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QUALIFYING EXAMINATION
Harvard University Department of Mathematics Tuesday September 1, 2020 (Day 1)
- (AG) Let X be a smooth projective curve of genus g, and let p ∈ X be a point. Show that there exists a nonconstant rational function f which is regular everywhere except for a pole of order ≤ g + 1 at p.
- (CA) Let U ⊂ C be an open set containing the closed unit disc ∆ = {z ∈ C : |z| ≤ 1 }, and suppose that f is a function on U holomorphic except for a simple pole at z 0 with |z 0 | = 1. Show that if
∑^ ∞
n=
anzn
denotes the power series expansion of f in the open unit disk, then
lim n→∞
an an+
= z 0.
- (RA) Let {an}∞ n=0 be a sequence of real numbers that converges to some A ∈ R. Prove that (1 − x)
n=0 anx n (^) → A as x approaches 1 from below.
- (A) Prove that every finite group of order 72 = 2^3 · 32 is not a simple group.
- (AT) Let X be a topological space and A ⊂ X a subset with the induced topology. Recall that a retraction of X onto A is a continuous map f : X → A such that f (a) = a for all a ∈ A. Let I = [0, 1] ⊂ R be the closed unit interval, and
M = I × I/(0, y) ∼ (1, 1 − y) ∀ y ∈ I
the closed M¨obius strip; by the boundary of the M¨obius strip we will mean the image of I × { 0 , 1 } in M. Show that there does not exist a retraction of the M¨obius strip onto its boundary.
- (DG) Let S be a surface of revolution
r(u, v) = (x(u, v), y(u, v), z(u, v)) = (v cos u, v sin u, f (v))
where 0 < v < ∞ and 0 ≤ u ≤ 2 π and f (v) is a C∞^ function on (0, ∞). Determine the set of all 0 ≤ α ≤ 2 π such that the curve u = α (called a meridian) is a geodesic of S, and determine the set of all β > 0 such that the curve v = β (called a parallel) is a geodesic of S.
Hint: To determine whether a meridian or a parallel is a geodesic, parametrize it by its arc-length and use the arc-length equation besides the two second- order ordinary differential equations for a geodesic. For your convenience the formulas for the Christoffel symbols in terms of the first fundamental form Edu^2 + 2F dudv + Gdv^2 are listed below.
Γ^111 = GEu − 2 Fu + F Ev 2(EG − F 2 )
. Γ^211 =
2 EFu − EEv − F Eu 2(EG − F 2 )
Γ^112 =
GEv − F Gu 2(EG − F 2 )
, Γ^212 =
EGu − F Ev 2(EG − F 2 )
Γ^122 =
2 GFv − GGu − F Gv 2(EG − F 2 )
, Γ^222 =
EGv − 2 F Fv + F Gu 2(EG − F 2 )
where the subscript u or v for the function E, F , or G means partial differ- entiation of the function with respect to u or v.
(a) Find the homology groups of X with coefficients in Z. (b) Find the homology groups of X with coefficients in Z/5.
- (DG) Suppose G is a compact Lie group with Lie algebra g. Consider an element g ∈ G, and let c ⊂ g be the subalgebra c = {X|Adg(X) = X}. Show there exists some > 0 such that for all X ∈ g with |X| < , there exists Y ∈ c such that g exp(X) is conjugate to g exp(Y ).
QUALIFYING EXAMINATION
Harvard University Department of Mathematics Thursday September 3, 2020 (Day 3)
- (AG) Let C ⊂ P^3 be an algebraic curve (that is, an irreducible, one-dimensional subvariety of P^3 ), and suppose that pC (m) and hC (m) are its Hilbert polyno- mial and Hilbert function respectively. Which of the following are possible? 1. pC (m) = 3m + 1 and hC (1) = 3; 2. pC (m) = 3m + 1 and hC (1) = 4.
- (RA) The weak law of large numbers states that the following is correct: Let X 1 , X 2 ,... Xn be independent random variables such that |μj | = |EXj | ≤ 1 and E(Xj − μj )^2 = Vj ≤ 1. Let Sn = X 1 +... + Xn. Then for any ε > 0
lim n→∞
P
Sn −
j μj n
| > ε
Now suppose that we don’t know the independence of the sequence X 1 , X 2 ,... Xn, but we know that there is a function g : { 0 } ∪ N → R with limk→∞ g(k) = 0 such that for all j ≥ i EXiXj = g(j − i) In other words, the correlation functions vanishing asymptotically. Do we know whether the conclusion (+) still holds? Give a counterexample or prove your answer.
3. (CA)
(a) Suppose that both f and g are analytic in a neighborhood of a disk D with boundary circle C. If |f (z)| > |g(z)| for all z ∈ C, prove that f and f + g have the same number of zeros inside C, counting multiplicity. (b) How many roots of
p(z) = z^7 − 2 z^5 + 6z^3 − z + 1 = 0
are there in the unit disc in |z| < 1, again counting multiplicity?
- (AT) Let S^1 = R/Z be a circle, and let S^2 be a two-dimensional sphere. Consider involutions on both, with an involution on S^1 defined by x 7 → −x
