Committee: Sarnak (chair), Zhang, Buckmaster Special Topics: Analytic Number Theory, Representations of Compact Lie Groups 4/30/2021 1:00-4:30 pm via Zoom Comments: I'll try to label who asked what question as best as I remember (Sarnak = S, Zhang = Z, Buckmaster = B). Most of these problems were phrased in a much more pleasant and conversational way than what I've written below. I'm also omitting much of the back and forth stumbling around that occurred during the some of the problems. Also I'll put some comments in brackets about fun things that were said or other thoughts on the problems. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ COMPLEX: (30 minutes) (B): Show that if f and \hat{f} are both compactly supported then f is trivial (S): First define the Fourier transform - defined Fourier transform for L^1(R) - \hat{f} is then entire and then by the identity theorem \hat{f} = 0 and so f = 0 (B): Prove the identity theorem - wrote Taylor series expansion around the accumulation point and argued via contradiction if there was some nonzero coefficient (B): Prove Liouville's theorem - wrote Cauchy integral formula and did the standard proof (B): Prove the fundamental theorem of algebra using Liouville - standard proof by inverting 1/p(z) to find a zero (B): Define the Gamma function and show that it is analytic for Re(s)>0: [Sarnak commented about how I used s to denote the variable instead of z] - split the integral to bound each part separately to show absolute convergence of the integral (B): If f_n are analytic on \Omega and converge uniformly to f on compact sets, is f holomorphic? - Yes by Morera's theorem (S): What do you know about Riemann mapping/what did you read? - I read Alhfors for Riemann mapping, I said I knew the proof from Alhfors [and I might have mentioned that I knew Riemann's original idea] (S): Explain Koebe's proof without writing anything - Create a family of holomorphic functions from \Omega to D with the desired properties - Show the family is nonempty by constructing a function using a branch of the square root function with a point not in \Omega - Use Montel's [was asked to specify the statement of Montel] - Obtain a subsequence converging to a holomorphic function where the derivative is maximized and show this function belongs to the family - Surjectivity using another square root construction (S): Draw a doubly connected region. [I drew an annuli and then was asked to draw a more blobby one] (S): Are these two regions biholomorphic? - Probably not these two, but doubly connected regions can be conformally mapped to annuli (S): When are two annuli equivalent? - When the ratios are the same [Started the Schwartz reflection proof and made it to C\{0} -> C\{0}] (S): Why does being bounded imply it is a removable singularity? [Completely mind blanked here but somehow we ended up discussing Laurent series and Casarati-Weierstrass] [I never finished the annuli problem because they were attempting to keep at 30 minutes per section at this point] -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- REAL ANALYSIS: (40-50 minutes) (B): Can you give a function in L^1(R) that isn't in L^2(R)? - 1/x^2 on 01 and 0 everywhere else (B): Prove Holder's inequality - Used Young's inequality [Sarnak looked up that this Young was some familiar relation to Mendelssohn who was related to Hensel, although I don't recall how] (B): If f,g are in L^1(R), can you bound inf_{\tau \in R} \int |f(x+\tau)g(x)| dx? - I first attempted to look the case of Schwartz functions which are dense in L^1 where this should be 0 - Was directed to look at an average and saw that \int \int |f(x+\tau)|g(x) dx d\tau \leq |f|_1|g|_1 and could bound things with Chebyshev's inequality [There was confusion here on whether or not the problem should have been stated where \tau ranged over some closed interval instead but I think it was agreed upon that as written the answer was 0 by looking at Schwartz functions] (B): Give an example of functions converging weakly but not strongly in L^2[0,1] (S): Also define weak and strong convergence - Used n*1_{[0,1/n^2]} [I got a bit confused when trying