This project mainly focuses on the isometric extension problem. This problem was proposed by D.Tingley in ‘Geom. Dedicata’ in 1987, which was to ask whether any surjective isometry between the unit spheres of real Banach spaces can be extended to a linear isometry. The importance of this problem is that if it is true, the local geometric property of a map on the unit sphere will determine the property of this map on the whole space. However, there was no progress at all on this problem for many years.

Prof. Guanggui Ding is a famous and leading mathematician in China. He has made significant contributions to this research field in the past decade. Around 2002, Prof. Ding found many effective methods in this problem and made a surprising progress. He made a breaking through and solved isometric extension problem in many classical Banach spaces, including Hilbert space, \ell^p (p>1), \ell^\infty and so on. Since then, Professor Ding and his students have made a lot of work and proved that any surjective isometry between a classical Banach space and a general Banach space can be extended to a linear isometry. These results were published in many journals, including ‘Proc. AMS’, ‘Nonlinear Anal.’, ‘Illinois J. Math.’, ‘J. Math. Anal. Appl.’, ‘Sci. China’ and so on. Prof. Ding also extended isometric extension problem to “into” isometries and Lipschitz maps. Some meaning results were obtained.

Besides the research work on isometric extension problem, Prof. Ding is also responsible for establishing a very strong and active Chinese research group in Banach space theory. Indeed, many currently active Chinese mathematicians in this field are either his former students or his postdocs in Nankai University. This is a very special and significiant contribution which deserves to be recognized.