Kenz Kallal's Generals Committee: Akshay Venkatesh (chair), Christopher Skinner, and Ruobing Zhang Special topics: algebraic number theory and representation theory of compact Lie groups 25 April 2023, 1:30P.M. - 4:00P.M. REAL ANALYSIS Zhang: What is the Lebesgue measure ? Zhang: What can you say if you replace "countable union" with "finite union" in the definition of the measure ? Zhang: What is an example of an uncountable measurable set ? Zhang: What is a measurable function ? Zhang: In what sense are measurable functions almost continuous ? Zhang: Are continuous functions dense in L^1 ? Venkatesh: What is the Fourier transform on L^2 ? Why is it an isometry ? Why is Schwartz space dense in L^2 ? [I mentioned offhand during my answer that the Fourier transform from L^1 lands in the continuous functions that vanish at infinity, and offered to give a proof that it is not surjective onto this space. The committee did not take me up on my offer.] Zhang: What is weak convergence ? What is an example of a sequence in a Hilbert space that converges weakly but not strongly ? ALGEBRA Venkatesh: Given a symmetric matrix over the real numbers, what can you say about it ? How would you prove the spectral theorem ? How would you compute the spectral decomposition of a matrix on a computer ? Venkatesh: Classify the quadratic forms over R. Classify the quadratic forms over a finite field. Skinner: What are some examples of PIDs ? What is the structure theorem for finitely generated modules over a PID ? Skinner: What is your favorite application of the structure theorem of finitely generated modules over a PID ? [I said it was the fact that a finite subgroup of the multiplicative group of a field is always cyclic] Skinner: What is rational canonical form ? What are the conjugacy classes in GL2(F_p) ? What is another interpretation of this number ? Professor Skinner was about to ask about the representation theory of GL_2(F_p), but Professor Venkatesh said we should save it for the representation theory special topic. Also at some point during this section there was a discussion about Sheldon Axler's book on linear algebra. COMPLEX ANALYSIS Zhang: If a sequence of holomorphic functions converges uniformly on the open unit disc, does it converge to a holomorphic function ? Zhang: What is the classification of isolated singularities ? Zhang: What are the meromorphic functions on the Riemann sphere ? Zhang: If an entire function is bounded by a polynomial, what can you say ? Zhang: What is the order of an entire function ? Zhang: For an entire function of finite order, what can you say about the locations of its zeroes ? Zhang: State the Riemann mapping theorem. Is the biholomorphism unique ? [I said that it is not unique, and stated what the automorphisms of the unit disc were. I also asked if the committee wanted to see a proof of the Riemann mapping theorem. They did not take me up on my offer, though Professor Skinner remarked that Gunning used to ask what the most beautiful theorem in mathematics is, and would refuse to accept any answer other than "Riemann mapping theorem."] Skinner: Prove the classification of automorphisms of the unit disc. Zhang: Do you know about Riemann surfaces ? Zhang: If you have a meromorphic function on a compact connected Riemann surface with exactly one pole, and that pole is simple, what can you say ? Zhang: Let Lambda be a lattice in C. Write down a nonconstant meromorphic function on C/Lambda. ALGEBRAIC NUMBER THEORY Skinner: What is the class group of K = Q[sqrt(39)] ? What is the Hilbert class field ? [I made a few mistakes in the computations for the above question that made it more complicated and led to a longer line of questioning than would be typical, but that was basically what was asked] Venkatesh: What are the adeles and ideles ? Why are they defined the way they are ? What does A_Q/Q look like ? [I defined A_Q, A_Q* and their topologies, and said that A_Q is a topological ring and that A_Q* is a topological group. I wrote down the fundamental domain for A_Q/Q. I also said that A_Q/Q and the norm 1 ideles mod Q* were compact, and said that + volume computation was responsible for finiteness of the class group, unit theorem, and analytic class number formula. I also mentioned that the topology being locally compact is useful for doing Fourier analysis a la Tate's thesis]. At some point someone asked me to state the adelic version of class field theory, but they got cut off by another question, and it wasn't asked again. REPRESENTATION THEORY Skinner: Where do the complex representations of GL_2(F_p) come from ? Most of the rest of the questions were about the theory of the Frobenius--Schur indicator in the special case of finite-dimensional representations of SU(2). At some point I made a remark that we had talked about essentially the same thing during the Skinner--Venkatesh seminar. They claimed that they did not remember due to old age. Venkatesh: When is a representation of a finite group self-dual ? Venkatesh: What are the irreducible representations of SU(2) ? Why is SU(2) simply connected ? Are the irreducible representations of SU(2) self-dual ? Venkatesh: Why is an invariant nonzero symmetric bilinear form on a representation necessarily nondegenerate ? Venkatesh: Which irreducible representations of SU(2) admit an invariant nonzero symmetric bilinear form ? Venkatesh: Can every vector space have a nondegenerate alternating pairing ? Note that this offers an alternative answer to the previous question. Venkatesh: What is the character of Sym^2(V) in terms of the character of V ? Skinner: What is your favorite theorem in representation theory ? [I said theorem of the highest weight] Zhang: What is Sp(1) isomorphic to?