Evan O'Dorney's generals, April 7, 2017 (by special arrangement), 1:30 p.m. -- 4:00 p.m.
Committee: Manjul Bhargava (chair), Shou-wu Zhang, Assaf Naor
REAL ANALYSIS:
What is the Baire category theorem?
[A countable union of nowhere dense sets in a complete metric space cannot cover the space]
And what's the your favorite application of the Baire category theorem?
[I replied that one could prove that most (i.e. a nowhere dense set of) L^1 functions are L^2. I stumbled in trying to prove it and asked for a hint. Naor reminded me that I myself had picked this problem.
Eventually it became clear that the proof was not very hard and that the Baire category theorem would come into play, if at all, only in a very degenerate form. I had not quite finished the details, but he seemed happy with the proof. I asked if he would have picked a different theorem. He named a few I had not heard of.
Then he named the theorem that any pointwise limit of continuous functions must be continuous somewhere. I again stumbled in the proof, and Naor reminded me that to use the Baire category theorem, one needs to write the "bad" set as a countable union of nowhere dense sets.
That sounds obvious, but I really felt taught at that moment; I had not seen the unifying thread concerning what space to apply Baire on in various problems. With a few hints, I got the gist of the proof --
to consider {x : max_{|x-y| <= 1/k} f(y) - min_{|x-y| <= 1/k} f(y) <= 1/n} and approximate this max and min by the pointwise convergent functions. As half an hour had elapsed and I had expressed desire to move on from real analysis, we moved on.]
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COMPLEX ANALYSIS:
Classify all conformal maps from the upper right quadrant to the unit circle.
[Square, apply any real-coef Mobius transformation of positive determinant, and then take a Mobius transform to the unit circle.]
And what is the Mobius transform from the upper half-plane to the circle?
[I scribbled that it was something like (z+i)/(z-i) and remarked, "I can be careful if you want me to." Manjul directed my attention to the image of i. I quickly realized it should be (z-i)/(z+i).]
And why are these all?
[The automorphisms of the half-plane are just PSL_2(R), e.g. by the Schwarz reflection principle]
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How would you prove the maximum modulus principle?
[Assume that |f(z)| had a local max in the interior of a region, and use the mean value theorem to derive that f must have been constant in a disk]
And how would you prove the mean value theorem?
[Evaluate the contour integral of f(z)/z using the residue theorem]
And why is the residue theorem true in this case?
[The function f(z)/z has a Taylor series that begins a_0/z + a_1 + a_2z + ..., and all but the first term is analytic and integrates to 0]
But in most treatments, the existence of Taylor series comes after the mean value theorem. How do you avoid using them?
[Use Cauchy-Goursat to shrink the circle of integration, and use differentiability to argue that (f(z) - f(0))/z is bounded near 0, hence its integral around a tiny circle contributes nil in the limit]
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What can you say about the zeros of an analytic fn?
[They form a discrete set]
Why?
[If an analytic fn has a sequence of zeros converging to a limit point, it must be identically 0]
Why?
[Factor out the leading term of the Taylor series at the limit point. This elicited laughs but was correct]
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What is the Riemann zeta function?
[\sum_n n^{-s}; converges for Re(s) > 1, has a meromorphic continuation to C...]
And why does the series define an analytic function for Re(s) > 1?
[because it converges uniformly on compact sets]
And how do you analytically continue it?
[Extend to Re(s) > 0 by integration by parts, then prove the functional equation. I showed the first step on the board, and I mentioned some of the main steps of the second.]
Where are its poles and zeros?
[Simple pole at s = 1, trivial zeros at the negative even integers, and nontrivial zeros that seem to be on the line Re s = 1/2]
What is Poisson summation?
[I was caught off guard because I didn't remember this was part of the proof of the functional equation. I stated it -- sum of f(n) equals sum of \hat{f}(n) over integers n, provided that e.g. f is Schwartz -- and then we went through the proof of the fnl eq step by step.]
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ALGEBRA:
Classify all integral domains of size 2017.
[Just Z/2017Z, the field of prime order]
And why must a finite integeral domain be a field?
[Multiplication by a nonzero elt is injective hence surjective]
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Prove that x(x+1)(x+2) + (x+3)(x+4) is irreducible over F_5.
[It is cubic and has no root]
For the same reason, r(x+a)(x+b)(x+c) + s(x+d)(x+e) is irreducible over F_5 for any nonzero scalars r,s and any permutation {a,b,c,d,e} of {0,1,2,3,4}. Can any irreducible cubic be put into this form?
[Yes, and the proof follows]
How many irreducible cubics are there over F_5?
[|F_{125} \ F_5| / 3 * 4 = 160]
And how many polynomials of the given form?
[4 * 4 * 5!/2!3! = 160]
Now it's enough to show that two polynomials of the given form cannot be equal.
[We worked through the cases when {a,b,c} and {a',b',c'} have 1, 2, or 3 common elements. Afterwards I discovered a more elegant proof that I leave the reader the pleasure of discovering]
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Name a Dedekind domain that isn't a UFD.
[Z[\sqrt{-5}]]
Name an example over a function field, say, is C[x,y]/(x^2 + y^2 - 1) a UFD?
[Apparently not: y^2 = (1 + x)(1 - x)]
Can you write C[x,y]/(x^2 + y^2 - 1) in a different form?
[The circle is complex affine equivalent to the hyperbola xy = 1, so it's also C[x,1/x]]
Is that a UFD?
[Yes!]
[The examiners were going to ask another question, but I was wondering where my initial analysis had gone wrong, and I was able to verify that y is not actually irreducible.]
