Generals exam: Henry Thackeray
May 11, 2015, 1:30 pm to 4:30 pm
Examiners: Prof. C. Skinner (chair), Prof. P. Sarnak, Prof. A. Naor
Special topics: Algebraic number theory, representation theory
Algebra
State the structure theorem for finitely-generated modules over a PID. How does one deduce the theorems on canonical forms of matrices from this? Why is F[x] a PID for F a field?
Solve e^{A} = B for the square matrix A given the square matrix B.
State the Sylow theorems.
Find all groups of order 21.
How does one determine whether the Galois group of a polynomial is in the alternating group?
Find a polynomial whose Galois group is S_{5} (more generally, S_{p} for prime p).
Real analysis
When is one L^{p} space contained in another?
Given a real number p at least 1, find a function f in L^{p}(R) such that for all q not equal to p, f is not in L^{q}(R). Is the set of all such functions big or small? In what sense (Baire categories)? Show it.
Give a function on R that is everywhere continuous but nowhere differentiable.
Define the Fourier transform. (I chose on the torus, as opposed to the real line.)
In what sense does the Fourier series of an L^{1} function on the torus converge to the function? Show it.
Complex analysis
What is the order of an entire function?
How many zeros can an entire function have, in terms of cardinality? Where can the zeros be in the complex plane?
Given an entire function of given finite order, bound the number of zeros in an open disc of radius r centered at the origin.
Show that an entire function of fractional (that is, finite and non-integer) order has infinitely many zeros.
The Riemann zeta function was discussed, including its analytic continuation to a meromorphic function on C, its growth and its infinitely many zeros.
Algebraic number theory
Analytic class number formula and bounding the class number in terms of L-functions, structure of units in ring of integers of a number field, prime-degree cyclotomic and quadratic extensions of Q, splitting of prime ideals, proof of quadratic reciprocity via Galois theory (quadratic extension inside cyclotomic extension), Dedekind zeta function and factorization in terms of L-functions, principal result of class field theory.
Representation theory
Show that a compact Lie group has a finite-dimensional irreducible representation.
Find all irreducible representations of SU(2).
State the Weyl character formula and explain all terms occurring in the theorem.
Note: This list of questions and topics is what I remember from the exam. The list may be incomplete and there may be inaccuracies. I stumbled on many of the questions but passed the first time.