Assistant Professor of Physics

C.N. Yang Institute for Theoretical Physics

Stony Brook University

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]]>I met Professor Eugenio Calabi in the Fall of 1989, a couple of months after I arrived on the University of Pennsylvania campus. We hit it off quite nicely, as he was a zealous lecturer and loved to go to the blackboard, while I was an eager listener and loathed to go in front of people, for fear that they might find out that I was not as good as my transcripts might have indicated. In retrospect, he must have found out early on, in his infinite wisdom, that I didn’t understand much of what he was talking about, but that apparently didn’t dampen an iota of his enthusiasm for talking to me. In my recollection, we spent 4-5 hours each day “talking” mathematics to each other—be it in his office, in the mailroom, or in our tea room. This is extraordinary luck for any graduate student to be blessed with. I endured growing pains for a long period of time after my graduation and Professor Calabi was with me during the whole process. His wit, humor and focus on the fundamentals have helped me regain balance in shaky times, and keep cool in sunny times. I cherish his invaluable teachings and wish to share these with the community. What follows are a few dialogues we have had over the years which hopefully will help elucidate the lasting wisdom of Eugenio Calabi.

**Like any ambitious young person, I was
curious about how to be famous:**

**XC:** How many papers do I need to write in order to be famous?

**EC:** One.

**Eugenio Calabi stresses the importance of
being original and doing what you love: **

**XC:** It seems to be relatively easy to write papers on a trendy subject?

**EC:** Be original and follow your own heart and

intuition.

**XC:** How do I be original?

**EC:** Read classical papers which have withstood the test of time. Like a dog, smell the smell miles away before anyone else has noticed.

**I often complained about the hardship
of getting my papers published: **

**XC:** Will being in a famous university help publish my papers?

**EC:** Maybe. Do you want your address to become famous because of you, or you to become famous

because of your address?

**XC:** A mediocre paper gets published in a top journal, what should I do?

**EC:** Nothing. If your paper is of fundamental importance, people would find it even if it were published in the corner of earth; if your paper is of mediocre quality, you are better off to be published in an obscure journal so no one will notice it.

**When I was relatively young, I was shocked to find out that a friend had “cheated” on me. It was a tough pill to swallow, but Calabi steered me away from bitterness: **

**XC: **A friend has stolen my idea. What should I do?

**EC:** Congratulations, now your idea is worth stealing!

**XC:** ???

**EC:** Will you have new ideas?

**XC:** Yes.

**EC:** Will you allow him to steal again?

**XC:** No.

**EC:** Then you win since you continue to have new ideas and he cannot continue to steal from you.

**For many, including myself, going to ICM is an ego trip. Calabi helped put it into perspective: **

**XC:** I was invited to ICM 2002.

**EC:** Congratulations, it is important to you personally!

**XC:** Just personally?

**EC:** Yes. We will know if it is important to geometry in 10 years. Think about who you remember among ICM speakers in differential geometry in 1994 or earlier.

Ever since my student years, Eugenio Calabi has stressed to me over and over again that it is mathematical problems, not he, that is my teacher. I didn’t quite understand initially, but I have gradually gained an appreciation and become a faithful follower of his philosophy. Indeed, with limited talent myself, I was blessed with many extremely gifted students and hence opportunities to put his philosophy into practice. We find good problems together and learn mathematics from the problems we work on. As a side benefit, I have learned, though with some struggles, quite a bit from my students over the years. Calabi is correct that problems and gifted students are my teachers. This is the biggest secret of my moderately successful career and I wish to share it with future generations.

]]>Christopher H. Browne Distinguished Professor of Physics

University of Pennsylvania

Matter can arrange itself in the most ingenious ways. In addition to the solid, liquid and gas phases that are familiar in classical physics, electronic phases of matter with both useful and exotic properties are made possible by quantum mechanics. In the last century, the thorough understanding of the simplest quantum electronic phase—the electrical insulator—enabled the development of the semiconductor technology that is ubiquitous in today’s information age. In the present century, new “topological” electronic phases are being discovered that allow the seemingly impossible to occur: indivisible objects, like an electron or a quantum bit of information, can be split into two, allowing mysterious features of quantum mechanics to be harnessed for future technologies.