to show weak convergence and at some point we had a discussion about lim_{\epsilon-> 0} \int_0^\epsilon |g|^2 = 0 for g \in L^2 ] [Not sure what the problem statement was here, but there was discussion about Fourier series coefficients being in l^2(Z)] (Z): Prove the spectral theorem for compact self-adjoint operators from L^2([0,1])-> L^2([0,1]) - \|T\| or -\|T\| will be attained as an eigenvalue since \|T\| = sup || (using compactness) - Decompose the space into + orthogonal complement (can do from self-adjointness) (Z): Use the spectral theorem to talk about Fourier series - If we get a compact operator T with eigenvalues {e(nx)}_{n\in \Z} then it will be an orthonormal basis - Constructed a convolution operator with kernel k(x,y) = \mu(-x+y) then T(e(nx)) = \hat{\mu(n)} e(nx) - Want \mu to be a bump function around 0 so that the eigenvalues are nonzero [Zhang and Sarnak asked more problems around this Fourier series spectral theorem problem that I don't remember, except at one point I tried to cite Fejer's theorem and Zhang objected (wanting me to use the spectral theorem only) and Sarnak made the funny comment that Zhang wanted me to do Fourier analysis with no analysis] -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ALGEBRA: (40-50 minutes) (Z): Classify the groups of order 8 [Luckily, we had played the finite group game at the dining hall recently so I rattled them all off the top of my head] - Z/8, Z/4xZ/2, Z/2xZ/2xZ/2, D_8, Q (Z): Classify all the conjugacy classes of these groups - D_8: {1}, {\sigma, \sigma^3}, {\sigma^2}, {\tau, \tau\sigma^2}, {\tau\sigma, \tau\sigma^3} - Q: {1}, {-1}, {i, -i}, {j, -j}, {k, -k} (Z): Write the characters for all of these groups - Cited structure theorem to do characters of abelian groups as A^\hat ~ A for abelian finite groups - Character table for D_8 with characters from G/[G,G] and induced from Z/4 - Character table for Q with characters G/[G,G] and used orthogonality for 2 dimensional rep [At some point, I stated |G| = \sum \dim(V_i)^2] (S): What is this 2 dimensional representations of Q? - Pauli matrices (S): e^A = B for complex nxn matrices - Taylor expansion of e^A and convergence with respect to norm of A as a linear operator - Invertibility is a necessity - Jordan block decomposition (at some point became 2x2 matrices) (Z): Prove Jordan-Chevalley decomposition over the reals. [I got completely stuck here as I'd only seen this over the complex numbers where you just use Jordan form] - defined nilpotent matrices and was told definition of semisimplicity to use is that there are no repeated roots in the minimal polynomial of X - explained rational canonical form from structure theorem - decomposition of rational canonical form block for 2x2 matrices [Took a 5-10 minute break here around 3 pm] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ANALYTIC NUMBER THEORY: (50-60 minutes) (S): Prove Dirichlet's theorem on primes in arithmetic progressions [I asked him if he wanted Siegel-Walfisz or just infinitely many, and he said infinitely many was good] - converted sum into \sum\sum \Lambda(n)\chi(n) [Oops Dirichlet actually proved that \sum_{p = a mod q} 1/p diverged so I was actually not doing the elementary way but was allowed to continue] - got up to Mellin inversion with L'/L [After this I wrote the Euler product for L(s,\chi) down and there was some follow up question I don't remember] (S): What does the proof boil down to? - Showing L(1,\chi)\neq 0 (S): How do you do this? - Wrote down the class number formula - Comments about only really needing to know this for real characters since the issue with using the technique for zero-free region of zeta function is when \chi^2 = \chi_0 (S): What is a Landau-Siegel zero? - defined it via a picture - mentioned something about \chi_163 (S): Prove Landau-Siegel's theorem [Siegel's theorem] - wrote down zeta_K(s) for K = Q(\sqrt{d},\sqrt{d'}) - explained why noneffective [We moved on before I got to shift the line of integration] (S): Talk about Waring's problem - Did Hardy-Littlewood's method for r_{s,k}(N) and wrote asymptotic down - Talked about Weyl inequality and minor