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

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

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

5. 海森堡不等式

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

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 $$\Bbb RP^3$$.

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

team 2014

2013 年第四届丘成桐(Shing-Tung Yau)大学生数学竞赛(S.T. Yau College Student Mathematics Contests)已经落下帷幕. 决赛已经于 8 月 11 日和 12 日在北京中国科学院数学与系统科学院思源楼和晨兴中心举行, 颁奖典礼也已于 8 月 12 日在清华大学举行.

Analysis and differential equations 2013 Individual

Geometry and topology 2013 Individual

Algebra and number theory 2013 Individual

Probability and statistics 2013 Individual

Applied Math. and Computational Math. 2013 Individual

Team 2013

A new book A History in Sum: 150 Years of Mathematics at Harvard (1825-1975) has just been published by Harvard.

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard’s mathematics department was at the center of these developments.A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics–in algebraic geometry and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose.

The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics–an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce’s successors–William Fogg Osgood and Maxime Bôcher–undertook the task of transforming the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Birkhoff produced a dazzling body of work, while training a generation of innovators–students like Marston Morse and Hassler Whitney, who forged novel pathways in topology and other areas. Influential figures from around the world soon flocked to Harvard, some overcoming great challenges to pursue their elected calling.

A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.

### Review

This book tells the tale of how mathematics developed at Harvard–and by extension in the United States–since early days. It is filled with fascinating stories about some of the legendary names of modern mathematics. Both fans of mathematics and readers curious about the history of Harvard will enjoy it. (Edward Witten, Professor Of Physics, Institute For Advanced Study)

A History in Sum is a beautiful tribute to a beautiful subject, one that illuminates mathematics through the lens of some of its most remarkable practitioners. The authors’ love of mathematics shines through every chapter, as they use accessible and spirited language to describe a wealth of heady insights and the all-too-human stories of the minds that discovered them. There is perhaps no better book for immersion into the curious and compelling history of mathematical thought. (Brian Greene, Professor Of Mathematics & Physics, Columbia University)

The book is written in a leisurely style, the scope is remarkably broad, and the topics covered are explained astonishingly well. Once I started the book, I simply couldn’t put it down and I was ecstatic to easily understand important mathematics far from my own research interests. (Joel Smoller, Professor Of Mathematics, University Of Michigan)

A History in Sum contains a wealth of good stories, stories that go to the heart of the development of mathematics in this country. The authors succeed in humanizing and enlivening what might otherwise be a dry treatment of the subject. (Ron Irving, Professor Of Mathematics, University Of Washington)

• Author: Steve Nadis and Shing-Tung Yau
• Hardcover: 280 pages
• Publisher: Harvard University Press (October 7, 2013)
• Language: English
• ISBN-10: 067472500X
• ISBN-13: 978-0674725003
• Price: \$39.95
• Product Dimensions: 6 x 9 inches

2012年第三届丘成桐(Shing-Tung Yau)大学生数学竞赛(S.T. Yau College Student Mathematics Contests)笔试已于7月1, 2日成功举行.

Analysis and differential equations 2012 Individual

Geometry and topology 2012 Individual

Algebra and number theory 2012 Individual

Probability and statistics 2012 Individual

Applied Math. and Computational Math. 2012 Individual

Group 2012