ANDY MAYER'S GENERALS (January 1991)
ALGEBRA
1) What is Jordan Canonical Form of a matrix?
Modules over a PID, examples.
2) What is a ring of integers? What does "integral over Z" mean?
3) Do you know about Dedekind domains and class numbers?
4) Is Z[t]/(t^p-1) --> Z(w) by t-->w where w is a pth root of unity an
isomorphism?
5) Galois theory. What is the Galois Group of x^5-2 over Q? Which inter-
mediate fields are normal? semi-direct product structure? etc.
REPRESENTATION THEORY
1) What is Schur's Lemma? What is dim HOM(V,W) where V and W are G-modules
which FAIL to be irreducible?
2) State and prove the theorem of Peter and Weyl. What is the structure of
L^2(G) where G is a compact lie group?
3) Write the character table of the alternating group A4.
4) When is a representation induced up from a normal subgroup irreducible?
What are the irreducible representations of a semi-direct product of
groups? State and sketch a proof of the Mackey Irreducibility
Criterion for induced representations.
DIFFERENTIAL GEOMETRY
1) Given a differentiable map f:M-->N, when is the inverse image of a
point p in N a submanifold of M? Why? Explain? Compare situations
when dim M > dim N and dim M < dim N. Intuitively explain the rank
theorem. Is there a preferred inclusion of the image of the tangent
space of M in that of N (dim M < dim N)?
2) Discuss the Frobenius integrability condition for a family of
tangent hyperplanes. Does this generalize to arbitrary codimension?
Reformulate this result in terms of vector fields.
3) State the Gauss Bonnet Theorem (case of a piece-wise differentiable
closed curve on a regular surface in R^3). Sketch a proof.
REAL ANALYSIS
1) When is L^p contained in L^q for p different from q? What are the
Holder and Minkowski formulas?
2) What is the Fourier Transform. State the Fourier inversion formula.
What about L^2? [no proofs required]
3) Give an example of an everywhere continuous, nowhere differentiable
function on the unit interval.
COMPLEX ANALYSIS
1) What is an analytic function? How many derivatives must be required?
How many does it possess?
2) How many zeros can an analytic function have in an open set? [no limit
point] prove this.
3) Residue Theorem? How is it related to index of a vector field?
What is the index of a vector field? How do you prove the residue
theorem?
4) When are two annuli conformally equivalent? Prove it.
5) State and prove the Schwartz Reflection Principle.
RULE OF THUMB: you can fail to intelligently answer many questions and
still
pass! Just show you know something, and STAY CALM! [of course I did not
do this.]
Committee: HSIANG,BIEN,PRATO