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QUALIFYING EXAMINATION. Harvard University. Department of Mathematics. Tuesday September 1, 2020 (Day 1). 1. (AG) Let X be a smooth projective curve of ...
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Harvard University Department of Mathematics Tuesday September 1, 2020 (Day 1)
∑^ ∞
n=
anzn
denotes the power series expansion of f in the open unit disk, then
lim n→∞
an an+
= z 0.
n=0 anx n (^) → A as x approaches 1 from below.
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.
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 )
2 EFu − EEv − F Eu 2(EG − F 2 )
GEv − F Gu 2(EG − F 2 )
EGu − F Ev 2(EG − F 2 )
2 GFv − GGu − F Gv 2(EG − F 2 )
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.
Harvard University Department of Mathematics Thursday September 3, 2020 (Day 3)
lim n→∞
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.
(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?