have to prove anything since they moved on to
another question as soon as I started.

Geodesics in the sphere, give a Jacobi field.

Gauss-Bonnet (with proof, for surfaces), Euler Characteristic... Then I
had to generalize to other dimensions, I didn't know all the details but
she didn't mind...

Sphere theorem.... Fortunately, without proof!!!
 
I expected a lot of more questions, but they decided to stop here, and
they sent me out for few minutes. They came smiling and told me that I
had passed. And this was the end.......

I have to say that my commitee was very nice. They helped me out if I
got stuck with something, sometimes by giving examples or by making the
questions more specific.

They never asked for rigorous proofs. And usually, when they saw
that I knew the subject, they moved to another thing, so actually I
didn't finish any of the proofs they asked me.

Anyway, Good luck for future generals!!! They are not as terrible as they
may seem...