Modern mathematics

17 June 2016

Professor Ngaiming Mok of Hong Kong University admits that the mathematical concepts he grapples with are too complex for most people. After all, his research into Complex Differential Geometry, holomorphic geodesic cycles on Hermitian locally symmetric space, or Fano manifolds does not make for easy dinner conversation.

However, on a recent return trip from Taipei, the professor got into talking with his taxi driver about how mathematics can relate to real-life situations. By the end of the hour-long trip to the airport, the driver had become so delighted in learning about the research methodology that he wanted his children to take up mathematics.

“Mathematics is not a monologue, it can be made relatable by understanding the background and intention of audience, their interest or, in some cases, lack of interest,” Prof Mok explained.

Application of mathematics

It is no surprise that mathematics is everywhere, in different forms, often times hidden from view. Calculations on probability and statistical modelling can be used to make trend forecasts in weather or the stock exchange, for example. More complex theories in physics about gravitational waves, or general relativity all rely on knowledge in pure mathematics.

Technology is advancing at breakneck speed today, leading to new innovations in all spheres of life. However, because mathematics is so esoteric and abstruse, most people don’t realise how much the newer and more efficient ways to do things are built on it. Mathematics is an essential skill set for the information age.

Prof Mok’s field is pure mathematics, but he says applied mathematics carries profound importance in today's technology-driven world. Mathematical theories are used to solve industrial problems, for instance the application of number theory (factorisation of composite numbers into products of primes) is an important method in cryptography. As another example, algebraic coding theory is one of the most significant tools in digital communications, and forms the basis for data compression, error-correcting codes, and for network coding.

During an interview in his office at the Department of Mathematics at HKU recently, Prof Mok grew animated as he explained the myriad uses of mathematics that make everyday tasks that we often take for granted possible.

“Mathematics is simply beautiful, and it is gratifying to find perspectives and being able to understand the phenomenon from the theoretical side and then applying it,” he said, “The development of mathematics should be such that it allows for constant feeding of methods to the people who are making applications with mathematics.”

Prof Mok

Transition

The world renowned mathematician recalled how he was natural at the discipline from an early age, and how the competitive tests to enter secondary school in Hong Kong in the late 1960s were actually an excuse to further develop his interest and skills in mathematics.

Those were the days when Hong Kong was in transition from old mathematics to a new one. From classical Euclidean geometry as an axiomatic theory, etc., the curriculum shifted to more modern language involving logic, set theory, probability, matrices, linear algebra, among others, in more formal and abstract formulation.

Being a student during this transition, Mok was a pupil who was able to learn both classical and modern mathematics, and he says that is what made him realise that both forms were important. He credits the intensive training he got in his primary and secondary school from teachers, but adds that he worked on self-learning at an early age to give himself a strong mathematical foundation.

Resolving of Generalised Frankel Conjecture

In 1988 he resolved the Generalised Frankel Conjecture, which deals with the notions of curvature, symmetry, and homogeneity to characterise already known model spaces by their curvature properties.

Even though he tried to simplify it for a layperson, Prof Mok's explanation of his theory would go over the heads of most people: "For a certain class of closed manifolds (like a sphere for example) the curvature is equal to zero or more and this property known as ‘semi positivity’ implies much more about the manifold, and in this case that the manifold has a local symmetrical structure."

Prof Mok faced several challenges while trying to solve this problem. However, as a geometer, he didn’t only rely on geometry to solve the problem but also brought to bear different subject areas within mathematics. He proved that such manifolds were symmetric by following three steps.

Firstly, he took the theory of evolution of Riemannian metrics and used nonlinear partial differential equations. The second element was a known theory in algebraic geometry which says one can insert (non-trivial) spheres inside certain closed manifolds of “positive curvature” in some sense. Lastly he used another known theory in Riemannian geometry which was the characterisation of symmetry in terms of invariance under parallel transport.

“In this way I had to prove many things on the technical side and make statements. But we might need to use many different methods to solve problems, I call it intra-disciplinary,” explained Prof Mok.

He has also made significant contributions in the field of algebraic geometry. His most noted work is the Lazarsfeld problem which he solved by collaborating with fellow mathematician Jun-Muk Hwang. He has also developed the theory of spheres inside projective manifolds with Hwang and other collaborators since the mid-1990s.

Developing new theories

Prof Mok is back to working with spheres inside manifolds with collaborators to develop a theory which unifies various mathematical aspects which were previously treated separately. He is also trying to develop a theory of calculus without metrics, which will make the theory broader and intrinsic to algebraic geometry.

He wishes to use his perspective of geometry to study and answer the questions of functional transcendence theory. Currently he is also mentoring PhD students at the University of Hong Kong.

Asked about his future plans, Prof Mok is characteristically understated: “I’m still doing mathematics, but there are different subject areas in mathematics. I make use of them. My mathematics is driven by attempt to solve concrete problems.”

Prof Mok was Alfred Sloan Fellow in 1984 and was awarded the Presidential Young Investigator Award of the United States in 1985 for his contribution in the field. He was Croucher Senior Fellowship Award recipient in 1998. In 2007 he received State Prize for Science in China, followed by the Bergman Prize of the American Mathematical Society in 2009. In 2015 he was elected Academician of the Chinese Academy of Sciences. Prof Mok began his career as a mathematician in 1980 at Princeton University after graduating from Yale (M.A.) and Stanford (Ph.D.), specialising in complex differential geometry, several complex variables and algebraic geometry. He was also a professor at Columbia University and at the University of Paris-Sud (Orsay). He returned Hong Kong in 1994 and has been with the University of Hong Kong since then, where he was first a chaired professor. He is currently Edmund and Peggy Tse Professor in Mathematics. Since 1999, Prof Mok has been working as the director of the Institute of Mathematical Research at the University of Hong Kong which is promoting international collaboration in mathematics in Hong Kong with research centres in different parts of the world including China, France, Germany, Japan, Korea, Singapore and the United States.

To view Prof Mok's personal Croucher profile, please click here.