Li Banghe

Li Banghe was born in 1942, in Yueqing City, Zhejiang Province. He graduated from the University of Science and Technology of China in 1965, and then had a job in Chinese Academy of Sciences, up to now. He was elected an academician of Chinese Academy of Sciences in 2001.
He devoted great efforts to the unity of mathematics, had quite systematic contributions in several fields which are far from each other. Published two books and over one hundred papers.
1.Quantum invariants and lowdimensional topology
Since 1980s, topological quantum fields theory has brought vital force to the researches of lowdimensional topology, yielding gauge invariants, magnetic monopole invariants and Witten type invariants of 3dimensional manifolds. Li has works for all of these invariants.
1) One of the typical problems in 4dimensional topology is to represent a homology class in a 4dimensional manifold by spheres. Studied by famous mathematicians like Milnor, passed through over forty years, it is still a difficult problem. The mathematical theory of gauge fields brought breakthroughs for it. So far, for eight simplyconnected 4manifolds, the problem has been solved completely. Li contributed five of them, and used various methods.
2) A problem more general and more important than representing by spheres is the problem of minimal genera.The magnetic monopole theory of physicists Seiberg and Witten brought inflexion to this problem. In the abstract of the paper“Minimal Genus Problem” published in 1997, Lawson wrote: In the resent five years, there have been several major breakthroughs on this problem, it is solved completely in some cases. The breakthroughs written therein include five theorems given by Li himself or Li collaborating with the other.
3)For Witten invariants, defined new invariants, cleared the relation of various invariants; completely solved the problem of generalized Gauss sums in algebraic number theory, then calculated the Witten invariants for all lens spaces, and answered a problem of Kirby and the other.
2. Differential topology
Li’s works in differential topology concern immersion, cobordism, embedding and foliation, while mainly in immersion.
 1)One of the two most fundamental theorems in differential topology is: Any nmanifold can be immersed in (2n-1)-Euclidean spaces. This theorem was generalized by Li and Peterson from the simplest manifolds (Euclidean spaces) to arbitrary manifold, that is: any map from a nmanifold to a (2n-1)-manifold is homotopic to immersions.
2)For classifying the immersions homotopic to a given map, Li’s theory concerns two steps. The first step is similar to classify immersions in Euclidean spaces, while the second steep relies heavily on the complexity of target manifolds. Under this framework, he solved the problem of classifying immersions for all maps from nmanifolds to 2n- manifolds, and those from 2-manifolds to 3-manifolds. Some famous topologists did not notice still having this more difficult “second step,” which led them to some mistakes.
3)Li and his collaborators solved the difficult problems of classifying immersions from nmanifolds into (2n-2)-Euclidean space, and classifying immersions from kconnected nmanifolds into (2n-k)-Euclidean space. Moreover, he had contributions in nonstandard analysis, theory of generalized functions, qualitative study on the solutions of single conservation laws, and mathematical biology.
He won Chen prize and Hua prize of Chinese Mathematical Society in 1989 and 2009, respectively.