Dan Fess – General Exam
Date/time: 25th May 2017 – 1.30pm to 4.20pm
Examiners: Sarnak, Skinner, Pusateri
Location: Skinner’s office (501)
There’s inevitably stuff that I’ve forgotten from my exam, but hopefully what’s below is relatively complete and useful for future students taking their general exam.
Real Analysis
P: Define the Fourier transform. (I gave the L1 form.) What properties does it have? Prove Riemann-Lebesgue. (I proved it for a nice dense class of functions and then invoked continuity of the Fourier transform from L1 to C0. I think I showed it for compactly supported smooth functions using integration by parts, which is probably way stronger hypotheses than necessary.)
Sa: Why are compactly supported smooth functions dense in L1? (Approximate the characteristic function of an interval and we’re done.)
Sa: So we need to know one compactly supported smooth function [and then we can play with it so it approximates the char. function of an interval]. Do you know one? (I mentioned e^-(1/x) is infinitely flat at 0. Sarnak asked me to prove this and I totally floundered; I couldn’t even show the derivative at 0 was 0. Eventually I used L’Hopital’s rule on x/e^x and the examiners all chuckled, but I was just happy to have got it out the way.)
Sa: Okay so how do we make this guy compactly supported? (Reflect it in, say, the line x=1 and multiply the two functions together.)
Sa: Is the Fourier transform of a compactly supported L1 function smooth? (I said no, to momentary silence, and figured I was wrong…. Sarnak told me to try differentiate under the integral sign, I responded saying that I had forgotten when that was permitted, and he said something along the lines of “if you do it and it makes sense, then you’re good”, which I found kinda funny.)
Sa: [Very briefly into Complex] Now consider the Fourier transform as a function on the complex plane. This is an entire function. What determines the number of zeroes of an entire function? (Order, defined it) What’s the order of the complex Fourier transform? (At most 1.)
Before getting too deep into Complex, Sarnak turned to Pusateri to take us back to Real, and said that we would return to this problem later.
P: What’s Lp space? What relations are there between these spaces? (Only when we are in a finite measure space or when the measure is discrete may these be subsets of one another.)
Sa: Is the unit ball in C([0,1]) compact? (No, e.g. take functions of norm 1 supported on [1/(n+1),1/n]. Furthermore, it’s true than the unit ball in a normed vector space is compact iff it is finite dimensional.) What does it mean for a linear operator to be compact? (Maps bounded sets to precompact sets.) Is the identity map on C([0,1]) compact? (No, this question is identical to the first one.) Suppose I have an integral operator from C([0,1]) to itself with a continuous kernel. Is this compact? (Yes, limit of finite rank operators given by approximating the kernel uniformly by step functions.)
Complex Analysis
Sa: State the Riemann Mapping Theorem. When are two doubly connected regions conformally equivalent? (I said I didn’t know and thought about the problem for a moment before Sarnak said that I wasn’t going to reinvent this part of math…okay fair enough.) How would Riemann understand conformal equivalence of such regions? (I said via the Dirichlet problem but that I hadn’t read anything in detail about this. At this point Sarnak was surprised and didn’t believe me when I said Stein & Shakarchi didn’t have anything about this. Skinner mentioned he had a copy on his shelf while laughing a little and Sarnak climbed out of his armchair to go grab it. He rifled through the pages, and I got a little nervous that maybe I had skipped over this part of the book….but sure enough there was nothing about it in there.)
Sa: What’s the maximum principle for harmonic functions? (I said I only knew it for harmonic functions on R^2, and wrote that down assuming the same thing held in general. He asked me about solving the Dirichlet problem but I said I only knew it in the cases that Riemann Mapping can tackle. Sarnak wasn’t pleased but I think his disappointment was with the content of Stein & Shakarchi rather than me. He’s more a fan of Ahlfors.)
I think we moved on to Algebra at this point, although looking back on it the Complex Analysis section seems very short. We never returned to the Fourier transform question, so most of the Complex Analysis was spent with Sarnak reading Stein & Shakarchi…
Algebra
Sk: Can you construct a polynomial with Galois group S_n? (Cycle type of Frobenius is given by factorisation of f mod p, which can be chosen so that we get an n-cycle, and (n-1)-cycle and a transposition, then these generate S_n.)
Sk: Tell me about Rational Canonical Form. (Use Structure Thm for f.g. modules over PID.)
