Zhao Yu’s generals Committee: Will Sawin (WS) (chair), Manjul Bhargava (MB), Ravi Shankar (RS) Special topics: analytic number theory and algebraic geometry May 8 2pm-430pm in Sawin's office Text in square brackets are my answers. REAL AND COMPLEX ANALYSIS: ——————————————————————— These two topics went together, I think RS planned for the unifying topic to be about convolutions. In this section * means convolution, not multiplication. RS: What is the convolution of two functions f*g? Take the integral around a contour in the complex plane [f*g(z)=\int_C f(x)g(z-x) dx.] What if both f and g are holomorphic? [I said it’s holomorphic, then realized it is actually zero as the integrand is zero by Cauchy Theorem.] What is a theorem that is of the above form? [Cauchy integral formula, g=1/x.] What happens when g has a pole inside C? [Write g=a/x+holomorphic and combine the two statements above.] RS: Take g(z)=e^{iwz} and C=R to be real line, what is f*g? [f*g(z)= \int_R f(x)e^{iw(z-x)} dx = e^{iwz} \hat f(w).] Isn’t it interesting that e^{iwz} appears again? This means that the convolution operator (f*- have eigenfunctions e^{iwz} with eigenvalue \hat f(w). Can you explain this on the fourier transform side? [Fourier transforming f*g gives \hat f(k)\hat g(k), fourier transform of e^{iwz} is a delta function, so we get \hat f(w) times a delta function. Fourier transforming back gives e^{iwz}.] RS: Compute \int_R 1/(1+z^2)e^{iwz}. [I was in real analysis mode, so I said by parts?] Use the complex plane. [Oh right it’s the standard contour question. Move it to upper semicircle with big radius and compute residues.] RS: Tell me intuitively what the fourier transform of 1/\sqrt(|x|) is. [I drew something wavy thing like |sin(x)/x^k|.] Why do you say so? [Uh… it looks pretty close to a dirac delta function which should transform to a sine wave?] A delta function supported at zero has fourier transform 1. [I was confused. After generals I realized he was trying to tell me to remove the sin(x) term to get |1/x^k|. Anyway we continued along a different approach.] What are some properties of 1/\sqrt(|x|)? [Even, goes to zero… turns out what they want is that it satisfies a scaling property f(ax)=a^cx, at this point I realized that the fourier transform also has such a scaling property and is of the form 1/|x|^k.] RS: Give conditions on when f*g is continuous [When g is continuous and f is compactly supported.] What about when f is in Lp and g is in Lq? What are appropriate values of p,q that we should choose? [Let’s take 1/p+1/q=1, then using Holder f*g(z) is well defined and bounded by ||f||_p||g||_q for each z. For continuity f*g(z)-f*g(z+eps)<= ||f-f_eps||_p ||g||_q where f_eps is the epsilon translation of f. Then ||f-f_eps||_p -> 0 by approximating f with compactly supported continuous functions.] Can you give another argument for ||f-f_eps||_p->0 for p=2? [Take fourier transform of f-f_eps since the L2 norm is the same by Plancheral, rewrite it as \int f(x)(e^{ik(x+eps)}-e^{ikx})dx and we see that an epsilon term comes out of e^{ik(x+eps)}-e^{ikx}.] RS: You mentioned Plancheral, is there a Lp analogue for that? [No, I don’t think so.] How about when p=1? [I said L1->C0 \subset L^inf, and that ||f||_1>=||f||_inf.] In general there will be such an inequality for 1/p+1/q=1. [Cool. After generals I saw this is the Hausdorff-Young inequality.] WS: What kind of test functions f would you try to see if an inequality like that is true? [polynomial, square bump functions, said something about the Dirichlet kernel and choosing signs based on that. He was happiest with square bump functions with varying height and width.] RS: Back to f*g(z)=\int_C f(x)g(z-x) dx, where C is a non-closed contour on the complex plane. Say g is holomorphic but not f. Is f*g holomorphic? [Yes, differentiate under the integral, I said f compactly supported to be safe.] What are the conditions allowing you to differentiate under the integral? [Couldn’t really remember, said something like look at sequence of quotients and use dominated or monotone convergence theorem.] What if f is a distribution? [I don’t know distributions well.] Say f is a delta function. [In this case you recover the original function, so f*g is holomorphic I guess it should be true for other distributions too..?] What if the contour looks very rough. [If f is holomorphic I can smooth the contour...I don’t think this is what they wanted though.] RS: I’m done asking questions. WS: Let’s ask some more complex analysis questions MB: Is the quarter sector biholomorphic to unit disc? [Yes, z->z^4-i/z^4+i.] MB: Should we ask about the Riemann Mapping Theorem? Everyone laughs. RS said he just asked it a week ago for someone’s generals. I said everyone knows this and gave a quick verbal sketch