After 29 years, Prof. Xujia Wang has returned to Hangzhou, the city of his birth.
Before the age of 6, Wang lived in Chun'an, which now lies under the waters of Qiandao Lake. Following the damming of the Xin'an River, he moved with his parents and fellow villagers from Chun'an to Jinxi, in Jiangxi province. His hometown was a tiny village with no more than 10 families – so small that it could be easily packed up and moved somewhere else.
For the past 29 years, Wang worked at the Australian National University in Canberra, researching partial differential equations. During his time there, he won the Morningside Medal of Mathematics, was elected a fellow of the Australian Academy of Science, and earned the prestigious Australian Laureate Fellowship. Then, in September this year, Wang joined Westlake University full time as a chair professor of mathematics.
It is very difficult for those outside the field of mathematics to understand the work that a mathematician does. Maybe it is because math looks very complicated. But after experiencing math's myriad complexities, Wang said, "I just like simplicity."
A Village Childhood
When Wang was 6, he and his family moved from the mountains of Chun'an to the flat plains of Jinxi. For Wang, it felt as if the world had suddenly opened up.
They were migrants from the Xin'an River project, and the whole village – comprising only a few dozen people – moved together. For the first year, they lived scattered among local families as they waited for new houses to be built. The migrant children attended school together, but the venue for classes was usually not fixed. It could be a warehouse, cowshed or the edge of the field. The "teachers" were just bigger children – teenage girls who had graduated from middle school.
Wang said: "Growing up in a rural village that could not be any smaller and where there were only a few dozen people, I had no exposure to the outside world and didn't understand how things worked. My parents were illiterate farmers, and life was very simple – free from complicated interpersonal dynamics."
That makeshift learning environment continued until middle school. Wang attended Hengyuan Middle School, another temporary school, which was established to give rural children more educational opportunities. At the village high school, far from the city and town, Wang demonstrated exceptional mathematical talent, becoming one of the best math students in the county.
With his outstanding grades, Wang had the opportunity to transfer to a high school in the city. But after much persuasion from his teacher, Wang eventually stayed at Hengyuan Middle School, believing that it didn't matter which school he studied at.
After the college entrance exams, Wang did not know which university to choose. A fellow villager – 20 years his senior – advised him to study mathematics at Zhejiang University in Hangzhou. Wang remains thankful to him to this day.
His upbringing gave Wang a straightforward approach to life, and his life choices and trajectory remained uncomplicated.
But going to university really opened up Wang's world view. Before that, he barely had access to any books that were not textbooks. After completing his undergraduate degree, he continued to pursue his graduate and doctoral studies, and also worked at the university, spending a total of 15 years there.
In 1994, he went to Austria as a visiting scholar, where he attended several concerts. During one, a Chinese soprano performed an a cappella piece, and the sounds were so powerful that it felt like the entire building was vibrating. Wang was left deeply moved by the experience, because sound waves encapsulated the principles of his research – partial differential equations.
He moved to Australia after his visiting scholar stint in Europe.
"I've lived a simple and natural life since my childhood and never had grand ambitions," he said. "In the path of life, every step that I've taken, I've just let nature take its course. Research-wise, I've also let my feelings and interests guide me into choosing which topics to pursue. After publishing several papers and gaining the recognition of my peers, it is what it is."
As a young man, Wang often pondered what makes good mathematics. Later, his respected mentor and collaborator, Prof. Shing Tung Yau, from the Chinese University of Hong Kong, told him, "Simple mathematics is good mathematics."
Wang wholeheartedly agrees.
"The fewer the conditions, the cleaner the results. If a result can't be described in simple terms, then it isn't good mathematics," he said.
While the world may appear simple at first glance, unfolding it reveals an intricate complexity, much like the mathematics Wang studies.
A Sheet of White Paper
At Westlake's School of Science, there's a massive white marble-like wall resembling a glacier. When I walked past it on Sept. 30, there was a sheet of white A4 paper taped to the wall. It read: Entrance to "New developments in fully nonlinear elliptic partial differential equations".
Half a month later, at our first meeting, Wang wore a red-and-black cap embroidered with the words "Great Wall", giving off a nostalgic 1990s vibe. "My forehead has been sensitive to sunlight lately," he explained softly, his accent still carrying traces of his rural upbringing.
He unfolded a curled-up piece of A4 paper that he had brought with him. Wang told me that he had done some preparation for the interview, and had the main ideas written down. The subject was "simple and natural", and the first sentence was "I just like simplicity".
I had also brought along a piece of A4 paper – a printout of the college entrance exam math questions from 1979, five questions in total. When Wang unfolded the paper, he laughed, saying that he had completely forgotten them. He then folded the paper and stuffed it back into the envelope.
"On that piece of paper, there was one problem that you got wrong at the time," I said, referencing a reply in an interview he had done more than a decade earlier.
With a frown, Wang immediately reopened the envelope and quickly scanned the sheet. His sensitivity to mistakes was apparent, and his serious demeanor as a mathematician emerged instantly.
Wang said he likes simplicity, but in reality, partial differential equations are really difficult.
When Amazon founder Jeff Bezos was doing his undergraduate studies at Princeton, he decided to transfer from the physics department simply because partial differential equations were too difficult. One particular problem tormented him for three hours. Bezos is highly intelligent, but that was when he realized he was not cut out for such study.
Equations, differential equations, partial differential equations, elliptic partial differential equations, nonlinear elliptic partial differential equations, and fully nonlinear elliptic partial differential equations. Mathematics is like a black hole that can suck you in.
