# QUALIFYING EXAMINATION

**Harvard University
Department of Mathematics
Tuesday 20 September 2005 (Day 1)**

1. Let X be the CW complex constructed as follows. Start
with Y = S^{1}, realized

as the unit circle in C ; attach one copy of the closed disc D = {z ∈ C : |z| ≤
1}

to Y via the map ∂D → S^{1} given by e^{iθ}
e^{4iθ} ; and then attach another copy

of the closed disc D to Y via the map @D → S^{1} given by e^{iθ}
e^{6iθ} .

(a) Calculate the homology groups H _{*}(X,Z).

(b) Calculate the homology groups H_{*}(X,Z/2Z).

(c) Calculate the homology groups H_{*}(X,Z/3Z).

2. Show that if a curve in R ^{3} lies on a sphere and has
constant curvature then

it is a circle.

3. Let X P^{5}
be the space of conic curves in
that is, the space of nonzero

homogeneous quadratic polynomials F ∈ C[A,B,C] up to scalars. Let Y
⊂ X

be the set of quadratic polynomials that factor as the product of two linear

polynomials; and let Z ⊂ X be the set of quadratic polynomials that are

squares of linear polynomials.

(a) Show that Y is a closed subvariety of X
P^{5}, and find its dimension

and degree.

(b) Show that Z is a closed subvariety of X
P^{5}, and find its dimension

and degree.

4. We say that a linear functional F on C([0, 1]) is
positive if F (f) ≥ 0 for all

non- negative functions f. Show that a positive F is continuous with the norm

||F|| = F(1), where 1 means the constant function 1 on [0, 1].

5. Let D_{8} denote the dihedral group with 8 elements.

(a) Calculate the character table of D_{8}.

(b) Let V denote the four dimensional representation of D_{8} corresponding

to the natural action of the dihedral group on the vertices of a square.

Decompose Sym^{2}V as a sum of irreducible representations.

6. Let f be a holomorphic function on C with no zeros.
Does there always exist

a holomorphic function g on C such that exp(g) = f?

**QUALIFYING EXAMINATION
Harvard University
Department of Mathematics
Wednesday 21 September 2005 (Day 2)**

1. Let n be a positive integer. Using Cauchy’s Integral
Formula , calculate the

integral

where C is the unit circle in C. Use this to determine the value of the integral

2. Let
be a smooth plane curve of degree d > 1. Let
be the dual

projective plane, and C^{*}
⊂ the set of tangent lines to C.

(a) Show that C^{*} is a closed subvariety of
.

(b) Find the degree of C^{*}.

(c) Show that not every tangent line to C is bitangent, i.e., that a general

tangent line to C is tangent at only one point. (Note: this is false if C is

replaced by a field of characteristic p > 0!)

3. Find all surfaces of revolution S
⊂ R^{3} such that the
mean curvature of S

vanishes identically.

4. Find all solutions to the equation y" (t) + y(t) = (t +
1) in the space D'(R)

of distributions on R . Here δ(t) is the Dirac delta-function.

5. Calculate the Galois group of the splitting field of
x^{5} − 2 over Q, and draw

the lattice of subfields.

6. A covering space f : X → Y with X and Y connected is
called normal if for

any pair of points p, q ∈ X with f(p) = f(q) there exists a deck transformation

(that is, an automorphism g : X → X such that g ◦ f = f) carrying p to q.

(a) Show that a covering space f : X → Y is normal if and
only if for any

p ∈ X the image of the map f_{*} : π_{1}(X, p) → π_{1}(Y, f(p)) is a normal

subgroup of π_{1}(Y, f(p)).

(b) Let Y S^{1} ∨ S^{1} be a figure 8, that is,
the one point join of two circles.Draw a normal 3-sheeted covering space of Y ,
and a non-normal three-

sheeted covering space of Y .

**QUALIFYING EXAMINATION
Harvard University
Department of Mathematics
Thursday 22 September 2005 (Day 3)**

1. Let f : CP^{m} → CP^{n} be any continuous map between
complex projective

spaces of dimensions m and n.

(a) If m > n, show that the induced map
is

zero for all k > 0.

(b) If m = n, the induced map

is multiplication by some integer d, called the degree of the map f. What

integers d occur as degrees of continuous maps f : CP^{m} → CP^{m}? Justify

your answer.

2. Let a_{1}, a_{2}, . . ., a_{n} be complex numbers. Prove
there exists a real x ∈ [0, 1]

such that

3. Suppose that ∇ is a connection on a Riemannian manifold
M. Define the

torsion tensor via

where X, Y are vector fields on M. ∇ is called symmetric if the torsion tensor

vanishes. Show that ∇ is symmetric if and only if the Christoffel symbols with

respect to any coordinate frame are symmetric, i.e.
Remember that

if {E_{i}} is a coordinate frame, and ∇ is a connection, the Christoffel symbols

are defined via

4. Recall that a commutative ring is called Artinian if
every strictly descending

chain of ideals is finite. Let A be a commutative Artinian ring.

(a) Show that any quotient of A is Artinian.

(b) Show that any prime ideal in A is maximal.

(c) Show that A has only finitely many prime ideals.

5. Let X ∈ P^{n} be a smooth hypersurface of degree d > 1,
and let A P^{k} ∈ X a

k-dimensional linear subspace of P^{n} contained in X. Show that k ≤ (n−1)/2.

6. Let
denote the space of bounded real sequences {x_{n}},
n = 1, 2, . . . .

Show that there exists a continuous linear functional
with the

following properties:

a) inf x_{n} ≤ L({x_{n}}) ≤ sup x_{n},

b) If
then L({x_{n}}) = a,

c) L({x_{n}}) = L({x_{n+1}}).

Hint: Consider subspace
generated by sequences {x_{n+1} − x_{n}}.

Show that
and apply Hahn-Banach.

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