When you looked at the world through the lens of geometry and topology, what did you learn?
That nonlinear equations were fundamental because in nature, curves abound. Climate isn’t linear. If the wind blows stronger that way, it may cause more trouble over there; it may even depend on the geometry of the earth. Usually you see the stock market described by linear equations and straight lines, but that is not really correct. The stock market fluctuates up and down in a nonlinear way. The Einstein equation described the curvature of the universe, and it was nonlinear. I ended up learning nonlinear equations from a master, although I didn’t know he was a master at the time. His name was Charles Morrey, and he was a classical gentleman. He always dressed in suits in class. He was a very nice man. Even if I was the only one there, he would lecture to me, just as if he were lecturing to the whole class.

Wait—you were sometimes the only one in his class?
Why should people care about ancient days? Morrey didn’t use modern notations. His book was hard to read. Kent State had just happened. The students and the faculty were all on strike, but Morrey still gave lectures. Soon everyone had dropped the class but me.

What happened next in your mathematical explorations?
It was Christmas break and I couldn’t go home, so I spent my time at the library reading all the journals and looking at rare books. That was the first time I met my wife, although only much later were we formally introduced. Through all this reading about topology I came across a theorem that talked about loops where curvature is everywhere negative—where the curve goes in like a saddle. The theorem says that when we have two such loops with vertices at the same point, they can’t deform into each other by bending or twisting unless they are equal to or multiples of each other. I came up with a related theorem: If the curvature is either negative or zero and if the loops conform, then there must be a lower-dimensional surface—specifically a torus—sitting somewhere inside.

How can a lower dimension sit inside a higher one?
Imagine attaching a rubber band to the handle of a coffee cup. The cup has three dimensions, but the rubber band, which is just a curved line, effectively has only one.

Why should anyone other than a mathematician care about a torus or a string hidden within higher dimensions?
Because topology can affect and constrain geometry in the physical world. If water flows around a sphere, for example, there must be two points where the water is totally still. On a planet covered with an ocean, the water can’t all flow in the same direction, say east to west, everywhere, without hitting a snag. In the case of another topology, the torus, water can flow around and around and there’s no point at which the flow stops because the hole eliminates the impasse. For each fixed topology, the geometry follows different laws.

In other words, you realized that topology sets the basic rules for geometry, which in turn affects the world around us. But then you went further, asking whether the underlying structure of space might explain the laws of physics. How so?
I started to look into complex manifolds. A manifold is just a space, with each point immediately around you looking like Euclidean space—the familiar kind of space that we see around us. Imagine the earth is covered with a checkerboard or a grid, like latitude and longitude. This is the kind of coordinate system that Descartes introduced to geometry in the 17th century. At each point on the grid the space appears flat and finite, but it’s actually curved, a sphere. Instead of being measured with real numbers, though, we measure complex manifolds with complex numbers, in which one of the coordinates includes a real number multiplied by the square root of negative 1—an imaginary number that we call i. [Since the product of two negative numbers is positive, ordinary math suggests the square root of negative 1 cannot exist—hence the moniker “imaginary.”]

How can complex manifolds and complex numbers help us understand the structure of space?
Space is not necessarily something you see in day-to-day life. You can define geometry locally, but globally you cannot visualize the big picture, you can only imagine it and represent it through coordinates. We draw lines of latitude and longitude in a coordinate system for the continents. But that system doesn’t work well at the north or south poles, where all the lines converge. In order to get a more complete picture in those regions, we need another, more localized coordinate system for more detail. In the end, we need several such coordinate systems patched together to get a detailed picture of the entire globe.

More generally, in describing any space, we are not restricted to the three dimensions we experience in our lives. Mathematically, we can suggest any number of dimensions: two, three, four, five, ten, just by drawing additional coordinates on a grid. In complex space, every number in a coordinate system describes not just one dimension but two. Most important, complex numbers make it simpler to move from one coordinate system to another, a necessary step when working in the higher dimensions necessary for string theory.

You are best known for your work on the Calabi conjecture, which at the time was a major unsolved problem in higher-dimensional mathematics. What attracted you to it?
I was drawn to important problems that gave insight into geometry and space-time. Sometimes solving a problem creates a new kind of thinking, sometimes the math itself is beautiful. The problem I went with, the Calabi conjecture, is a very elegant statement about the curvature of complex manifolds.

What does “curvature” mean in this context, since you aren’t talking about the kind of curves we normally experience?
Curvature is second-order information—for instance, suppose I am driving a car around a curved freeway. The car’s velocity will change as you go, so you can measure the curve in terms of changes in velocity along that one-dimensional line. Then there is Gauss curvature, which gives you the curvature of a two-dimensional surface by multiplying the largest and smallest curvature of the family of all curves tangential to the surface at a given point. For higher-dimensional space, such as the three-dimensional space around us, we calculate the curvature of all the two-dimensional surfaces passing through the point where curvature is in question. Finally there is Ricci curvature, which we measure by averaging the curvature of all two-dimensional surfaces tangential to each other along a common direction. In essence, Ricci curvature is an average of part of the total curvature of a space. It is an abstract geometric concept, but it is fundamental.

