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In 1956 he came back to his motherland out of patriotism and taught at Peking University ever since. Besides teaching as a professor instructing master and doctoral candidates, Professor Liao also actively worked as a member on the editoral committees of China's leading journals in the fields of science and mathematics, such as Science in China (Scientia Sinica), Chhwse Science Bulletin, Chinese Annals of Mathematics and Applied Mathematics and Mechanics. Professor Liao devoted all his. life to the progress of mathematics.
A world renowned mathematician, Professor Liao made great contributions to the development of topology and differential dynamics. Before the 60's he had worked at periodical transformation, homotopy theory and fiber bundle, etc. In order to solve the f.~.,~ous Sroith problem, he first introduced multiplicational structure into singular homology group. In tae study of fiber bundle theory, he successfully solved the Chern and Spanier's guess by using his own creative method of symmetrization to deal with the calculation of the second layer of barriers. And in the research of homootopy, he unified the duality theorems by Poincar6, Alexander and Leifuchez by proposing a new local group and loca! upper and Cartesian products. His work was highly evaluated by the master mathematicians all over of the world.
When differential dynamic system began to develop in the early 60's, Professor Liao found immediately this field was full of challenge and started the research on it. After two decades' efforts, he proposed a creative theoretical system based on a model system of equations and a barrier set, which was quite different from that widely used by western researchers. By using these theories and methods, he made great historical contributions to a series of important problems such as smooth ergodic theory, colliery of closure, and the guess of stability, which made him become one of the main representatives in the field of differential dynamic system. In order to site his autsdanding contribution in both topology and differential dynamic system, he was given the second award of national natural science in 1982, the award of mathematics by the Science Academy of the Third World in 1986, the first award of national natural science in 1987 and the award for the progress in science and technology offered by He Liang and He Li in 1995.
Professor Liao paid great attention to teaching and training young scientists. He insisted on elieitation method in teaching and stressed the training in quality and the ability of solving
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