Our Eminent Scientists @ CUHK:
A Beautiful Legacy of Geometric Research
Prof. LI Man Chun Martin
Department of Mathematics
The idea of minimal surfaces could be observed from the curves of architecture to soap bubbles. Professor LI Man Chun Martin, a geometer from the Department of Mathematics, has achieved substantial progress towards establishing the existence of minimal surfaces thus using them to understand the geometry and topology of their surrounding space. LI is also an alumnus of CUHK Mathematics, and is witness and testament to the endeavours of mathematical research over generations of scholars.
The geometry of soap bubbles
Plateau's laws, formulated in the 19th century by a Belgian physicist Joseph Antoine Ferdinand Plateau, describe the shape and configuration of soap bubbles. The configurations of the soap bubbles other than those of Plateau's laws are unstable, and the film will quickly tend to rearrange itself to conform to the laws. These laws for minimal surfaces, however, remained an observed experimental fact without a rigorous mathematical proof for more than a century. It was not until 1976, when Jean Taylor, an American mathematician, reported the findings on the structure of singularities in soap bubble-like and soap film-like minimal surfaces, that Plateau's laws were finally proved mathematically using geometric measure theory. Thus, it was scientifically proven that tiny soap bubbles hold a vast array of far-reaching geometric structures, with influence in research ranging from general relativity in physics to the endoplasmic reticulum in biology, just to name two examples.
A "typical" soap bubble structure for the demonstration of minimal surfaces. (Photo Credit: McNeel Forum)
LI is an expert on studying the geometry of space using minimal surface theory. In Classical Euclidean geometry, we learn about problems related to straight lines or other geometric objects like polygons or spheres in two or three dimensions. For research about minimal surfaces theory, geometers would study the analogy of straight lines in higher dimensions. The conformations of minimal surfaces may vary with dimensions. Apart from the more detailed study of minimal surfaces, LI is also interested in pursuing further development in the theory of minimal surfaces with free boundary, where the location of the boundary of the minimal surface is not fixed in advance but is allowed to move freely on another prescribed surface. These minimal surfaces serve as a means to study the geometry of the underlying space together with the boundary effects. The research findings could serve as a powerful tool to solve existing geometric problems.
Endeavours of Geometric Research over Generations
Li’s contributions in geometric research have been recognised by China's Excellent Young Scientists Fund 2020. Based on the established foundational work, Prof. LI will continue to pursue more detailed study of min-max theory, geometric flows and geometric partial differential equations; he will also explore new applications in three-dimensional topology, conformal geometry and calibrated geometry. The breakthrough would have significant influence in the advancement and understanding of physics, materials science, architecture, biology. Concentrating on pure mathematics research may seem tedious at times. However, he often receives inspiration from advances made in other disciplines, shedding new light on his own research.
The tent-like structure, 2010 Shanghai EXPO Boulevard “Sun Valley Pavillion” is the world’s largest membrane roof to date. Minimal surfaces inspired the design of the membrane roof. (Photo credit: © Thomas Ott, INHABITAT)
The research breakthroughs in pure mathematics or other forefront fundamental research may not see their applications in the same era. LI considered the long-running research journey as a “marathon relay”. Li’s journey also exemplifies the endeavours of generations of mathematicians. He had conducted research under the supervision of Professor Richard Schoen, who was one of the first students of Professor YAU Shing Tung, Distinguished Visiting Professor-at-Large of CUHK. The supervisor of Professor YAU was Professor CHERN Shiing-shen, the Mathematics Maestro who effectively reshaped the fields of geometry and topology. Mathematics is a deductive science with diverse research areas, and its problems may take decades, even centuries, to be resolved by generations of scholars. Once the mathematical theory has been proved, it will never be outdated and will continue to yield influence over vast subject areas and leave a lasting impact on society.
“Imagination will take you everywhere.”
From the time studying at CUHK Mathematics as an undergraduate to LI’s return to his alma mater for teaching and research, it has been a sort of passing a tradition. LI has observed that as time goes by, the campus, as well as the attitudes of students, have gone through significant changes. The Internet has long been a necessity for learning and doing research, but now students could access information from around the world with the snap of a finger. The value of “knowledge” seems to have depreciated over the years. However, LI wanted to remind students who wish to pursue a research career, that they must have a critical thinking mindset to look for truth and reality; it’s the only way to transform massive amounts of information into ‘knowledge’ to meet the challenges ahead.”
Professor LI pointed out that the problems of minimal surfaces first arose two centuries before the implementation of this concept in vast subject areas and architecture, revealing the endeavours of geometric research over generations.
Professor LI was promoted to the rank of associate professor in 2020. Even before achieving tenureship, he has been taking good care of his physical and emotional health amidst his ardent devotion to research. Only maintaining a keen interest in the subject could a researcher persist in this extended journey. LI often goes hiking during his spare time, and being in nature would refresh his mind, sometimes even bringing him inspiration in overcoming bottlenecks in his research. LI also shared one of his favourite mottos, “Logic will get you from A to B. Imagination will take you everywhere.” - Albert Einstein. Professor LI extends Einstein’s insight to suggest that only with the breadth of perspective could one open up a progressing new dimension.
Text: Angela HUNG | Editing: Christine LING