Why is Ricci curvature fundamental?
In physics, Ricci curvature is analogous to matter. Space with zero Ricci curvature is space without matter—a vacuum.

And how does all this relate to the Calabi conjecture?
Calabi said that certain topological conditions call for the existence of nonflat, closed, complex spaces without Ricci curvature anywhere. Such spaces would enjoy many beautiful properties. You might find the sub-dimensional loops or the torus I described in that very first paper I wrote—or you might find intersecting branes [short for “membrane,” another topological shape]. I was 100 percent sure that the spaces Calabi called for could not exist. No mathematician or physicist had ever found an example of one, and most geometers considered them too good to be true.

So what did you do next?
I spent a lot of time thinking about how to disprove Calabi. By 1973 I was teaching at SUNY–Stony Brook and planning to move to Stanford. That May I put my belongings into this little Volkswagen and drove across the country on Highway 80. I thought America was a country where everyone traveled around, but to my amazement, a lot of the people I met told me they had never driven more than 10 miles from their town. I crossed the Rocky Mountains. The car broke down at one point. By the time I was at Stanford for a few months, I thought I had finally proven Calabi wrong.

Disproving the Calabi conjecture would have been a major achievement; how did you announce it?
In August there was a big conference at Stanford with the top geometers in the world, including Calabi. I talked to Calabi and told him my idea. He said, “That sounds great. Why don’t you give a discussion about it to me?” It was scheduled for 7 p.m. Calabi brought a few colleagues from the University of Pennsylvania, and then a few others heard about it, and a few others. There was a little crowd. I talked for about an hour, and Calabi was excited. “I’ve been waiting for this for a long time, and I hope it’s right,” he said. All the other people said, “Great, finally we can stop the wishful thinking that Calabi is true.” Then Calabi wrote to me in October. He said, “I’m trying to reconstruct your argument, and I’m having some difficulty. Could you explain the detail?” I started to reconstruct it and I found a problem too. I got totally embarrassed. I did not respond to Calabi at that moment and instead tried extremely hard to patch up the proof. I couldn’t, so I looked around to find other examples where Calabi was wrong. I didn’t sleep for two weeks. But every time I found an example that was close, the proof fell apart at the last minute. Finally I said, gee, this cannot be such a delicate matter. Now I had much deeper insight into the issue and felt there must be some truth to the whole thing. I determined that it had to be right.

So after all that work trying to prove that Calabi’s conjecture was wrong, you decided it was correct after all?
I began developing the tools to understand it, and by 1975, only one part of the proof was left. That year my wife got a job in Los Angeles. I moved to UCLA. All in a short time, we got married, bought a car, bought a house in the Valley, and had to look for furniture. My mother came from Hong Kong for the wedding, and then her parents came—they all stayed under one roof and got into fights; it was complicated and crazy. I was fed up, so I locked myself in the study and thought about Calabi instead of the family problems, and I solved the whole thing. I went over the proof three times in detail, and I went to see Calabi in Pennsylvania. On a snowy Christmas Day, he came with me to visit mathematician Louis Nirenberg at New York University. We spent all day Christmas going over it, and I spent the next month writing up the proof for publication.

The implications were enormous. You were instantly famous.
It solved some major problems in algebraic geometry—about a dozen of them. A lot of people offered me jobs.

Some of the higher-dimensional spaces now called Calabi-Yau spaces proved fundamental to string theory. What is the connection?
When Einstein published his general theory of relativity in 1915, there was an immediate urge to unify the force of gravity with the other forces known at the time, with electricity and magnetism. Mathematicians thought they could do this with five dimensions, four of space and one of time. But then physicists found new particles and needed extra dimensions for the strong force and the weak force. When they worked it all out, they determined they could explain the universe with something they called string theory, which replaces the pointlike particles in particle physics with tiny, elongated vibrating strings. To be consistent with quantum theory, the strings needed 10 dimensions in which to vibrate: three of space, one of time, and six dimensions of “compact space.” Dimensions in compact space are so small you can’t detect them through any conceivable experiment. They amount to pure structure. It so happens that Calabi-Yau spaces with six dimensions also have specific topological traits corresponding to the requirements of string theory. If these spaces truly modeled the six-dimensional space called for in string theory, they would help us deduce the geometry and, by extension, the physical laws of the universe.

