Two Essays by a Mathematician, Written Twenty Years Apart

Jaigyoung Choe, Korea Institute for Advanced Study

1. Waiting for a ship

"Has your ship come in?" A friend of mine used to ask me this question when he saw me in the mathematics building. This was while I was working hard on my thesis problem, and his question seemed to be half greeting and half inquiry as to whether I had succeeded; whether I had found the key idea of the problem. Seeing me walking around the department with unfocused eyes and pale complexion, he might have known what my answer would be even before he asked me the question!

This friend knew that I had recently started my mathematical adventures at graduate school. I had loaded my ship with a large cargo of hard work and sent it out into the vast ocean of uncertainty toward the island of the truth, hoping for its successful return.

One day, to my surprise, it looked as if my ship was coming back. I drove hurriedly for an hour to the nearby university where my thesis advisor was visiting. He listened quietly to my explanation of my new idea for thirty minutes. After I had finished, he started to speak to me slowly. "I do not want to discourage you, but two mathematicians called me last week saying that they had solved the problem with a better idea."

Then, I had to send another ship out to sea. My advisor kept moving from place to place, and my family and I were busy packing our bags and following him. Even during this time of frequent moves, I spent most of the time searching for my ship, wondering from where it would come.

It was in the place where the Vikings had lived long ago that my long- awaited ship came slowly sailing toward me. At the time I was visiting that faraway place with my advisor, his students, and their families. This was the institute in a suburb of Stockholm where for almost 100 years many mathematicians had come to visit and try to solve their problems. My ship was approaching me little by little while, together with my colleagues, I was studying and discussing in the antique office and library, drinking wine and coffee in the yellow villa, cooking and eating in the kitchen, and walking around near the institute.

Like everyone else, I have read many essays which express regret about the brevity of life. Among these, I can still remember one in particular. It said that even if you saw and met a large number of people in your life, those with whom you had true friendship would be only a few among them. I also would like to comment on the shortness of our life in my own way. Why is a mathematician's life short? It is because a mathematician knows a large number of theorems from books and papers, but his own theorems are only a few.

After the ship of my thesis came in, I continued waiting for other ships. Overall, one ship has come in and been unloaded in my office every year. There have been a great many times when I fretted that I had seen no trace of a ship for a long time. There were as many times when a ship drew near at first but then disappeared over the horizon, leaving me dejected. There were even times when the ship, coming toward me, got caught in a stormy wind and was wrecked near my port.

However, with the passage of time, my walk to the little port as I wait for a ship of deep theorems has become more lighthearted. Whether a ship comes in or not, and whether the ship coming in is large or small, all I have to do every day is to enlarge my little port, to gaze at the horizon calmly, and to lead my seaward life pleasantly, dreaming of beautiful ships of truth.

2. The problems of Queen Dido

Tyre is an ancient Phoenician city in today¡¯s Lebanon. A king of Tyre had a daughter, Dido, and a son, Pygmalion in the 9th century BC. Dido married her uncle. Even though the king named Dido and Pygmalion his joint heirs, upon his death the people of Tyre chose Pygmalion alone as their ruler. The ruthless, greedy brother killed Dido¡¯s husband in hopes of taking his wealth. To save her life and fortune, Dido led her attendants and followers and set sail for a new land. Eventually they arrived on the coast of North Africa. There, Dido tried to buy land from the local king to establish a new homeland. They bargained that she would obtain only as much land as she could enclose by the skin of an ox. So Dido cut the oxhide into thin strips and tied them together to form a closed cord of great length. Thus, she supposedly solved a famous mathematical problem called the isoperimetric problem: how to enclose the maximum area with a closed cord of fixed length. Dido¡¯s solution was the circle and her cord encircled an entire nearby hill. There, Dido and her followers built a city named Carthage, which became very prosperous for centuries to come.

In the early 20th century some mathematicians started wondering whether Dido¡¯s problem could be generalized to a curved surface. Suppose Dido had her followers enclose a region with closed cords of fixed length on a curved surface. Could they acquire a bigger region than they did in Carthage? To make this question more plausible, it should be assumed that the curved surface is a minimal surface, like a soap film, because only such a surface can mimic the flat plane of the territory of Carthage. The answer to this question, known as the second problem of Dido, is still unknown, but is conjectured to be ¡°No¡±.

This problem has been one of my favorite problems for a quarter century. I have been working on this problem off and on, sometimes by myself and other times with a collaborator. Only two times so far have I obtained partial answers, assuming some kind of connectivity of the boundary cords. Dozens of times, I thought I had found a remarkable proof for the problem. However, in less than two days each time, an unexpected gap would arise, always to my utter disappointment. Melodramatically phrased, my life as a mathematician up to now seems to be filled with lots of failures, like that of Dido. Nonetheless, undaunted, I still devote part of my time and energy to delving into the second problem of Queen Dido.

According to Virgil¡¯s Aeneid, Aeneas was the son of Venus and second cousin of the king of Troy. After Troy was defeated by the Greeks, he gathered the surviving Trojans and left, looking for a new home. After six years of wandering, a fierce storm forced Aeneas and his fleet to make a landing at Carthage. He met Dido, and recounted the Trojan War to her. After this, Aeneas had a year-long affair with the queen, who proposed that the Trojans settle in her land and that she and Aeneas reign jointly over their peoples. But Jupiter reminded Aeneas of his destiny in Italy: to become the legendary founder of Rome. Aeneas reluctantly accepted the divine command and left Dido secretly. When Dido learned of this, she cursed Aeneas¡¯ descendants and committed suicide by stabbing herself and falling onto a burning pyre.

The legend of Dido has remained popular throughout the ages and inspired many operas and dramas. When I was deeply frustrated by Dido¡¯s second problem, I once wrote a poem unexpectedly. This soothed me a bit. Sometimes, I fall into a reverie, musing that if Aeneas had not forsaken Dido, her second problem would not have been so difficult. But in reality, my frustration with her problem often reminds me of her despair.
Dido fell in love with Aeneas despite a vow of eternal fidelity to her dead husband. For this reason, in The Divine Comedy Dante places her in Hell among the sexual sinners. But I dare to presume that if Dante had appreciated her contribution to mathematics, he would have written differently. Mathematical originality could be as valuable a human virtue as fidelity.

In Aeneid, Virgil writes that after Aeneas arrives in Italy, he descends into the underworld where he meets Dido, but she turns away from him to return to her husband. If I meet a future mathematician who succeeds in solving Dido¡¯s second problem I will shake hands with him or her and exchange a hug with great pleasure. And I will feel relieved. A mathematician may sometimes be jealous. But a beautiful theorem is beyond anybody¡¯s jealousy; it just gives us bliss, and then leads us to the next attractive problem.

(Appeared in The Seoul Intelligencer, SEOUL ICM 2014)