arcs [Didn't get into major arcs, singular series or singular integrals beyond the written final asymptotic formula] (S): Explain how Vinogradov's Mean Value Theorem comes in - Define the theorem - Replacement of Hua's identity [It also gives an improvement on Weyl's inequality but I didn't mention it because I knew that I didn't remember that proof at all] - Explained that it lowers to s>= k^2 log(k) + O(k) - Said a little about connections to the moment curve and decoupling [Sarnak wanted Buckmaster to ask me more about decoupling but that didn't end up happening, so I handwaved reducing to the moment curve and didn't end up saying anything about decoupling the curve itself] (S): Give a lower bound on the L^1 norm of \sum e(\alpha x^2) - Use Holder's and the divisor bound for the L^4 norm [This surprised me because we had discussed this problem in a previous meeting and he told me he might ask me about doing the L^1 norm of \sum \Lambda(n) e(n\alpha) similarly on my generals] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ [At this point it was past 4 and I think the committee wanted to wrap things up] REP OF LIE GROUPS: (20-30 minutes) (Z): What are the representation of U(1), U(2), U(3)? - U(1) = S^1 so one dimensional representations corresponding to e^{2*pi*i*n*x} for n\in Z - U(2) = U(1) x SU(2) (S): What are the representations of SU(2)? - Started homogeneous polynomials proof, calculated the weights - Wrote down the characters of ((\alpha, 0),(0,\alpha^{-1})) as \sum_{k=0}^{n} \alpha^{n-2k} [Sarnak noted that the character here was enough to show irreducibility since \int |\chi|^2 dg = 1 before I started doing irreducibility using X and Y and weight lowering/raising] (S or Z): Are these all of the representations of SU(2)? - Peter-Weyl statement, matrix coefficient map [They moved on before discussing U(3)] (S): State the Weyl character formula - defined roots, Weyl chamber, positive roots, dominant weights (the way with \lambda(H_\alpha)>0 for positive root \alpha but also mentioned the way with fundamental dual Weyl chamber from Adams), alternating sums and the formula (S): How would you look at L^2(SL_2(C))? - discusssed Weyl unitary trick -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- More comments: - I think my Lie groups section was unusually short (due to the fact we were already running long and Sarnak had office hours at 4:45) and probably not indicative of what other Lie group sections might be like - What I used to prepare: - Complex: Stein and Shakarchi, Ahlfors for Riemann mapping, Dirichlet problem, multiply connected regions - Real: Rudin's Real and Complex Analysis, Eugenia Malinnikova's lecture notes from Stanford Math 205A, Paul Garrett's course notes for functional analysis - Algebra: Fulton and Harris's Representation Theory, Keith Conrad's expository papers, Paul Garrett's notes for reps of GL_2(F_p) - Analytic Number Theory: - Davenport's Multiplicative Number Theory - Davenport's Analytic Methods for Diophantine Equations and Diophantine Inequalities - Wooley's lecture notes on arithmetic harmonic analysis (there are nice problem sets to go with these, also has content about VMVT and treatment about reducing the number of variables needed) - Koukoulopolous's The Distribution of Prime Numbers (this contains Granville and Sound's proof of the zero free region which is much more intuitive than the cosine trick, it also has a more motivated exposition of Vaughan's identity) - Representations of Lie Groups: - Adams's Lectures on Lie Groups (read this first and it was a struggle. Kirillov's course notes might be a better starting off point) - Fulton and Harris's Representation Theory (a very algebraic approach, good for practicing decomposing representations) - Knapp/Knapp and Trapa for infinite dimensional representations of sl_2(R) and sl_2(C) - Zhang's lecture notes from Mat 449 from Spring 2021 - I also met with Giorgos, Trajan, and Maciej every weeknight to discuss problems from previous exams and towards the end we started quizzing each other/doing practice exams. This was very helpful and also much more enjoyable than studying alone!