Sk: How many conjugacy classes are there in GL_2(F_p)? (Use rational canonical form. There are (p-1)(p+1).) How many irreps of this group are there? (=no. of conj. classes)
Sk: What are they? (Induce irreps from the Borel subgroup and the subgroup of unipotent matrices.) What are the one dimensional reps of the Borel subgroup? (I guessed they just came from characters of the diagonal, but I wasn’t sure.) Well, let’s consider the structure of the Borel subgroup? ((F_p)*x(F_p)* semidirect product F_p. And from writing out the product of two matrices it’s clear that the commutator subgroup is the subgroup of unipotent matrices i.e. the copy of F_p. So the one-dim reps, being precisely the characters of the maximal abelian quotient, are just characters of (F_p)*x(F_p)*, which themselves are products of characters of (F_p)*. Skinner stopped me before calculating anything.)
Sa: Let’s say we’re working over C. What’s Jordan canonical form? What is the image of the exponential map on n x n complex matrices? (GL(n,C), since it suffices to show all Jordan canonical forms are in the image; write J = D*U, D diagonal, U unipotent, and then J = exp(log(D)+log(U)).)
Algebraic Number Theory
Sk: How would you solve y^2 + 13 = x^3? (Study class group of Q(sqrt(-13)) and consider this as an equation of ideals, invoke prime factorisation of ideals and see that (y+sqrt(-13)) is the cube of a principal ideal, then follow your nose.) [They asked me about Minkowski while I was in the process of finding the class group too.]
Sk: Let’s take a quadratic extension of the rationals. What proportion of prime ideals are principal in this field? (1/#Cl(K)) How do you prove this? (Using L functions and seeing that primes are equidistributed in ideal classes. Alternatively, these are exactly the primes splitting completely in the Hilbert class field, and we can show that the polar density of such primes is 1/[H:K].)
Sk: What’s the structure of the units in a real quadratic field? Why does the Pell equation have a solution? (I was kind of stumped because I knew a proof of Dirichlet’s Unit Thm but Skinner wanted a more intuitive explanation. Eventually I figured out that what they were looking for me to say is that we want to use Minkowski to pick up loads of lattice points of bounded norm (i.e. beneath a pair of hyperbolae xy=+-M), by choosing rectangles to avoid the points we have already discovered. Then because there are only finitely many ideals of bounded norm, some of these guys have to differ by a non-torsion unit. I found drawing a picture helped me see what’s going on.)
Sa: Fix some squarefree integer d. For what values of m is there a solution to x^2 - d*y^2 = m? (I also recall Sarnak asking me about solutions to the norm equation in a cubic field, I got stuck and we just considered a quadratic extension. I was still stuck so Skinner said what if the class number is one? In this case norms of elements are the same (up to sign) as norms of ideals so it suffices to see when prime powers are norms, and this depends on the splitting type of each prime, and the sign is accounted for by considering norms of units. We didn’t return to any of the more general cases.)
Representation Theory of Compact Lie Groups
Sa: If I give you a compact group, can you exhibit a finite dimensional representation? (Take an integral operator on L^2(G) with real, symmetric, G-invariant kernel, then this is a compact self-adjoint operator, so by the Spectral Theorem it has some finite dimensional eigenspace and the G-invariance of the kernel says this is in fact a subrepresentation.)
Sa: Does every finite dimensional representation appear in L^2(G)? (Yes, because characters arising from f.d. subreps of L^2 form an orthogonal basis for the L^2 class functions on G by Peter-Weyl.)
Sa: Are f.d. reps of a compact group always unitary? (Yes, stick an arbitrary inner product on G and integrate.)
Sa: Why do we talk about representations of SL_n(C) as the same thing as representations of SU_n(C)? (Weyl’s Unitary Trick)
Sa: Write out the Weyl Character Formula and explain each term. (Standard.) I assume you know how to prove that maximal tori are conjugate? (I seem to recall Sarnak winking at me as he asked this and I said that indeed I did know a proof. He replied that we needn’t go over it.) Work everything out in the case SU(3).
This was the end of the exam. I waited outside the room for a couple minutes while the examiners talked. I could hear muffled noise through the wall and I’m sure they were chatting about their favourite Complex Analysis books, which I found amusing. Soon enough Skinner opened the door and told me I’d passed.