GENERAL EXAM QUESTIONS Victim: Timothy P. Barber Examiners: Trotter, Conway, Nelson, Kohn, Darmon Adv Topics: Representations of discrete groups, Computational complexity theory The list of questions is divided into the five topics, and each topic is divided up by examiner (with initial). I had to retake Algebra and Complex, so that's why there are five examiners total. ALGEBRA: D: Let G be the group of 3x3 matrices over GF(p), p prime. What does basic group theory tell us about G? How many conjugates does the p-Sylow have? Give a matrix form for the elements in this subgroup. T: Explain the conjugacy in terms of eigenvalues and eigenvectors. D: Give a matrix form for the normalizer of the p-Sylow. T: What is Galois theory? State the main theorem. What is the splitting field of x^5 - 2 over Q? What are the intermediate extensions? D: Which extensions are normal, which are not, and why? What are the Galois groups (over Q) of all extensions? C: How many groups are there of order 15? Prove it. How many abelian groups are there of order 36? How is the structure theorem for finitely generated modules over a PID related to the Jordan Canonical form? Give the 4x4 Jordan forms with min poly (x-1)(x-2)(x-2). COMPLEX: K: What would you like to discuss? (I said Cauchy's Thm) Prove Cauchy's Thm. What theorem of multivariable calculus is this similar to? What is Stokes Thm? (I gave a vague description) Prove the Cauchy Integral Formula. Prove Liouville's Thm. How many zeros can an analytic function have in the disk? Can the zeros have a limit point on the boundary? Give an example. C: What is special about a conformal map? What happens in the neighborhood of a pole? (f -> oo) In which directions does it go to infinity? What kinds of singularities are there? Explain each. What happens near z=0 for the function (log z) / z. Is z=0 a singularity? What is it? T: How many zeros can an entire function have? Can the zeros be ANY set with no limit point? How is this related to the order and genus? Suppose we have an entire funtion with a poles at 1 and 2i. Given a power series for this function about the origin, where does it converge? How many power series are there? Why must there be more than one? How can we compute the coefficients for each? N: An entire function f has Re(f) + Im(f) bounded. What can you say about f? REAL ANALYSIS: T: When is L^p contained in L^q? Prove it. For what measure is this containment reversed? Prove it. N: Is there a function on [0,1] which is discontinuous at every rational point? Prove it. How about at every irrational point? Prove that. What can you say about the Fourier transform of an L^1 function? Prove it. REPRESENTATIONS OF DISCRETE GROUPS: C: Give the character table for the symmetric group on 3 letters. What does the character table of an abelian group look like? From the character table of a group G, how can you identify.. ..a normal subgroup? ..an abelian subgroup? ..a p-subgroup? Why are the entries in the character table algebraic integers? What does the Frobenius-Schur indicator tell you? Do you know how to prove Mackey's criterion? (I said yes, then he went on to the next question.) Given the character tables for groups G and H, what is the character table for G x H? COMPUTATIONAL COMPLEXITY THEORY: T: What is a regular language? What is a finite automaton? Prove that NFA = DFA = regular languages. N: Explain the difference between uniform and non-uniform complexity classes. How are these related to the arithmetic hierarchy in mathematical logic? State Shannon's Theorem. (Most functions on n bits can't be computed with fewer than 2^n/n logic gates.) Prove this theorem. C: What is Kolmogorov complexity? Why must all minimal programs be algorithmically random strings?