The building blocks of matter are largely known. Matter is composed of fundamental particles with electric charges that are precisely quantized in units of the indivisible fundamental charge *e*, and whose behavior is governed by the laws of quantum mechanics. In an atom, electrons with charge *-e* orbit the positively charged nucleus, similar to planets orbiting the sun. However, according to quantum mechanics, electrons can only exist in orbits with discretely quantized energy. This can make atoms electrically inert. Electrons are locked in place like legos: dislodging them requires a big enough kick to overcome the energy gap to the next discrete energy level.

This simple picture allows a cartoon-like understanding of the insulating phase. In a crystal of inert atoms, all of the electrons are stuck on their home atom as in Fig 1a, allowing no flow of electric charge. There is a sense in which the insulating state is the most boring state: nothing happens. But if you add an electron to an insulator, that added electron can move around and conduct electricity. Alternatively, you can *remove* an electron. The missing electron, known as a “hole,” can also move when another electron takes its place, as in Fig 1b. Holes behave just like fundamental particles with charge *+e*, but they are not among the original building blocks. They are *emergent* particles that are fundamental excitations of the insulating state. They are also *useful*: most of modern electronics technology is founded on our exquisite control over electrons and holes. A semiconductor, like silicon, is an insulator to which electrons and holes can easily be added.

Of course, real materials are more complicated than the above cartoon picture. One of the triumphs of 20th century physics was the quantum theory of solids, which provides a detailed understanding of materials like silicon. However, there are things that the cartoon gets right. There is still an energy gap in silicon, and the fundamental excitations are still charge *-e* electrons and charge *+e* holes. The sense in which the cartoon “gets it right” leads to the deep and beautiful idea of topology. In your mind’s eye, you can imagine smoothly transforming a silicon crystal into a trivial atomic insulator by, for example, slowly moving the atoms apart. If along the way it remains an insulator with a finite energy gap, then there is a sense in which it has stayed the same. Topology is the mathematical study of objects that can be continuously deformed. The classic example for topology is the sense in which a doughnut is the same as a coffee cup. If they were made out of clay, you can imagine continuously molding one into the other. The thing that stays the same is the number of holes: the hole in the doughnut turns into the handle of the coffee cup. But not everything is the same. You can’t mold a ball into a doughnut without poking a hole in it—and you can’t do that smoothly. Insulators that can be continuously deformed into one another are *topologically equivalent*. Thinking about it this way poses a very interesting question that was overlooked for decades in the quantum theory of solids. Are there “topological insulators” that can *not* be smoothly deformed into trivial atomic insulators? The remarkable answer is yes, and they have fascinating, and potentially useful properties.

The simplest version of a topological insulating phase occurs in a one-dimensional (1D) polymer called polyacetylene, which is an electrical insulator that consists of a chain of atoms with alternating strong and weak bonds. However, there are two possible configurations (“strong-weak-strong-weak” and “weak-strong-weak-strong”) which are topologically different in the above sense. The cartoon picture for this insulator is that there are twice as many spaces for the electrons than there are electrons, so in the A-phase (B-phase) the electrons occupy only the black (red) sites in Fig. 1c. Importantly, there are just enough electrons to exactly compensate the positive charge of the nuclei, so that with half the spaces filled the system is overall electrically neutral.

Now, something magic happens when you add an extra electron, say, to the A-phase. In Fig. 1c there is one added electron, so the net charge is *-e*. In Fig. 1d, by simply shifting electrons over one space, that added charge *-e* splits in half! The only places where there is any net charge is in the regions where the electrons are bunched together. Since there are two of them they each have charge precisely *-e/2*. These *-e/2* charges can move and are emergent particles in the same way that holes are.

How can this be? The electron is indivisible and can’t be split. But notice that between the *-e/2* charges the insulator is in the B phase. The impossible” *-e/2 *charges exist on the boundary between topologically distinct insulators. This is the essence of topological phases of matter. They allow the indivisible to be split by putting the “impossible” on the boundary. There are many examples of this general phenomenon, and the impossible” things that they allow are truly remarkable.

Our first example is motivated by a problem: the flow of electrons in conductors is disorganized. It’s like trying to navigate a crowded hallway. You are constantly bumping into people, which makes it hard to get where you are going. This problem gets worse when the conductors are smaller, so this poses a serious obstacle to the miniaturization of electronics technology. How can we make the flow more organized? To organize traffic, we build divided highways. Can we do the same for electrons?