of the proof [Consider injective family of maps to disk, use Arzela Ascoli, do the square-root trick]. WS: I think we haven’t covered measure theory yet. RS: What does convergence mean for measures? [Weak convergence of measures is \int f d mu_n -> \int f dnu for all f.] They made me define convergence in measure: fn->f with respect to a measure dmu if mu({|fn-f|>eps})->0 as n->infty for any eps>0. Compare this to L1 convergence and almost everywhere convergence. [Almost everywhere convergence clearly implies this (After generals: only on finite measure spaces). L1 implies this by Markov, but this does not imply L1 (take a sequence of boxes with area 1 where width ->0).] ALGEBRA: ——————————————————————— MB: What are the orientation preserving symmetries of a cube [I prove that this is S4 as follows. The symmetry group G acts on the four diagonals, so G->S4. G has size 8*3=24 as after fixing a vertex (8 of them), there are 3 rotations. This is an injection as the four diagonals determine the orientation of the cube.] Using this, find a surjective map S4->S3 [G=S4 acts on the three axes of the cube, so G->S3, this is surjective as one can always map any permutation of axes to another.] Write the character table of S4. [5 conjugacy classes, 5 reps: trivial, sign, regular, regular twisted by sign, the last one is the regular of S4->S3.] Consider the permutation representation of S4 on the vertices of the cube. How do you decompose this? [Calculate trace = # fixed vertices, and take inner product with the irreps.] MB: Classify all groups of order 25. [I proved that groups of order p^2 are abelian by showing the center is nontrivial by looking at conjugacy classes, so by structure theorem of Z-modules its Z/25 or (Z/5)^2.] Classify all rings of order 25. [If underlying additive group is Z/25 the ring is Z/25 because multiplication is uniquely determined, if it’s (Z/5)^2 take 1 and x, multiplication is determined by what x^2 is, if x^2=ax+b then the ring is F5[x]/x^2-ax-b, completing the square reduces to F5[x]/x^2-a. If a=0 we get F5[x]/x^2, a=square is F5*F5, a=non-square is F25.] MB: Find a degree 25 Galois extension. [Take Q(zeta_125) which is degree 100 Galois, with Galois group Z/100, take the fixed field of the normal subgroup of order 4.] Find a degree 25 non-Galois extension. [Anything works, really. I gave Q(2^{1/25}).] What is the Galois closure of this? Degree of the extension? [Q(2^{1/25},w) where w=zeta_25, degree is 25*20=500.] What is the Galois group? [View as a subset of S_25 via the action on the roots {w^j 2^{1/25}}. An element of the Galois group looks like 2^{1/25}->w^a2^{1/25}, w->w^b. These can be identified with affine transformations x->ax+b.] What is the normal subgroup? [A normal subgroup of affine transformations are translations, so this is the semidirect product of Z/25 and Z/20.] WS: You assumed something when you said deg(Q[2^{1/25},zeta_25])=500 [That the Galois closure of Q(2^{1/25}) and Q(zeta_25) don’t intersect.] This might not work if 2 and 25 are replaced with other numbers, can you come up with an example? [Something along the lines of zeta_6+zeta_6^2 = i\sqrt 3 should work.] WS: Just now you used the structure theorem of Z-modules, this also holds for PID, what else can you use this for? [Rational canonical form, Jordan normal form.] Do this for k[T]. [Any module decomposes into a free part and a sum of k[T]/f_i(T).] What can you say about the f_i(T)? [Two ways of arranging it: either f_i(T) divide each other or f_i(T) are powers of irreducible polynomials.] WS: How many monic irreducible polynomials of Fp are there of degree d? [Each irreducible poly has d-1 roots in Fp^d so it should be (5^p-5^{p-1})/(d-1).] The p-1 is not quite right. [Oops, Fp^{d-1} is not inside Fp^d, the largest subfield of Fp^d is Fp^{d/2} so it should be approximately (5^p-5^{p/2})/(d-1).] RS: I have a linear algebra question. WS: When I took generals I got exactly one algebra question, which was also a linear algebra question. RS: Go ahead and ask! WS: What can you say about an orthogonal symmetric positive definite matrix [Spectral theorem says eigenvalues are real, positive definite matrix says eigenvalues are positive, (after some prodding) orthogonal matrix says eigenvalues are 1. Hence it must be the identity!] RS: Here’s my question. Let v and w be vectors in R^10. What is det(1+vv^T+ww^T)? WS: Do it for A=1+vv^T first. [WLOG assume v=||v||e_1, then A sends e_1 to (1+||v||^2)e_1, e_i to e_i, so product of eigenvalues is 1+||v||^2. For the 1+vv^T+ww^T case we should expect 1+||v||^2+||w||^2+2(v,w). You can prove this by letting v=a*e1 and w=b*e1+c*e2.] ANALYTIC NUMBER THEORY: ——————————————————————— WS: What are Dirichlet L-functions and why are they useful [They are periodic mod q so they can count stuff in arithmetic progressions, say prime number theorem along a mod q.] MB: What is a Dedekind zeta-function? How are they related to Dirichlet L-functions? [If K is abelian, zeta_K is a product of Dirichlet L-functions, in general it is a product of Artin L-functions.] WS: Do this for Q(\sqrt 3, \sqrt 7). [zeta_K(s) = zeta(s)L(1,chi_3)L(1,chi_7)L(1,chi_3chi_7).] 
