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??) ------ 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!