There exists a topological phase, called the quantum Hall state, that accomplishes this. This is the “mother of all topological phases:” the one we understood first and the one we understand the best. It was discovered in the 1980’s, when experimentalists making electrical measurements on semiconductor devices found a striking quantization of the Hall resistance when electrons were confined to a 2D plane in the presence of a strong magnet. This prompted theoretical physicists to think deeply about how that could be and to introduce the notions of topology that now underlie the field.

The quantum Hall state has an energy gap in the 2D interior, so the electrons are locked in place like in an insulator. However, it is topologically distinct from a trivial insulator. It necessarily has a “one-way” electrical conductor on its 1D boundary. These one-way edge states are remarkable because an electron in them has no choice but to go forward. This makes the electrical conduction *perfect* and leads to an electrical Hall resistance that is precisely quantized in units of the fundamental constant *h/e²* (*h* is Planck’s constant). Experimentally, this quantization is measured to one part in a billion. It is so accurate, that these measurements now serve to *define* the standard unit of resistance called the Ohm, which is a key part of the international system of units used by physicists.

The one-way edge states are also remarkable because they are *impossible*. If they were to exist in isolation, deep principles of charge and energy conservation would be violated. An ordinary 1D conductor allows motion in both directions, with counterpropagating lanes that are inextricably tied to each other. The quantum Hall state allows those indivisible lanes to be split, as shown in Fig. 2a. Once you have a one-way edge state it is impossible to get rid of it. If you did, then the other edge would be an impossibly isolated one-way conductor. Thus, the edge states are *topologically protected*.

The one-way edge states could be useful for organizing the flow of charge in tiny electrical conductors. However, a difficulty is that using present technology a strong magnet is required to create them. There is another topological phase called a three-dimensional topological insulator (3D TI), which could provide a route towards that goal without the magnet.

A 3D TI is a material that is an insulator on its interior, but is a very special kind of electrical conductor (called “helical”) on its 2D surface. The helical surface conductor can be viewed as splitting an ordinary 2D conductor into two pieces: one on the top and one on the bottom (Fig. 2b). These helical conductors are impossible in isolation. They possess a special property called a “single Dirac cone” that is impossible in a purely 2D system. Once you have them, though, they are impossible to get rid of. Like the edge states in the quantum Hall effect they are topologically protected.

Unlike the quantum Hall effect, 3D topological insulators were conceived of by pure thought. Theorists realized that such a phenomenon was possible, and developed techniques to predict specific materials where it would occur. Experimentalists then created the materials and performed measurements that verified their properties. For example, a landmark experiment in 2008 observed the single Dirac cone on the surface of a crystal of Bi₂Se₃ using a technique called angle resolved photoemission spectroscopy. This helped initiate a vast subfield of condensed matter physics with strong interplay between theoretical physics, computational physics and experiment. Nowadays there are many more materials that are known to be 3D TI’s and related topological states.

One of the potential applications of topological insulators is to use them to organize the flow of electric charge in the tiny conductors. A second, more speculative but also more ambitious, proposal is to use topological insulators and related materials to construct a quantum computer. This problem motivates our third example of a topological phase.

Ordinary computers perform logic operations on bits: the 0’s and 1’s that can be combined to encode binary numbers. A quantum bit (or qubit), can be 0 or 1 or *both at the same time*. This combination of 0 and 1 is enabled by a mysterious feature of quantum mechanics called superposition. This allows a quantum computer—a device that performs operations on qubits—to be much more powerful than an ordinary computer. Unfortunately, a quantum computer is very hard to make because qubits are fragile. If you measure a qubit, you lose most of the information that it encodes. The difficulty is therefore making sure that the quantum computer doesn’t accidentally measure itself. This fact makes constructing a quantum computer one of the grand technological challenges for the coming century. One approach is to try to make the qubits very well isolated. There is another approach which takes advantage of a topological electronic phase.

The idea is to *split the qubit*. Qubits are ordinarily pointlike (0D) objects, like atoms or spins, that can exist in two distinct states. However, just as a

topological insulator can split an electrical conductor into two, there exists a related topological phase called a topological superconductor that can split a qubit into two pieces that reside at the two ends of a 1D material, as shown in Fig. 2c. The beauty of this is that the qubit is shared between the two ends and cannot be measured with any local measurement on one end. The quantum information is thus topologically protected, and immune from accidental measurement. One possible route to creating a topological superconductor is to combine an ordinary superconductor (which can easily be found) with a topological insulator, or a related topological material. There is promising experimental evidence that such topological superconductors can be created. Demonstrating that they have the capacity to store quantum information remains a challenge—but one which seems likely to be solved.

