第2届丘成桐大学生数学竞赛试题

S.-T.Yau College Student Mathematics Contests 2011

Analysis and Differential Equations

Individual

2:30–5:00pm,July 9,2011

(Please select 5problems to solve)

1.a)Compute the integral: ∞−∞x cos xdx (x 2+1)(x 2+2)

,b)Show that there is a continuous function f :[0,+∞)→(−∞,+∞)such that f ≡0and f (4x )=f (2x )+f (x ).

2.Solve the following problem: d 2u dx 2

−u (x )=4e −x ,x ∈(0,1),u (0)=0,du

dx

(0)=0.3.Find an explicit conformal transformation of an open set U ={|z |>1}\(−∞,−1]to the unit disc.

4.Assume f ∈C 2[a,b ]satisfying |f (x )|≤A,|f

(x )|≤B for each x ∈[a,b ]and there exists x 0∈[a,b ]such that |f (x 0)|≤D ,then |f (x )|≤2√AB +D,∀x ∈[a,b ].

5.Let C ([0,1])denote the Banach space of real valued continuous functions on [0,1]with the sup norm,and suppose that X ⊂C ([0,1])is a dense linear subspace.Suppose l :X →R is a linear map (not assumed to be continuous in any sense)such that l (f )≥0if f ∈X and f ≥0.Show that there is a unique Borel measure µon [0,1]such that l (f )= fdµfor all f ∈X .

6.For s ≥0,let H s (T )be the space of L 2functions f on the circle T =R /(2πZ )whose Fourier coefficients ˆf n = 2π0e

−inx f (x )dx satisfy Σ(1+n 2)s ||ˆf n |2<∞,with norm ||f ||2s =(2π)−1Σ(1+n 2)s |ˆf n |2.

a.Show that for r >s ≥0,the inclusion map i :H r (T )→H s (T )is compact.

b.Show that if s >1/2,then H s (T )includes continuously into C (T ),the space of continuous functions on T ,and the inclusion map is compact.1

S.-T.Yau College Student Mathematics Contests2011

Geometry and Topology

Individual

9:30–12:00am,July10,2011

(Please select5problems to solve)

1.Suppose M is a closed smooth n-manifold.

a)Does there always exist a smooth map f:M→S n from M into the n-sphere,such that f is essential(i.e.f is not homotopic to a constant map)?Justify your answer.

b)Same question,replacing S n by the n-torus T n.

2.Suppose(X,d)is a compact metric space and f:X→X is a map so that d(f(x),f(y))=d(x,y)for all x,y in X.Show that f is an onto map.

3.Let C1,C2be two linked circles in R3.Show that C1cannot be homotopic to a point in R3\C2.

4.Let M=R2/Z2be the two dimensional torus,L the line3x=7y in R2,and S=π(L)⊂M whereπ:R2→M is the projection map. Find a differential form on M which represents the Poincar´e dual of S.

5.A regular curve C in R3is called a Bertrand Curve,if there exists

a diffeomorphism f:C→D from C onto a different regular curve D in R3such that N x C=N f(x)D for any x∈C.Here N x C denotes the principal normal line of the curve C passing through x,and T x C will denote the tangent line of C at x.Prove that:

a)The distance|x−f(x)|is constant for x∈C;and the angle made between the directions of the two tangent lines T x C and T f(x)D is also constant.

b)If the curvature k and torsionτof C are nowhere zero,then there must be constantsλandµsuch thatλk+µτ=1

6.Let M be the closed surface generated by carrying a small circle with radius r around a closed curve C embedded in R3such that the center moves along C and the circle is in the normal plane to C at each point.Prove that

M

H2dσ≥2π2,

and the equality holds if and only if C is a circle with radius √

2r.Here

H is the mean curvature of M and dσis the area element of M.

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