Emma Carberry's general's.
Gunning, Fefferman and Conway.
Compact Riemann Surfaces and Several Complex Variables.
COMPACT RIEMANN SURFACES (Gunning)
Tell us about the Riemann-Roch Theorem. For what is it useful? (I stated
it in terms of line bundles, and began to justify my definition of
characteristic class as the degree of the divisor of a non-trivial
meromorphic cross-section, but was stopped).
Just say why that is well-defined.
What is a divisor?
For what values of the characteristic class does the Riemann-Roch Theorem
give you complete information (in terms of the number of linearly
independent holomorphic sections)?
What is the generic situation?
How would you justify this? (I introduced the Brill-Noether matrix).
What is the connection between the Brill-Noether matrix and the space of
holomorphic sections? Prove it.
For a given characteristic class, what are the minimum and maximum
numbers of l.i. holomorphic sections? Justify this.
Talk about the case when the divisor is of the form r.p (ie a single point).
What is Abel's Theorem?
What does it mean for g=1? (I said that it lead to the result that a torus
was analytically equivalent to its Jacobi variety, but this wasn't what he
wanted, so he made his question more explicit.)
Explain the statement of the theorem more concretely for the case g=1.
You mentioned earlier that Abel's Theorem tells us that a Riemann
Surface embeds into its Jacobi variety. How do you know that this map is
non-singular? (The collection of holomorphic abelian differentials don't
have a common zero).
Prove this.
What do you know about automorphisms of Compact Riemann Surfaces?
(there's finitely many for g>1)
Why?
Fefferman left the room for a while during this part of the exam, and
Conway was absorbed in some notes he'd brought with him. Gunning asked me
what area I'd like to do next, and when I said 'Algebra' Conway looked up
from his reading and muttered something about needing to think up some
questions. Conway being Conway, this didn't take long.
ALGEBRA (Conway)
State the main theorem of Galois theory. (I began a proof, but he wasn't
interested. He told me that I should answer more quickly - I hadn't
hesitated, but had taken the time to write the statement down - so after
that most of my answers were verbal, and things went quite fast.)
What condition on the Galois group is given by irreducibility of the
polynomial?
What happens when the polynomial has a root in the base field?
What Galois group would you expect a cubic to have?
Draw the subgroup lattice for S3.
Give an example of a cubic with Galois
group S3. (I gave x^3 -2).
Draw the field diagram for its splitting field... (more easy questions
that already I cannot remember).
What is the quaternion group?
Draw its subgroup lattice.
He picked one of the subgroups containing four elements. To what kind of
field extension does it correspond? (quadratic)
Write this field extension as Q(a). Can you say whether or not a is real?
Show that if f is an irreducible quintic with precisely two non-real
roots, then its Galois group is S5.
What are the Sylow theorems? Prove them.
Talk about Jordan canonical form. (I asked whether he wanted me to prove
it from the Structure theorem for f.g. modules over a PID, but he just
wanted me to define it so that he could ask more questions).
Do you know what the three different types of order are that an eigenvalue
can have? (I think 'order' was the term he used - I answered that I didn't
know any names for them, but explained how to calculate the number of
'blocks' associated to a given eigenvalue, the maximum size of each such
block and the sum of their sizes, and he seemed happy. He told me the
names of these invariants, for which I thanked him, but the knowledge
didn't visit me for long).
What happens when the field is not algebraically closed?
REAL ANALYSIS (Fefferman)
Define F(s):= integral from 0 to infty of f(x)exp(-sx).dx, where f is
'nice' (say smooth with compact support, and you may take that support to
be bounded away from 0).
Assume first that s is real. Is F always defined?
Is it continuous?
Differentiable?
What is its derivative?
What about when s is complex?
When is F analytic?
If s is purely imaginary, how do you recover f from F?
What does this tell you about the general case?
SEVERAL COMPLEX VARIABLES (Gunning and Fefferman)
Gunning:
What is a domain of holomorphy?
What other characterisations do you know? (He wasn't interested in proofs
of their equivalence).
Of what are domains of holomorphy in Cn examples? (Stein spaces)
Define a Stein space.(I said it was a space on which the higher
cohomology groups vanished for any coherent analytic sheaf).
What is a coherent analytic sheaf?
Can you 'weaken' your definition of a Stein space?
Suppose that you know that the first cohomology group is trivial for the
ideal sheaf of a particular holomorphic subvariety. What does this tell you?
(You can extend holomorphic functions defined on that subvariety to global
functions).
Suppose now that you have this information for all holomorphic subvarieties.
What does that tell you?
What are the Cousin problems? (I described additive Cousin and showed
that it could always be solved on a Stein space).
What about multiplicative Cousin? (I stated it, showed that on a Stein
space there are only topological obstructions to its solution, and
mentioned Oka's principle).
Suppose that you have a function that is holomorphic in each variable
separately. What can you say about it? (It's holomorphic).
Prove this.
Fefferman:
Suppose you have two holomorphic functions g1 and g2 on C^2. Do there
exist holomorphic f1, f2 such that (f1)(g1) + (f2)(g2) = 1?
The exam lasted about 100 minutes and was conducted most pleasantly.