Bhargav Bhatt's Generals
(total time: 1:22)
Committee: Wiles (chair), Pandharipande, Gunning
Special topics: Algebraic number theory, commutative algebra
Note: For most statements below, the proofs required were far closer to "main
idea sketches" rather than actual proofs. Also, I was allowed to choose what
topic to start with. The account that follows is, to the best of my knowledge,
chronological.
Commutative algebra (Pandharipande)
- compute Tor_1^Z(Z/p,Z/q)
- classify finite dimensional commutative algebras over C with (vector space)
dimension d
- for finite modules over a (noetherian) local ring, prove free = projective =
flat; give counterexamples over non-local rings
- what's Hilbert's syzygy theorem? how does this relate to Serre's theorem on
regular local rings? give two different definitions of K_0 of a regular local
ring (either use all f.g modules, or just the projective ones) and prove their
equivalence
Algebraic number theory (Wiles)
- "prove" that Q has no unramified extensions
- compute the class group of K = Q(sqrt(-31))
- what's a Hilbert class field? what happens to primes of K in the Hilbert
class field of K? what about the principal ideals? why? (somewhere along the
way I was asked to write down the statement of Artin reciprocity)
- "prove" the unit theorem. can you use this to find a fundamental unit of
Q(cubert(2))?
- do you know complex multiplication or modular forms? (no.)
Complex analysis (Gunning)
[Gunning started off by saying "Now, for some comic relief, let's do this thing
called analysis. So clear all this nastiness from the board!"]
- what happens to analytic functions on the punctured unit disk at the origin?
(this led to the standard discussion on singularities)
- what are Laurent series expansions? how are these related to the Cauchy
integral formula? how does one compute the Laurent coeffecients?
- state the Riemann mapping theorem (stated it for subsets of C).
- [Pandharipande: didn't Riemann do this for Riemann surfaces? (no.) can you??
(yes.)]
Real analysis (Gunning)
- relations between L^1, L^\infty and pointwise convergence, counterexamples in
the opposite directions, completeness of L^p
- state the Radon-Nikodym theorem. how does this give a Lebesgue analogue of
the fundamental theorem of calculus?
Algebra (Wiles)
- state the Sylow theorems
- classify groups of order 35
- classify groups of order 21 (this took me the longest as I didn't realise 7
divides 21 for about 10 minutes!!)
- [Pandharipande: how many irreducible representations does S_n have? what
classical function in mathematics does this number relate to?]
Commutative algebra (Pandharipande)
- what's the automorphism group of k[x]?
- do you know what the automorphism group of k[x,y] is? (isn't this hard??)
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It lasted an hour and 22 minutes.
I was fortunate to have a really nice committee -- the questioning was very
casual. Also, they gratuitously, and with amazing frequency, dropped generous
hints (ex: to speed up computations of the class group, to let me know if the
question was hard, and, in some cases, to even tell me the answer) -- it
hardly felt like an exam!