S.T. Yau College Student Mathematics Contests 2014

第五届丘成桐大学生数学竞赛笔试已于 2014 年 7 月 12 日至 13 日举行. 竞赛组委会组织专家集中阅卷后, 评选出参加决赛(面试)的团队和个人名单. 第五届丘成桐大学生数学竞赛决赛(口试)将于 2014 年 8 月 2 日和 3 日在北京举行.

分析与方程

1. Let  \(f \colon\Bbb R\to \Bbb R\) be continuous function which s.t.

\[\sup_{x, y\in \Bbb R} |f(x+y)-f(x)-f(y)|<\infty\]

if we have \(\lim_{n\to \infty}\frac{f(n)}n=2014\), Prove \(\sup_{x\in \Bbb R}|f(x)-2014x|<\infty\).

2. Let \(f_1\), \(f_2\), \(\dotsc\) , \(f_n\in\) \(H(D)\bigcap C(\bar{D})\) , where \(D=\{z: |z|<1\}\). Prove

\[\phi(z)=|f_1(z)|+|f_2(z)|+\dotsb+|f_n(z)|\]

achieve maximum value on \(\partial D\).

3. Prove that if there is conformal mapping betwwen the annulus \(\{z:r_{1}<|z|<r_{2}\}\) and the annulus \(\{z:\rho_1<|z|<\rho_{2}\}\)

then

\[\frac{r_{2}}{r_{1}}=\frac{\rho_{2}}{\rho_{1}}\]

4. 设\(U(\xi)\) 是 \(\Bbb R\) 是有界函数且有有限多个不连续点, 证明

\[P_U(x)=\frac1\pi\int_{\Bbb R}\frac y{(x-\xi)^2+y^2}U(\xi)\,\mathrm d\xi\]

是调和的(Harmonic function)在半平面 \(\{z \in \Bbb C\colon\Im z >0\}\), 若 \(\xi\) 为 \(U\) 连续点

\[P_{U}(x)\to U(\xi), z \to \xi\]

5. 海森堡不等式

\[\int_{-\infty}^{+\infty}x^2|f(x)|^2\,\mathrm dx\int_{-\infty}^{+\infty}\xi ^2|\hat{f}(\xi)|^2 \,\mathrm d\xi \geq \frac{(\int_{-\infty}^{+\infty}|f(x)|^2\,\mathrm dx)^2}{16\pi^2}\]

几何与拓扑

1.  Let  \(X\) be the quotient space of \(S^2\) under the identifications \(x \sim -x\) for \(x\)  in the equator \(S^{1}\). Cmpute the homology groups \(H_{n}(X)\). Do the same for \(S^{3}\) with antipodal points of the equator \(S^{2} \subset S^{3}\) identified.

2.  Let \(M \to \Bbb R^3\)  be a graph defined by \(z=f(u,v)\) where \(\{u,v,z\}\) is a Descartes coordinate system in \(\Bbb R^3\). Suppose that \(M\) is a minimal surface.

Prove that:

(a) The Guass curvature \(K\) of \(M\) can be expressed as

\[K=\Delta \log (1+\frac1W),W:=\sqrt{1+(\frac{\partial f}{\partial u})^{2}+(\frac{\partial f}{\partial v})^{2}}\]

(b) If \(f\) is defined on the whole \(uv\)-plane, then \(f\) is a linear function. (Bernstein theorem)

3.  Let \(M=\Bbb R^2 / \Bbb Z^2\) be the two dimensional torus, \(L\) the line \(3x=7y\) in \(\Bbb R^2\), and \(S=\pi (L) \subset M\) where \(\pi :\Bbb R^2 \to M\) is the projection map. Find a differential form on \(M\) which represents the Poincare dual of \(S\).

4. Let \((\tilde M,\tilde g) \to (M,g)\) be a Riemannian submersion. This is a submersion \(p: M \to M\) such that for each \(x\in \tilde{M}, \ker^{\bot}(Dp) \to T_{p(x)}(M)\)  is a Linear isometry.

(a) Show that p shortens distance.
(b) If \((\tilde{M},\tilde{g})\) is complete, so is \((M,g)\).
(c) Show by example that if \((M,g)\) is complete, \((\tilde{M},\tilde{g})\) may not be complete.

5. Let \(\psi :M \to \Bbb R^3\) be an isometric immersion of a compact surface \(M\) into \(\Bbb R^3\).

Prove that

\[\int_MH^2 \,\mathrm d\sigma \geq 4\pi\]

where \(H\) is the mean curvature of \(M\) and \(d\sigma\) is the area element of \(M\).

6. The unit tangent bundle of \(S^2\) is the subset

\[T^1(S^2)=\{(p,v)\in \Bbb R^2\, | \, \|p\|=1, (p,v)=0,\|v\|=1\}\]

Show that it is a smooth submanifold of the tangent bundle \(T(S^2)\) of  \(S^2\) and \(T^1(S^2)\) is diffeomorphic to \(\Bbb RP^3\).

感谢博士数学论坛的网友 zwb565055403 提供的两套试题, 网友数函分享的 PDF 试题

个人赛试题

Analysis and differential equations Individual 2014

Geometry and topology Individual 2014

Algebra and number theory Individual 2014

Probability and statistics Individual 2014

Applied Math. and Computational Math. Individual 2014

团体赛

team 2014