'Solving Math Problems' Professor Kim Sanghyun of the Institute for Basic Science... July Scientist of the Month

Professor Kim Sang-hyun, Department of Mathematics, Institute for Advanced Study

[Asia Economy Reporter Junho Hwang] The Ministry of Science and ICT and the National Research Foundation of Korea have selected Professor Sanghyun Kim of the School of Mathematics at the Institute for Basic Science as the recipient of the July Science and Technology Person of the Month Award. The Science and Technology Person of the Month Award is given monthly to researchers and developers who have contributed to the advancement of science and technology through outstanding research and development achievements, awarding the Minister's Prize and a cash prize of 10 million KRW.

Professor Kim received this award for discovering a finitely generated group that can precisely identify the singular regularity of the group of diffeomorphisms over the entire real number range, thereby solving difficult problems in geometric group theory and topology. The group of diffeomorphisms refers to the set of symmetries of one-dimensional spaces such as circles and lines, where these symmetries smoothly deform the space and are called diffeomorphisms. Singular regularity is a value that measures how smooth each diffeomorphism in the group is.

Professor Kim analyzed the relationships among the three aspects of analysis, dynamics, and group theory inherent in the group of diffeomorphisms and proved how partial groups within the diffeomorphism group determine singular regularity. This is regarded as building a new bridge connecting the purely algebraic perspective of space symmetries (such as operations like addition and multiplication) with the analytical perspective (the degree of smoothness of motions flowing over the space).

Professor Kim’s research also completely resolved a challenging problem raised by mathematician A. Navas in his invited lecture at the 2018 International Congress of Mathematicians (ICM) concerning the dynamical constraints of diffeomorphisms. The related results were published in the world-renowned mathematics journal Inventiones Mathematicae in March of this year.

The Ministry of Science and ICT evaluated that big data handled in fields such as medical imaging, social network services, and jet engines can be described as mathematical spaces that change over time, which directly relates to the symmetry of one-dimensional spaces. Professor Kim pioneered a systematic research field connecting the algebraic properties of the symmetry of one-dimensional spaces with the differentiable degree of these symmetries.

Professor Kim explained, "This research is the first systematic result connecting the algebraic properties of differentiable symmetries of one-dimensional spaces with their differentiable degree," and added, "I hope that through the interaction among dynamics, low-dimensional topology, and geometric group theory, this can be developed into a rigidity theory within the group of diffeomorphisms."

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