How do you prove that these are the same? [Check that the Euler factor at p are the same] Do this for Q(\sqrt 3) [Wrote down 1/1-N(p)^-s for the 3 cases of splitting, inert, ramified. You can tell this from Fp[x]/x^2-3 which is determined by whether 3 is a square mod p which is a Dirichlet character.] MB: Tell me about some analytic properties of Dedekind zeta-functions. [pole at s=1 with residue given by class number formula, wrote this down, gave a quick sketch of the geometry of numbers proof.] How do you show that it is analytic? [I said I only knew the proof for Dirichlet L-functions, and described the proof for that instead.] WS: Tell me about Waring’s problem. [Can estimate # solutions of {N=x1^k+...+xs^k} for s>=2^k+1 using circle method, final result is N^{s/k-1} times a product of p-adic densities times a singular integral.] What is the heuristic for the leading term N^{s/k-1}? [xi<=N^{1/k} so N^{s/k} choices in total, they sum up to O(N) range so divide by N.] What is it about the x1^k+...+xs^k that makes the circle method work? What about other kinds of equations? [It is a sum of the same function so we can take the generating function. Circle method also works for prime sums and combinations like x1^2+p1+p2. But it doesn’t work for stuff like x^2y^2+z^2w^2. Somewhere we discussed Birch’s theorem that a homogeneous equation with enough variables will have a solution.] Surprisingly, they did not ask me about major and minor arcs at all. MB: What is the density of squarefree numbers? [\prod (1-p^{-2})=1/zeta(2).] Can you give a lower bound for this? [uh… the large sieve doesn’t seem to work.] They started guiding me through the proof. WS: Write down the generating function of squarefree numbers. [f(s)=\sum_{n squarefree} n^{-s}=\prod_p (1+p^{-s}).] Express this explicitly. [f(s) = \zeta(s)/\zeta(2s).] How do you truncate this? [Inverse mellin transform \int_{c-i\inf}^{c+i\inf} zeta(s)/zeta(2s) x^s/s ds. Pulling the contour to the left we see that the simple pole of zeta(s) gives a main term of x/zeta(2)! I said the error term should look like O(xexp(-c\sqrt\log x)) like prime number theorem.] Actually you can get power savings. [Oh right in PNT we looked at zeta’(s)/zeta(s), so zeros of zeta(s) mattered. Now only zeros of zeta(2s) matter, and these lie on the scaled critical strip 0P1 is hyperelliptic where L=p*O(1) then L^{g-1} is the canonical bundle.] You can get the canonical bundle of a plane curve too [If i:C->P2 then I guess it should be a power of M=i*O(1). Since I expect it to be degree 2g-2=d(d-3) then M^{d-3} should be the canonical bundle. Suffices to show there are at least g sections, but h0(P2,O(d-3)) is (d-1)C2=g! Also, these sections don’t vanish identically on the curve because it is degree d-3 while the curve is degree d.] At this point we moved on, and I guess we forgot to complete the argument. For completion here it is: [If C is both hyperelliptic and a plane curve, then L^{d-3}=M^2, but h0(C,M^2)=h0(P2,O(2))=6 but h0(C,L^{d-3})=h0(P1,O(d-3))=d-2.] WS: What does the degree-genus formula compute if it's not smooth [Arithmetic genus p_a=1-\chi(X,O_X).] Say you have a curve C and a double curve 2C. You would expect the Euler characteristic of 2C to be double that of C, but this is not the case. Why? [I didn’t really see the argument of why it would be double (I guess naively you expect h^i to double) so I just computed it via the exact sequence 0->O_X(-2C)->O_X->O_{2C}->0 with Hilbert polynomials and showed this is not the case.] There’s an exact sequence involving O_C and O_{2C} [After some thinking...on charts we have 0->I/I^2->A/I^2->A/I->0 which corresponds to 0->O_C(-C)->O_{2C}->O_C->0.] WS: Yup, that’s the right exact sequence [Mishearing...oh it’s only right exact?] Nono, that’s the correct exact sequence. This is exactly what happens with my kid, I can’t use the word “right” instead of “correct” [Haha. Anyway I guess the point is that in the exact sequence O_C is twisted on the left so the euler characteristic of 2C is not double of C.] WS: Compute the cohomology of the cotangent sheaf of P^n [Use the Euler exact sequence 0->\Omega_{P^n}->O_^{P^n}(-1)^{n+1}->O^{P^n}->0 and the corresponding LES in cohomology. Computed h^0 of second and third term, and said that h^i vanishes. Used Serre duality to show h^n(O(m))=h^0(O(-n-1-m)).] Why is the canonical bundle O(-n-1)? [Take determinant of Euler exact sequence.]