Knowing the fundamental building blocks of matter and the rules that govern them is only part of the story in physics. In 1687 Newton’s *Principia* laid out the fundamental rules of classical mechanics. It was much later that the concept of energy emerged as an organizing principle for understanding what matter that obeys Newton’s laws can do. Likewise, the rules of quantum mechanics were laid out in the early 20th century. Organizing principles, like topology, for understanding quantum matter are still emerging. This makes it an exciting time to be a physicist, because matter can arrange itself in the most ingenious ways! •

In July 2019, on behalf of the Simons Center for Geometry and Physics, Claude LeBrun, Stony Brook Mathematics Department, conducted an interview with Eugenio Calabi at the University of Pennsylvania Mathematics Department. Calabi is of course primarily famous for his pioneering work on Calabi-Yau manifolds, a class of solutions of Einstein’s equations that plays a central role in current research, both in mathematics (where they connect differential geometry and algebraic geometry in a remarkable new way) and in physics (where they are used to construct model universes via string compactification). This film is partly intended as a celebration of the recent award of the Veblen Prize to Simons Center and Stony Brook faculty members Simon Donaldson, Xiuxiong Chen, and Song Sun, for work answering questions first raised by Calabi. But, more specifically, it will also help us remember Calabi’s personal role in the story, both as Xiuxiong Chen’s thesis advisor, and as the mathematical grandfather of Chen’s student Song Sun.

**CL****: **To begin, Gene, before we get to mathematics, could you say a little bit about your childhood and how you came to the United States?

**EC:** I grew up in Milan, Italy. My father was a lawyer. And he realized early on that I was gifted in mathematics. I remember when I was in first or second grade trying to memorize the multiplication table. My father taught me what a prime number was, and he told me, “your job is to find the law—how prime numbers succeed one another.” And it was a standard question to me: “have you found the law of prime numbers?”

I went to public school in Milan, through what you might call tenth grade. I left when I was about 15, with my family, and we spent a year in France waiting for our American visas. This was 1938-1939, just before World War II broke out.

**CL****:** So you had already been through the horror of Mussolini’s racial laws and so forth?

**EC****:** Yes. It had just begun in 1938. My father had been getting alarmed and he started planning a possible exit from Italy around 1936, at the end of the Ethiopian war and the breakout of the Spanish Civil War, in which Italy and Hitler were allies. And when the racial laws came out he decided there and then, overnight, that he had to put the family safe. And, so, we eventually came here in the spring of 1939.

**CL****: **Well done. And how old were you when you got to the United States?

**EC****:** Sixteen. And I got admitted to MIT right away. It was a various “quirk” of scheduling—I had skipped a year both entering and another one exiting France. I majored in chemical engineering at MIT, but by the time I graduated I had decided to switch to mathematics. I applied for graduate school at both Harvard and Princeton in 1947. I was admitted to both, but at Princeton they offered temporary housing for graduate students so I went there.

**CL****: **(Laughs) Important decisions are often made on the flimsiest basis, right?

**EC****:** Oh yes.

**CL****: **But you do have to have a place to live.

**EC****:** I also spent two terms at the University of Illinois. That was my real start in pure mathematics. In 1947 for the winter and spring terms. Then that summer I moved to Princeton.

**CL****: **Did you start working with Salomon Bochner immediately?

**EC****:** About a year in—1948.

**CL****: **So Bochner was the person that knew something about differential geometry. In particular, he worked on Ricci curvature and he proved an important theorem about harmonic forms. He seemed to be very skeptical in that article about the Hodge** –**de Rham theorem. He talks about harmonic forms but he doesn’t talk about how they’re related to topology at all.

**EC****:** That I learned a little later. I was strictly studying differential geometry and the problem of selecting metrics—the manifolds of many possible metrics—and whether there are any problems about the function space of all metrics. Nonlinear analysis was still at its very infancy at the time. But it was the geometrical aspects that attracted me, and Kähler manifolds seemed to be the area to work in.