Some cosmology theories imply the existence of other universes. Could each Calabi-Yau space describe a different set of laws in each of those universes?
Yes, each isolated universe can be modeled by a different Calabi-Yau space. But some of my colleagues have also studied a beautiful concept called mirror symmetry, in which each space has a mirror image with the same quantum field theory and the same physics.

How many Calabi-Yau spaces are there?
Using a computer program, Philip Candelas at the University of Texas at Austin found up to 10,000 Calabi-Yau spaces, with almost half of them mirror partners of each other. Each member of a pair is topologically distinct but still conforms to the other algebraically and gives rise to the same forces, the same particles, the same rules. The resulting geometric structure can be used to determine physical quantities associated with each space, like particle mass.

String theory is often described as a mathematically elegant way to explain all of physics. But how can we know that it describes the real universe?
We cannot know for sure, but the mathematics inspired by string theory solves some old, longtime questions. That part is rigorous and its truth cannot be challenged. If the structure of the math is deep, it will solve something in nature one way or another; it is difficult to imagine that such deep structure corresponds to nothing. Everything fundamental in math has ultimately had a meaning in the physical world. If these spaces modeled the six-dimensional space called for in string theory, they would help us deduce the geometry and the physical laws of the universe.

You’ve long promoted mathematics in China. How have academic conditions changed there over the years?
I first went back in 1979, right after China opened up to outsiders. People were poor. Times were difficult. It was bedlam. I saw lots of people who were uneducated, and I felt I needed to help. By 1985 I’d taught about 15 Chinese grad students accepted to programs in the States. At first it was my adviser and mentor, Shiing-Shen Chern, who went to China and founded a mathematical institute there. I didn’t want to interfere with his work, but he was getting old and I started to go visit more often. In 1994 I was asked to give a speech. I said it’s great that China has an open policy; now we must start moving forward step by step, training young people to establish an intellectual base.

Eventually you founded four math institutes in China. How did that happen?
I met with Jiang Zemin, the future president, who wanted me to help build up math in China. After that, with the help of a donor, I built the Institute of Mathematical Sciences at the Chinese University of Hong Kong followed by three more institutes on the mainland—but China has always been run in a collaborative way, and other universities began demanding part of the funds. Still, the institutes have been able to carve out some independence, and today I go to China five or six times a year.

In the past decade, you’ve been critical of science and math in China. Why?
The university system is beset by academic politics, and it’s difficult for young people to move ahead. When China opened up, the people running things were in their fifties and sixties. The same people are still running things. Most do not follow modern developments because of their age. There are some brilliant young people, but it is a struggle for them to be recognized. Often that happens in China only after they are recognized by the outside world. I said, “Give some freedom to the young guys,” and people got upset.

You’ve commented that at the highest levels of accomplishment, Chinese mathematicians have far to go, and that the best of them have left the country. What are the prospects for math and science in China today?
The economy has been getting better and the government wants to invest more in science, so in the long run, I think the future is bright. Many more Chinese graduate students who come to study in the United States will be willing to return to China.

How does China’s relationship with the United States come into play?
I see a constructive relationship for academia. The U.S. gets human resources in the form of bright, young Chinese kids. The students learn well here, because the U.S. provides them with the freedom to research in their own way, and some of them will bring their knowledge back to China. But my goal is to train many more young mathematicians within China by providing an environment that allows them to focus on research and be recognized for their work.

You have criticized the academic system in the United States as well.
Young people are under too much pressure here. As a result, some of the proofs they publish are factually wrong. Before I published my proof of the Calabi conjecture, I checked it three times. Many young mathematicians don’t do that.

Most people don’t realize how political math can be: In 2006 The New Yorker accused you of taking credit from the Russian mathematician Grigory Perelman after he proved the famed Poincaré conjecture. What happened there?
In a process as intricate and daunting as proving the Poincaré conjecture, it is understandable that Perelman released his manuscript with several key steps merely sketched or outlined. One of my students tried to fill in some of the details, and I supported that. I also said that my friend Richard Hamilton, a geometer at Columbia University, laid much of the groundwork that Perelman ultimately relied on to construct his proof. For these things The New Yorker tried to accuse me of stealing credit, but that is ridiculous. What I think of as the Hamilton-Perelman proof of the Poincaré conjecture is a great triumph for mathematics, and I fully support the award of the Fields Medal to Perelman. Hamilton deserved the Fields Medal too, but he was ineligible because of the age restriction [you must be under 40]. To suggest that my position has ever been any different is completely untrue.

Physicists often talk about the beauty of math. What does that mean to you?
The first time I saw my wife, I thought she was charming—more than charming, shocking to me. I had great motivation to know her more. When I look at the Calabi conjecture, it shocks me too. It’s an elegant, simple construct and explains a great deal. It’s exciting when you go deeper and deeper into a complicated structure that you can spend most of a lifetime working on. It was shocking when it showed up in physics, and it’s beautiful whether it’s true or not.