On the afternoon of June 13,2019, Yiming Long, a academician of the Chinese Academy of Sciences and professor of chern Institute of Mathematics of Nankai University, visited Beiyang mathematics lecture hall, and began with Jule Verner's famous novel "Around the World in Eighty Days", and talked about the problem of closed geodesic. The scene gathered, teachers and students from Tianjin University, Tianjin Normal University and other universities came together to listen to the report. Xiaotao Sun, Dean of the school of mathematics of Tianjin University, presided over the event.
Yiming Long introduces the concept of "closed geodesic" in mathematical theory, which is the shortest local closed route, taking the protagonist Fogg's route around the earth as an example. Then he asked the question: is there a closed geodesic line on the two-dimensional sphere after adding mountains and basins (that is, under Riemannian metric)? Then he introduced to the students the pioneering achievement of American Mathematical master Birkhoff on closed geodesic, and gave a brief explanation of two methods introduced by Birkhoff to solve the problem of closed geodesic: Minimax method and Birkhoff mapping method. He also briefly introduced Morse theory, which was established by Morse, an American mathematician, to study geodesics.
Yiming Long took Russian mathematician katok's construction of a family of Finsler metrics on the sphere, which made there only two different closed geodesics on the two-dimensional sphere as an example, explaining to teachers and students that there may not be infinite closed geodesics on the sphere under Finsler metrics. He also introduced the result he proved in 2004 with German mathematician Bangert: there are always at least two different closed geodesics on a two-dimensional sphere equipped with any non reversible Finsler metric. According to this, it can be further obtained that there are at least two different closed geodesics on any two-dimensional compact Finsler surface.
Finally, academician Yiming Long concluded with Einstein's famous saying: "Don't worry about the difficulties you encounter in mathematics, I can assure you that the difficulties I encounter are greater than you". He encouraged the students to keep curiosity, laid a good foundation, and be brave in innovation. Through constant efforts, students should be ready to make contributions to the development and innovation and solving major scientific problems in the future.
At the end of the seminar, academician Yiming Long also answered the students' questions on learning interest, ability to discover problems, cultivation of geometric intuitive ability and so on, had in-depth exchanges with teachers and students, and encouraged students to devote themselves to mathematics learning and research.
Text: Youquan Zheng, Shen Zhao
Picture: Lili Li