**CL****: **I guess Bochner also started working on Kähler geometry not very long after his paper on harmonic one forms. But he always refers to them as “so-called Kähler metrics.” He apparently knew that Kähler had been a Nazi and was very unhappy with the man over this.

**EC****:** Well, I hadn’t noticed that. I had met Kähler when I just finished my doctorate in 1950, at the first International Congress of Mathematicians. I spoke with him briefly. My thesis was on Kähler metrics—the embedding problem. And I invented the word ‘diastasis’ in my thesis: the distance function. The normalized potential for Kähler metrics that works in the analytic case only.

**CL****: **In 1954, you spoke at the next International Congress in Amsterdam, I believe? And that was when you first announced your entire program about representing Kähler classes on compact Kähler manifolds?

**EC****:** Yes. I also wrote the first paper on the so-called generalization—on the theorem on the hessians, on real numbers—complex functions. At my first job at Louisiana State University, I bought a book on affine differential geometry, by Blaschke. And I was fascinated by it.

**CL****: **So this work on real the Monge-Ampere equation—was this before or after you started thinking about the complex Monge-Ampere and Ricci flat Kähler metrics?

**EC****:** Simultaneous. It was a step toward it. I was motivated by the other one but I also became interested in Affine differential geometry on its own right. And in fact, my current interest as it is—or at least pretends to be, at my age I can’t do much anymore—is still affine geometry.

**CL****: **Still it’s quite amazing that you’re still doing mathematics. I saw you gave a very nice talk a couple of years ago. So how old are you now?

**EC****:** 96

**CL****: **96. And still at it. That’s wonderful.

**EC****:** Well, now it’s what we call research simulation.

**CL****: **(Laughs) Well doing mathematics is such a kick, how could you possibly want to give it up?

**EC****:** My favorite hobby.

**CL****: **We’re very lucky that someone actually wants to pay us to pursue our hobbies.

**EC****:** To follow your hobbies as a profession is the extraordinary luck I’ve had in my life.

**CL****: **Yes. So, we were talking about some things that happened in 1954. I think that was the year you also submitted your paper for the Lefschetz Festschrift Volume, and that’s the one that lays out the Calabi conjecture. At the time, I think you thought at first maybe you could prove it, but then by the end of the paper you’re expressing grave doubts about one aspect of the continuity method.

**EC****:** I remember hearing for the first time in my life, in that period, a priori estimates.

**CL****: **(Laughs) Yes, so you wrote me a letter once in which you claimed that you had sent your paper for the Lefschetz volume to André Weil…

**EC****:** Ah yes. And it was at a meeting in Trieste, Italy where I actually met Nirenberg and Bers. And I was talking to them about this problem and finally they told me you cannot solve partial differential equations without a priori estimates. And I said “what’s that?”

**CL****: **(Laughs) I think this is a great tale for any young mathematician to hear because we all start off—even very, very talented people—start off from a position of relative ignorance, right? And then you have to talk to the experts and find out.

**EC****:** The point is that you don’t learn during the course, you learn after, when mulling over the course. That has been my experience at least.

**CL****: **Your paper on the Calabi conjecture is an amazing paper. It’s very, very impressive. But apparently you sent this to André Weil right? You must have been thinking that he was the expert on K3 surfaces?

**EC****:** Well I asked him whether it was of any interest because I knew very little about algebraic geometry at the time. And he answered me very quickly and he said “how do you prove it?”

**CL****: **He was a very brilliant man but not known for generosity. He was a very smart guy who liked to show that he was the smartest guy in the room.

**EC****:** Oh yes. I was intimidated by him, actually.

**CL****: **So you were Bochner’s student. And you had just given a talk at the Lefschetz Festschrift. This makes me wonder… I’ve heard all of these stories about how Bochner basically would not talk to Lefschetz. I’ve been told in particular Bochner did not want to be in the same room as Lefschetz.

**EC****:** That’s right. When we had invited speakers, Lefschetz liked to get the students acquainted with the speakers, so he encouraged them to have a reception for the speakers at the graduate college. And the usual routine was that Lefschetz would come very early. And leave early. And then we’d call Bochner.

**CL****: **So it’s a mystery that they were both very major mathematicians.

**EC****:** They respected each other. I mean, at least I’ve heard Bochner quoting Lefschetz. The other way around, I did not follow Lefschetz too closely, but I think it happened.