How do we think of this geometrically?
[As a Picard group / divisor class group]
What happens in the case C[x,y]/(y^2 - x^3 + 1)?
[Elliptic curve has infinite Picard group, even after removing a point]
What about the case R[x,y]/(x^2 + y^2 - 1)?
[This is again not a UFD, and the examiners were happy with my incomplete proof]
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REPRESENTATIONS OF FINITE GROUPS (a.k.a. 0-dimensional compact Lie groups):
What is Brauer's theorem?
[I had no idea and they moved on]
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Write the character table for S_4.
[I constructed the 5 irreps while writing their characters.]
Why are these all the irreps?
[Either because the number of irreps is the number of conjugacy classes or because the sum of the squares of the dimensions is |G|.]
Pick one of these and prove it.
[I chose the latter and explained that it comes from the decomposition of C[G] into matrix algebras.]
Why must every irrep appear in C[G]?
[Because it's a fin gen C[G]-module, hence a quotient of C[G]^n for some n, hence a direct summand]
Is there a more explicit way to embed rho tensor rho^*, for any irrep rho, into C[G], respecting the G x G-action by left and right multiplication?
[I had never seen one. Zhang taught me.]
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REPRESENTATIONS OF COMPACT LIE GROUPS AND LIE ALGEBRAS:
Tell me about the simplest compact Lie group.
[The circle?! Because my experience is mainly with semisimple Lie algebras, I worked out the rep theory from scratch, eventually finding that the irreps are indexed by Z and that complete reducibility holds, which surprised me, as it fails for the corresponding Lie algebra.]
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You've just shown that the "unitary dual" of S^1 is Z, where the unitary dual of a topological group G is Hom(G,S^1). Accordingly the unitary dual of Z is S^1. Here's a difficult question (from Zhang): What is the unitary dual of Q?
[I expressed it as an inverse limit and wondered if it had some nice structure]
Certainly R embeds into Hom(Q,S^1). In fact, R is the connected component. What, then, might the component group be?
[I guessed \hat{Z} and he told me my intuition was correct]
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Why do mathematicians talk as if the study of all semisimple Lie groups reduces to that of compact Lie groups?
[Unitary trick: Any ss Lie gp shares its (complexified) Lie algebra with a cpct gp]
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Classify the representations of SU_2.
[I shifted focus to sl_2(C), went through the decomposition into weight spaces and the argument that the weights must be integral, and constructed the reps as symmetric powers of the std]
Call the k^th power of the std rep V_k. For what k,m,n does V_k tensor V_m tensor V_n have an invariant vector?
[When k,m,n satisfy the triangle inequality and have even sum. I had worked this out in preparation, and had I not, the examiners would have been happy with a sketch of what one would do]
Decompose Sym^2(V_n).
[I described the procedure and illustrated it in the case n=4]
Your calculations show that there is a quadratic map from V_4 to V_4. What is it?
[I didn't know. I mentioned that I had done similar cases and that I knew that the quadratic map from V_2 to V_0 takes a binary quadratic form to its discriminant. Manjul directed my attention to the map from V_3 to V_2. It had come up in papers I've read: it is the Hessian. Only then did I realize that all three of these maps can be described in the same way.]
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What is the Weyl character formula?
[I showed an example for sl_3 while stating it]
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ALGEBRAIC NUMBER THEORY:
What primes are of the form x^2 + 2y^2?
[Since Z[\sqrt{-2}] is a PID, it's just the primes that split in this ring, which is those of the form 1,3 mod 8]
What primes are of the form x^2 + ny^2? What can you say in general?
[It happens if and only if the polynomial H_n defining the Hilbert class field splits completely or, equivalently, has a (simple) root mod p. There are subtleties with small primes / primes dividing n (I think), but the examiners didn't get into these.]
When are congruence conditions on p enough?
[When the class group is 2-torsion. Manjul reminded me that I meant the ring class group of Z[\sqrt{n}], not the class group of the ring of integers of Q[\sqrt{n}].]
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Why is the class group finite?
[Minkowski bound represents every ideal class by an ideal of bounded norm; then use unique factorization to say there are finitely many of these]
What does the bound look like?
[Roughly sqrt(Disc K), yielding a bound of (Disc K)^{1/2 + eps} for the class number]
Is it attained?
[Contrary to what I had thought, it is actually sharp for imaginary quadratic fields, and Manjul helped me prove it]
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Mod what proportion of primes is 2 a cube?
[2/3. Apply the Chebotarev Density Theorem to the Galois extension Q[2^(1/3), \omega]]
How do primes split in the extension Q[2^(1/3)]?
[I had mainly answered this. He wanted me to fill in the gaps, especially with regard to the ramified primes.]
What is the discriminant?
[Pairwise differences of roots of x^3 - 2 yields -108]
What is the ring of integers? You've been using this implicitly all along.
[The discriminant shows that Z[2^(1/3)] is maximal away from 2 and 3. Then the elements 2^(1/3) and 2^(1/3) + 1, respectively, satisfy Eisenstein polynomials at 2 and 3.]
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What is the analytic class number formula?
[I had seen some special cases, but not a proof. We ended with a short introduction to Artin L-functions.]
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[The exam lasted 2 1/2 hours. I was very happy to have started with Real Analysis, because it gave the opportunity for the examiners to ask questions in other topics that encompassed some real analysis and discover that my skill was not so bad after all. I remained calm throughout the exam (thank God!). The examiners deliberated behind closed doors for about 3 minutes before announcing that mine was a "great pass."]