Name: David Nadler
Committee: Christodoulou, Kochen (C), MacPherson
Topics: Algebraic Topology, Differential Geometry
Date: May 6, 1997
Duration: Two and a half hours or so



Kochen and I chatted a couple of minutes before MacPherson and Christodoulou 
arrived. Kochen mentioned that he had been a grad student at Princeton and 
had taken this exam when "men were men" and the exam was far more fearsome. 
I was not sure if this was a good or a bad sign... The others arrived, they 
sent me out of the room for a minute to get themselves organized, then called 
me back in and we got underway with Christodoulou somewhat sinisterly 
announcing, "We will begin with real variables..."



Real Analysis (Christodoulou):

What is a necessary and sufficient condition for a function to be Riemann 
integrable?

Is it possible that the characteristic function of an open set is not 
Riemann integrable?

What is a necessary and sufficient condition for a function to be the 
indefinite integral of its derivative?

If a function is continuous and of bounded variation is it absolutely 
continuous?

Suppose you have a sequence of functions f_n of bounded variation on a 
compact interval [a,b] (normalized so that f_n(a)=0 for all n) such that 
their total variations are uniformly bounded, i.e. there exists a constant 
c>0, independent of n, such that Tf_n(b)<c for all n. What can you conclude?

State the Arzela-Ascoli theorem. What is the analog for L^p spaces? In 
other words, characterize the compact subsets of an L^p space.

Do you know what a weak derivative is? Let u be a compactly supported 
function on R^2. Can you bound an L^p norm of u in terms of the integral 
over R^2 of the length of the gradient of u? 

Complex Analysis (MacPherson):

Give an example of a real-valued real analytic function on R whose power 
series at zero does not represent it everywhere.

Integrate a function over R by contour integration and Cauchy's theorem.

(Kochen) If f is meromorphic, what is the meaning of the contour integral 
of f'/f?

What is the universal cover of C minus a point? minus two points?

Classify all Riemann surfaces with first betti number equal to one.

Algebra (Kochen):

Classify finite fields and their subfields.

State the fundamental theorem of Galois theory.

What is the Galois group of x^4 - 2 x^2 + 9?

Consider the simple operator on C given by multiplication by a complex 
number. It decomposes into a stretch and a rotation. What is the 
generalization of this to operators on a Hilbert space?

Define p-adic numbers; what is a valuation?



Lemonade Break: Christodoulou requested we take a short break; Kochen took 
this as an opportunity to serve lemonade. I was happy with water, so I 
declined. But Christodoulou was so enthusiastic at being offered a 
glassful that I probably made a mistake. (The lemonade did appear homemade.)


 
Differential Geometry (Christodoulou):

Define the second fundamental form.

Consider a curve c on a hypersurface in a Riemannian manifold with unit 
normal field N along c. Let c_t be the curve given by exponentiating tN. 
Calculate the change in length of c_t with respect to t in terms of the 
second fundamental form of the hypersurface.

Fix a point p in a Riemannian manifold. Characterize a point q in the cut 
locus of p which realizes the distance from p to its cut locus.

Fix a point p in a Riemannian manifold and a vector X in the tangent space 
at p. Consider another vector Y in the tangent space at p but translate it 
to X (in other words, identify the tangent space at p and its tangent 
space...); call it Y*. If you pull back the metric from the manifold to the 
tangent space at p via the exponential map, what is the length of Y* in 
terms of X and Y?

Algebraic Topology (MacPherson):

Classify real vector bundles over a circle. What are their Whitney classes?

Give an example of two vector bundles with the same Whitney classes which 
are not isomorphic.

What is the homology of S^3 minus a trefoil knot? two linked circles?

Does Alexander duality preserve the cup product?  

What is the cohomology map induced by the inclusion of CP^2 minus a point 
into CP^2?