**CL****: **There’s certainly many results proved by later mathematicians that use ideas from Lefschetz combined with ideas from Bochner. So, for the benefit of our audience, it might be good to review just a little bit about the statement of what’s now called the Calabi-Yau theorem. That if you have a compact Kähler manifold then if you specify any volume form, provided it has the right integral, that you can represent, in a given Kähler class, you could find a unique metric with that volume form. And then this has the consequence that if the first Chern class of the Kähler manifold is zero, you can prove that there is a unique Ricci flat Kähler metric in each Kähler class. So, in your paper in the Lefschetz volume you give a truly beautiful geometric proof of the fact that the solution, if it exists, is unique. And this is just a beautiful argument. Basically, a maximum principle argument.

**EC****:** Sure, it’s the Bochner principle.

**CL****: **I definitely think that your paper is still worth reading because the uniqueness is so beautifully explained there. And of course, as it turned out, in terms of proving existence, you had one of the key ideas of the continuity method. But in terms of actually showing that the set of parameters for which you have a solution of the perturbed equation is closed, you needed other ideas and that’s only solved by Yau in the 1970s.

**EC****:** That’s right. That’s another dramatic episode because when he first announced it, he had earlier thought he had the counterexample. And that was wrong. He recognized it but then he announced the proof shortly after, maybe less than a year. There was a great deal of excitement. We had to hear the details, so we had arranged a meeting as soon as we could. The meeting took place on Christmas day in Nirenberg’s office in New York. We met there in his office and heard the first proof. Which I did not understand.

**CL****: **So there’s a C^{0} estimate which is done by iterating L^{p} spaces. But there’s also sort of a key C^{3} estimate that’s based on your earlier work in affine geometry. Did he discuss that in the same lecture?

**EC****:** Yes he quotes it.

**CL****: **You told me once that that was the first paper where you proved something with a priori estimates, right?

**EC****:** That’s right. It was a celebration of the first understanding of analysis.

**CL****: **So, with respect to your work on extremal Kähler metrics…Well actually one odd historical thing that some people even in the field may not know… There have been a lot of papers on Kähler-Einstein metrics with nonzero scalar curvature. And they have often put in the title that this is a proof of Calabi’s conjecture. Whereas you didn’t actually explicitly say that, although it follows from your assertions about constant scalar curvature.

**EC****:** Oh yes. Well I realized only later that constant scalar curvature is obstructed in some manifolds.

**CL****: **When did you first realize that?

**EC****:** When the paper by Futaki appeared.

**CL****: **Oh I see. So, you’d had these concerns about holomorphic vector fields – It’s there in your first article about constant scalar curvature Kähler metrics. You assume from the beginning that you’re on a manifold which supports no holomorphic vector fields, but you don’t say why that’s an important fact.

**EC****:** Oh yes it was of interest to me as a variational problem. And then the first examples of extremal metrics in Ricci curvature is not constant. I even wrote a short paper announcing it.

**CL****: **It’s a fairly long paper that’s in Yau’s conference volume. And there was a seminar on differential geometry and it’s a very beautiful article where you discuss the general variational problem, show what the Lagrange equations are, and then produce solutions on blow-ups. For example, CP2 blown up at a point supports solutions in every Kähler class, which have non-constant scalar curvature. When did you actually find those solutions? Was that just in the process of writing that paper?

**EC****:** Yes. In the process, in the early 1980s.

**CL****: **So those are now usually called extremal Kähler metrics, although it was not shown that they were minima for a long time. You had given a beautiful proof in the constant scalar curvature case that actually if you have a solution it is the minimum of the functional. It was, I think, your student Xiuxiong Chen that first proved that they were always minimizers in the general case.

**CL****: **I’m just thinking back. You really made great strides in Kähler geometry as a branch of Riemannian geometry in the ’50s, ’60s, ’70s and ’80s. But back in the 1950s how well did people really even understand what a Kähler metric was?

**EC****:** Well I understood it from the Bochner lectures. We asked him to give a course in differential geometry for the first time in my second year of graduate school, 1948-1949. And he defined it there.

**CL****: **In particular it turns out that Kähler geometry is an example of Riemannian geometry of special holonomy. And that fact seems to have just been ignored by many people on the kind of algebro-geometric end of things.