Wang described partial differential equations using a relatively simple example.
When you throw an apple into the air, it follows a parabolic trajectory. This motion is complicated: the trajectory changes, and so does its speed. But, regarding the rate of change in terms of speed, what doesn't change is gravitational acceleration. Each second, the downward, vertical acceleration due to gravity will be 9.8 meters per second.
Regardless of how the apple is thrown, that mathematical description remains universally applicable on Earth. Conversely, if you know the gravitational acceleration and the initial speed, you can calculate the apple's trajectory, or its position as a function of time. That is the basic idea of differential equations, which is similar to a type of inverse operation.
In other words, when describing changes in things, you can consider using differential equations. In the case of the apple, time is the only variable. That is an ordinary differential equation. But in reality, there is never just one variable – there are several variables that influence each other.
So, welcome to the world of partial differential equations. Maxwell's equations, the cornerstone of electromagnetism, are partial differential equations. The Schrödinger equation in quantum mechanics is also a partial differential equation. Research into fluid, heat conduction, light waves and sound waves all rely on the use of partial differential equations.
"Most partial differential equations are incredibly difficult to solve, if they can be solved at all," Wang said. "Fortunately, we only do research on a very small portion of them. The problems that we conduct research on often have real-world applications."
Nature at its core is simple, and the underlying logic actually makes sense, Wang said. It is like Newton's three laws of motion, or the axioms of elementary mathematics. But often, simple logic can become very complicated after deduction. When different disciplines are applied together, the results can become so complex they are almost incomprehensible.
"Research is like a process of clearing the clouds to see the sun," he said.
An Equation
Wang reaches into a drawer and pulls out a lens. He holds it in one hand, while switching on the flashlight function on his mobile phone with the other hand, trying to find a focal point on the white wall.
"Can you see it?" he asked.
"It's a Gaussian curve,"I replied after a few seconds, finally recognizing it.
The lens itself didn't have a pre-set Gaussian shape – it simply appeared to have an uneven curvature. By only changing the curvature of the lens, it further controls the refraction and convergence of the light, letting the Gaussian figure emerge.
The lens was produced using Wang's theoretical work. The same principle can be applied in astronomical observation. By optimizing the curvature of the reflective lens, it helps the starry sky to better converge to one point. Radar detection can also be based on this principle of optimization, because radar is also a type of wave.
Underlying the design of the lens, and a focus of Wang's research career, is a partial differential equation known as the Monge-Ampere equation.
"It looks very simple, doesn't it?" Wang asked. But behind that simplicity lies immense complexity. The left-hand side of the formula represents a Hessian matrix, a square matrix consisting of all second-order partial derivatives of the unknown function u. The right-hand side of the formula is a given function, which is dependent on the x position, the function u itself and the gradient Du.
Confused? Wang said spare yourself the self-blame. "If we invited Monge and Ampère here, they probably wouldn't understand the modern Monge-Ampère equation," he said.
And that Gaussian projection on the wall? That was an application of the Monge-Ampère equation in geometric optics. The concise equation, like the projection, has appeared in many areas. In differential geometry, the Monge-Ampère equation is used in researching the shape and properties of convex surfaces. In the discipline of fluid mechanics, it is used to describe certain types of fluid flow problems. In economics, the equation is used to solve optimal transmission problems, similar to transportation problems, to minimize transmission costs.
With the rise of artificial intelligence, the Monge-Ampère equation has appeared, once again, in the field of deep learning. So, the work of mathematicians can seem abstract, but it is actually closely related to our world.
Wang once proved the regularity of free boundaries in the Monge-Ampère equation. He gave the simple example of a block of ice in the water. The boundaries of the ice are dynamically changing, depending on the temperature distribution of the water and ice. Meanwhile, the boundaries of the ice will also affect temperature, so the change in temperature must be solved. And the so-called regularity is needed to prove this type of boundary change, which is smooth and continuous in mathematical terms.
From Complexity to Simplicity
Mathematicians can work anywhere, Wang said. If you see a mathematician sitting somewhere drinking coffee, he is very likely to be pondering a problem. "I had a peer who died in a car accident. When I heard the bad news, I knew he must have been thinking about a problem while driving," Wang said.
After that, Wang forced himself not to think of math problems while driving. Now, he has also forced himself not to think of math problems at bedtime. On that piece of A4 paper that he brought with him, he wrote:
"But there will also often be times when I meet with difficulties that I am unable to overcome. When I wake up in the middle of the night, the math problem again enters my brain. This kind of torture may carry on for many months, and you cannot just give it up. This has happened countless times in my life."
He added that the math pondered late at night is basically the "wrong" math.
"Because, for one problem, you need to carefully think about it for half an hour. It basically becomes clearer if you think about it for an hour or more," he said.
"Doing research requires focus. Focus on doing one thing well. If a person can do one or two things well in their lifetime, that's enough."
And what allows him to maintain such focus for so long? That is largely due to pure interest. Facing mathematical research, Wang said he and many other mathematicians are the same. First, they assess whether the problem is of any interest to them, and not if it is of any use. What is interesting is that the hardest math problems can often be described in deceptively simple terms, even though many have baffled humanity for centuries. The ultimate complexity is often rooted in pure simplicity.
Perhaps, when we have not experienced taking the mathematical route, we can never fully understand the complexities of a mathematician's brain. The simplicity that Wang talks about might be the relief he feels after experiencing countless complexities.
