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QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday October 1, 2002 (Day 1) There are six problems. Each question is worth 10 points, and parts of questions are of equal weight. 1a. Exhibit a polynomial of degree three with rational coeﬃcients whose Galois group over the ﬁeld of rational numbers is cyclic of order three. 2a. The Catenoid C is the surface of revolution in R 3 of the curve x = cosh(z) ab
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QUALIFYING EXAMINATION Harvard University Department of MathematicsTuesday October 1, 2002 (Day 1) There are six problems. Each question is worth 10 points, and parts of questions areof equal weight. 1a. Exhibit a polynomial of degree three with rational coeﬃcients whose Galoisgroup over the ﬁeld of rational numbers is cyclic of order three.2a. The Catenoid C  is the surface of revolution in R 3 of the curve x = cosh( z )about the z axis. The Helicoid H  is the surface in R 3 generated by straightlines parallel to the xy plane that meet both the z axis and the helix t −→ [cos( t ) , sin( t ) ,t ] . (Recall that sinh( x ) = e x − e − x 2and cosh( x ) = e x + e − x 2.)(i) Show that both C  and H  are manifolds by exhibiting natural coordinateson each.(ii) In the coordinates above, write the local expressions for the metrics g C  and g H  , induced by R 3 , on C  and H  , respectively.(iii) Is there a covering map from H  to C  that is a local isometry?3a. In R n , consider the Laplace equation u 11 + u 22 + ··· + u nn = 0 . Show that the equation is invariant under orthonormal transformations. Findall rotationally symmetric solutions to this equation. (Here u ii denotes thesecond derivative in the i th coordinate of a function u .)4a. Let C  denote the unit circle in C . Evaluate   C  e 1 /z 1 − 2 z 5a. Let G (1 , 3) be the Grassmannian variety of lines in C P  3 .1  (i) Show that the subset I  ⊂ G (1 , 3) 2 I  = { ( l 1 ,l 2 ) | l 1 ∩ l 2  = ∅} is irreducible in the Zariski topology. (Hint: Consider the space of triples( l 1 ,l 2 ,p ) ∈ G (1 , 3) 2 × C P  3 such that p ∈ l 1 ∩ l 2 , and consider two appro-priate projections.)(ii) Show that the subset J  ⊂ G (1 , 3) 3 J  = { ( l 1 ,l 2 ,l 3 ) | l 1 ∩ l 2  = ∅ , l 2 ∩ l 3  = ∅ , l 3 ∩ l 1  = ∅} is reducible. How many irreducible components does it have?6a. For the purposes of this problem, a manifold is a CW complex which is locallyhomeomorphic to R n . (In particular, it has no boundary.)(i) Show that a connected simply-connected compact 2-manifold is homotopyequivalent to S  2 . (Do not use the classiﬁcation of surfaces.)(ii) Let M  be a connected simply-connected compact orientable 3-manifold.Compute π 3 ( M  ).(iii) Show that a connected simply-connected compact orientable 3-manifoldis homotopy equivalent to S  3 .(iv) Find a simply-connected compact 4-manifold that is not homotopy equiv-alent to S  4 .2  QUALIFYING EXAMINATION Harvard University Department of MathematicsWednesday October 2, 2002 (Day 2) There are six problems. Each question is worth 10 points, and parts of questions areof equal weight. 1b. Let C [ S  4 ] be the complex group ring of the symmetric group S  4 . For n ≥ 1 let M  n ( C ) be the algebra of all n × n matrices with complex entries. Prove thatthe algebra C [ S  4 ] is isomorphic to a direct sum  i =1 ,...,t M  n i ( C )and calculate the n i ’s.2b. (i) Show that the 2 dimensional sphere S  2 is an analytic manifold by ex-hibiting an atlas for which the change of coordinate functions are analyticfunctions. Write the local expression of the standard metric on S  2 in theabove coordinates.(ii) Put a metric on R 2 such that the corresponding curvature is equal to 1.Is this metric complete?3b. Let C  ∈ C P  2 be a smooth projective curve of degree d ≥ 2. Let C P  2 ∗ be thedual space of lines in C P  2 and C  ∗ ⊂ C P  2 ∗ the dual curve of lines tangent to C  .Find the degree of  C  ∗ . (Hint: Project from a point.)4b. Let f  : R → R be any function. Prove that the set of points x ∈ R where f  iscontinuous is a countable intersection of open sets.5b. Prove that the only meromorphic functions f  ( z ) on C ∪ {∞} are rationalfunctions.6b. (i) Show that the fundamental group of a Lie group is abelian.(ii) Find π 1 (SL 2 ( R )).3  QUALIFYING EXAMINATION Harvard University Department of MathematicsThursday October 3, 2002 (Day 3) There are six problems. Each question is worth 10 points, and parts of questions areof equal weight. 1c. Let H  = { ( u,v ) ∈ R 2 | v > 0 } and B = { ( x,y ) ∈ R 2 | x 2 + y 2 < 1 } . For e 2 = (0 , 1) ∈ R 2 , map H  to B by the following diﬀeomorphism. v −→ x = − e 2 +2( v + e 2 )  v + e 2  2 . (i) Verify that the image of the above map is indeed B . (Hint: Think of thestandard inversion in the circle.)(ii) Consider the following metric on B : g = dx 2 + dy 2 (1 − x  2 ) 2 . Put a metric on H  such that the above map is an isometry.(iii) Show that H  is complete.2c. Let C  ⊂ C P  2 be a smooth projective curve of degree 4.(i) Find the genus of  C  and give the Riemann-Roch formula for the dimensionof the space of sections of a line bundle M  of degree d on the curve C  .(ii) If  l ∈ C P  2 is a line meeting C  at four distinct points p 1 ,...,p 4 , prove thatthere exists a nonzero holomorphic diﬀerential form on C  vanishing at thefour points p i . (Hint: Note that O C P  2 (1) restricted to C  is a line bundleof degree 4. Use the Riemann-Roch formula to prove that this restrictionis the canonical line bundle K  C  .)3c. Let A be the ring of real-valued continuous functions on the unit interval [0 , 1].Construct (with proof) an ideal in A which is not ﬁnitely generated.4
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