**EC****:** Well yes, it was really a remarkable discovery but it took years to focus from there—from its first announcement by Kähler.

**CL****: **My colleagues Xiuxiong Chen and Simon Donaldson were kind enough to provide some questions that I should ask. One interesting question that Simon raises is that in some of your early papers you were interested in non-Kähler complex manifolds, in particular with Eckmann. You found a beautiful set of examples of complex structures on products of spheres which you know are on manifolds that certainly can’t admit Kähler metrics because their second cohomology is trivial. Have you followed these areas of non-Kähler complex geometry at all?

**EC****:** No, just those questions that are on my mind, if there are any more examples other than the usual. Perhaps one being, after Milnor’s discoveries, whether you could also have twisted spheres?

**CL****: **Well questions about complex structures on the sixth sphere are one of those annoying things that just won’t go away. It’s unfortunately one of those problems that is basically still quite out of range of any technology we have, and may always be.

**EC****:** Yes, I’ve been bothered by that. But the obstruction to complex structures is basically unknown. Unless you have an almost complex structure. Obstruction to integrability.

**CL****: **Another very interesting general question that Simon Donaldson proposed was, “when did you see analysis starting to have a major impact in differential geometry?”

**EC****:** From the correspondence I had with André Weil.

**CL****: **That’s when you realized that it was an essential thing?

**EC****:** There was a problem. It was fundamentally analysis. I think that mathematicians very often what they do at a later age was originally in their mind much earlier.

**CL****: **Maybe with 20/20 hindsight though sometimes you know your ideas are so confused when you’re young.

**EC****:** Well I know I’m certainly confused when learning things. Learning is a confusing process and the real learning takes place after you’ve sort of digested the information. Learning is a digestive process, in a creative way.

**CL****: **So in your own case—we mentioned before this article on extremal Kähler metrics that came out in the 1980s. In a way, you’re writing about things that you had started to think about years before but actually being forced to sit down and write out the details you’ve discovered new things.

**EC****:** That’s right. Well that’s the digestive aspect.

**CL****: **So, in the course of your career you’ve seen these enormous changes in the way that mathematics is communicated. Looking back it seems like the number of mathematicians, particularly mathematicians that did something that would be relevant to your own area, would have been relatively small in fact, where you would have probably known almost everybody.

**EC****:** I used to.

**CL****: **So I mean mathematics was usually disseminated by mail and telephone. And in particular, anything that was in the direction of a collaboration, or dependent upon, had to be done in those ways. But also, you could only exchange ideas with people if you already knew them.

**EC****:** That’s why we went to meetings. Meetings were useful. I think they still are.

**CL****: **Some of your early work was on complex manifolds and you were asking questions that were alien to algebraic geometers. Were there ever any meetings where you would meet people that were interested in these topics?

**EC****:** Very few. The only one I knew was Bochner and his other students.

**CL****: **So when you collaborated with Eckmann. How did you meet Eckmann?

**EC****:** Heinz Hopf was visiting Princeton, and I told him about the construction. And Bochner balled me out for doing so. But Hopf told me that Eckmann had just done the same thing. And so, he put us in touch. I met him later, shortly after. We decided to write jointly by correspondence.

**CL****: **When you look back, was there a certain period of time when you really thought that what you were doing was particularly fun and exciting?

**EC****:** No. I was just finding my way.

**CL****: **I’m just wondering though, you’re now you at a point in your career when your name is famous. People probably don’t realize Calabi is a person. Calabi-Yau is a phenomenon. You were telling me earlier that there was a dance performance called Calabi-Yau in New York. And so, it’s kind of entered in popular culture. Do you ever have the experience that people want to come up and ask you about physics or something like that, or string theory?

**EC****:** No. I’ve heard about them, of course. But no, my favorite slogan to explain mathematics to the layman is “’it’s quintessentially science fiction.” I never quite understood the implications. But it was a piece of luck—unexpected.

**CL****: **Well but on the other hand sometimes when you just do what comes naturally in mathematics it pays off, right?

**EC****:** Yes. That’s my luck.

**CL****: **Well, Gene it’s been wonderful interviewing you, I’m so glad that we could do this.

**EC****:** Well, as I mentioned earlier it’s an ego trip for me.

**CL****: **It’s a well-deserved one. I think that many people will find this interview interesting and I’m glad we were able to do it. Thanks a lot