February 2011 10th issue

In this issue, we are glad to have the great mathematician, Prof. Shing-Tung Yau, to share his experience in investigating the geometry. In it he shares with us how he came to success in proving the Calabi-Yau Spaces.

About Shing-Tung YauProf. Shing-Tung Yau (1949 –) is a Chinese mathematician in the field of differential geometry. In his early years, he studied mathematics in the Chinese University of Hong Kong. Then he went to the University of California at Berkeley and studied with Prof. Shiing-Shen Chern, the legendary geometer. Prof. Yau has been a professor of mathematics at Harvard University since 1987 and is the current department Chair.

Prof. Yau received several awards for his contributions, including the 4 highest awards in mathematics.

The Giant of Maths, Prof. Shing-Tung Yau — Exploring the Geometry of the Universe

Prof. Yau’s first major contribution to differential geometry was his proof of the Calabi conjecture. It was a great achievement in mathematics, so how did he do it?

Prof. Shing-Tung YauNational Medal of Science

(1997)

Crafoord Prize (1994)

Wolf Prize (2010)

Fields Medal (1982)

Coming issue: New Topics in the NSS Mathematics

— Applications of the Dispersion of Data and

the StandardDeviation

Hong Kong Educational Publishing Co.

Mathematics Express February 2011

Road to Geometry

In secondary school, Prof. Yau was fascinated by the principles of Euclidean geometry (geometry which we learn in secondary mathematics). He loved the geometric figures built by straight lines, circles, planes and spheres. Yet, he realized that classical Euclidean geometry cannot describe a simple object like an apple. Soon, he found the power of calculus. Using calculus and Euclidean geometry, curved objects can be described.

In 1969, during his first year in the University of California at Berkeley, Prof. Yau learned about Riemannian geometry and tried to relate the curvature of a space to a more general way of characterizing shape with statements. His strong interest in geometry drove him to make an important proof in the future.

Riemannian geometry is a non-Euclidean geometry named after Bernhard Riemann (1826 –1866).

Bernhard Riemann

Physics with Geometry

When Prof. Yau read about Einstein’s equations, he wondered if one could construct a spacetime that is a vacuum, yet its curvature and gravity are still pronounced. He believed that such a space could not exist, so he started to think about this, in a mathematical way.

In the mean time, Prof. Yau found that a geometer Eugenio Calabi (1923 –) had posed almost the same question. Calabi framed the problem in complicated mathematical language, which was called the Calabi conjecture. In words, it is: ‘Can there be gravity, or the curving of space, in a closed vacuum – a compact space that has no matter?’ Prof. Yau and his friends thought that it was ‘too good to be true’, so they spent about 3 years, attempting to disprove the existence of such a space.

Such a space would be beautiful, but isn't it 'too good to be true'?

Can there be gravity, or the curving of space, in a closed vacuum – a compact space that has no matter?

The Calabi conjecture is a conjecture about the properties of multidimensional shapes, made by Eugenio Calabi in 1954. At that time, Calabi believed that he was correct. Yet, he did not prove it.

Calabi and Yau, 2004

Hong Kong Educational Publishing Co.

Mathematics ExpressFebruary 2011

Calabi-Yau Space

Calabi-Yau SpacesOne day, Prof. Yau discovered a way to disprove Calabi’s proposition and announced it at a big conference at Stanford. However, a few months later, he had some difficulty while reconstructing his argument. Thus, he tried to look for other examples where Calabi was wrong. When Prof. Yau looked for more examples, he had a much deeper insight into the issue. Finally, he determined that the Calabi conjecture had to be right.

In 1976, while completing the proof of the Calabi conjecture, Prof. Yau also found a general mechanism for constructing such spaces, which we call Calabi-Yau spaces.

Prof. Yau’s contribution earned him the Fields Medal in 1982. Soon, it was known that the proof of Calabi-Yau spaces was not only a mathematical achievement, but also related to physics. It helped the physicists Gary Horowitz and Andy Strominger develop a new idea for explaining the universe, string theory. After that, string theory also inspired some extremely powerful mathematicians.

Prof. Yau tells us that even a brilliant mathematician needs effort and training. One’s success does not come easily, it needs the work of predecessors.

ConclusionMathematics and other disciplines, for instance, engineering or physics, help and inspire each other. Important ideas flow through different disciplines and these flows result in the birth of new and important subjects.

Mathematics also helps us know more about our daily lives and, most importantly, it leads us to a deeper understanding of nature, which is the original intent of mathematics, according to Prof. Yau.

Fields Medal

About the Fields Medal

In 1924, at the International Congress of Mathematicians (ICM) in Toronto, a resolution was adopted that two gold medals should be awarded to recognize outstanding mathematical achievement at each congress. Prof. J. C. Fields, a Canadian mathematician, was the donor for establishing the medals. The gold medals were named in his honour.

The Fields Medal is awarded every four years at the congress. In 1966, it was agreed that up to four medals could be awarded at each congress. Therefore, at each Congress, there are two to four winners.

Design of the Medal

The medal was designed by a Canadian sculptor, Robert Tait McKenzie.

Front

The front of the medal shows the face of Archimedes, his name in Greek capital letters, the designer’s monogram and date. The inscription means ‘To transcend one’s spirit and to take hold of the world’.

Back

The inscription means ‘The mathematicians having congregatedfrom the whole world awarded because of outstanding writings’.

Calabi and Yau, 2004

By Stefan Zachow (ZIB) By Stefan Zachow (ZIB)

BackFront

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Mathematics Express October 2010February 2011Mathematics Express

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The Latest Publication of Prof. Shing-Tung Yau

Latest news

As for Prof. Yau, mathematics is not a wholly abstract discipline

disconnected from everyday phenomena. It is crucial to our

understanding of the physical world and also the beauty of nature.

Therefore, starting in 2006 autumn, Prof. Yau and Steve Nadis

wrote a new book, The Shape of Inner Space – String Theory and the

Geometry of the Universe’s Hidden Dimensions.

The book shows how mathematics and physics come together to

the benefit of both fields through the Calabi-Yau spaces and string

theory.

According to Prof. Yau, Steve Nadis mainly writes for an astronomy

magazine and does not have a strong mathematics background. In

the book, Prof. Yau explains his abstract mathematical thoughts to

Steve, then, Steve conveys the thoughts in words, hoping to give

people a sense of how mathematicians think and approach the world.

It took them 4 years to finish the book. In it they share the

experience of how different fields of study communicate and hope to

change people's point of view on mathematicians and other scientists.

The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions