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.