On March 21, 2018, the conference hall of the administration building of Peiyangyuan Campus was crowded with students and enthusiastic applause. Zhang Weiping, an academician of the Chinese Academy of Science, Professor of Shiing-Shen Chern Institute of mathematics of Nankai University and Deputy-director of the Center for Applied Mathematics of Tianjin University, delivered an academic lecture entitled ‘From Triangles to Manifolds’ to the students and teachers of the School of Mathematics. It was also the second class of the mathematic lecture series given by the School of Mathematics at Peiyangyuan Campus. The lecture was presided over by Sun Xiaotao, Dean of the School of Mathematics. 

At the beginning of lecture, Zhang told a story of Atiyah and Singer, winners of the 2004 Abel Prize to introduce the Atiyah-Singer Index Theorem which is known as one of the greatest mathematical theorems of the twentieth century.  

After that, Zhang talked about the deep friendship and intimate relationship between Chen-Ning Yang and Shiing Shen Chern. In particular, Zhang quoted a poem written by Chen-Ning Yang, which was widely spread throughout the science community to express his respect and admiration to Shiing Shen Chern as well as to bring out the history of the Atiyah-Singer Index Theorem in a special way. Then, he introduced the great contributions of five famous mathematicians: Euclid, Gauss, Riemann, Cartan and Chern. The story started with Euclid who found that the sum of inner angles of any triangle is invariant. Then Gauss introduced the Gauss curvature to a curved triangle and presented the Gauss-Bonnet Theorem. The Gauss-Bonnet Theorem is regarded as a bridge between local and global topology. The Gauss-Bonnet Theorem further explained one essence of mathematics--Change is hidden in steadiness, and the principle of changes is same. Based on Gauss' work, Riemann proposed the concept of Riemannian space, which generalizes the geometry of two dimensional space to high dimensional space. Then, the theory is extended to Cartan calculus on manifolds. Later, known as "The Father of Modern Differential Geometry" Shiing Shen Chern established the Gauss-Bonnet-Chern theorem and a Hermitian manifold to establish a new direction for differential geometry in a large range. 

Zhang said that the important reason for Mr. Chern achieving success was mainly due to three principles -- solving the right problems, using the right methods and developing the right direction. On the basis of the important results of the previous mathematicians, Atiyah and Singer continued to break through the limitation of the Dirac operator and to consider a wider ellipse operator, and as a result, Atiyah-Singer Index Theorem was put forward. At the same time, Zhang used the story of Mr Shiing Shen Chern as an example to tell students that the contributions of a mathematician are not to be viewed as easy and casual; on the contrary, every successful mathematician must make great effort and work hard in silence. He exhorted students to study, to research and to work hard despite having to endure  loneliness and an endless workload. We should enjoy our lives in hard times, and we should believe that all sufferings have their rewards. 

At last, Zhang summarized the characteristics of the recent development of mathematics which is more inclined to intercross, interpenetration and interaction between mathematics and physics. He said that he hopes to see that the two disciplines can complement each other, learn from each other and make progress together. He mentioned that in the field of mathematics and physics, professors such as Chengtong Qiu, Gang Tian, Jun Li, Kefeng Liu and Yongbin Ruan have made outstanding contributions. And he also hoped that the new generation could make an important impact on the development of mathematics. Zhang’s profound and humorous words got the enthusiastic applause from the audience several times. 

The students asked Zhang a number of questions about the study and research of mathematics. Zhang answered their questions one by one and encouraged them with inspiring words. He concluded that the foundation for learning is essential, but even more important is to be able to study and discuss problems with friends and get more profound knowledge by ourselves, and he said that the soul of  learning mathmatics is continually seeking truth . He also pointed out the current trend and ultimate goal of academic research in mathematics is to study the physical and mathematical properties of space for exploring the essence of differential geometry. 

The academic lecture successfully concluded with enthusiastic applause from the audience. Teachers and students not only understand the professional knowledge and the development of differential geometry, but also learn about the stories and contributions of great mathematicians. 

The Peiyang Math Forum is a high-end mathematics forum set up by the School of Mathematics of Tianjin University integrating ideological collisions, academic exchanges and cultural heritage. The Peiyang Math Forum aims to create a platform for young people who are interested in mathematical learning and research to inspire their interests, to broaden their academic horizons, to expand their ideological dimensions. The mathematics lecture series at Peiyangyuan Campus would provide a platform to guide those who would like to pursue research in mathematics and provide chances to meet experts in mathematics face to face. The mathematics lecture series at Peiyangyuan Campus also provides in-depth communication with students and teachers to make it become one of the brand activities of the School. 

By: the School of Mathematics

Editors: Qin Mian and Keith Harrington