2017 年沃爾夫數學獎得主 Charles Fefferman 自述

小時候我想知道火箭是如何運作的，於是從圖書館借了一本物理書來看。結果,我發現一句話也讀不懂。睿智的父親跟我解釋說，因為書中到處都是你未曾學的數學。

於是我開始學習數學課本，從小學四年級的算術開始。在學完微積分以後，父親把我帶到了當地的馬里蘭大學尋求指導。這是我與馬里蘭大學交往的開始。那裡的老師極好。這是一個很大的州立學校，但我感覺整個數學系都在為我提供私人輔導。我被馬里蘭大學作為本科生錄取了。這是不合規定的，因為當時我還只有十四歲，但數學系主任向校方施壓說，如果學校不錄取我的話他就辭職不幹了。倘若沒有他們從前的壯舉，今天你[譯者按：指攝影師庫克(Mariana Cook)]不會來拜訪我。

三年後，我到普林斯頓念研究生。能追隨施泰因(Eli Stein)學習，是我最大的幸運，他不僅是一位偉大的數學家，而且也許是我見過的最好的數學老師。施泰因的教學和榜樣對我的工作仍然是一個主要的影響。

我想描述我的兩項貢獻。第一個貢獻是掛谷集(Kakeya set)與傅里恭弘=叶 恭弘分析之間的一個聯繫。平面中的掛谷宗一集具有奇特的形狀。你可以將一枚細針在一個掛谷集的內部翻轉一整周；而這個掛谷集的面積可以要多小就有多小。傅里恭弘=叶 恭弘分析研究複雜的振動如何分解為簡單的振動。例如，小提琴弦的複雜運動由根音、第一泛音、第二泛音等構成。如果將高頻部分移去，小提琴弦的音調將會降低。部分原因是，小提琴弦是一維的。照片則是一個二維的影像，也是由類似於琴弦的根音和泛音這樣的簡單片段構成的。由於照片是二維的，它可能無法對焦，而當截去高頻部分時又會突然精準對焦。這就是因為掛谷集的存在。我在一九七〇年代發現了這一點。二維以上空間的掛谷集繼續呈現了具有挑戰性的問題。本書中的照片當然是完美對焦的。

其次，我花了很多年研究關於原子的數學問題。任何一本量子力學書都會解釋為何一個电子與一個質子聯合而成一個氫原子。但書上不會告訴你為何數以萬計的电子與數以萬計的質子一起結合成數以萬計的氫原子。這是一個困難得多的問題，需要許多數學；完全的解答仍然未知。我的貢獻是，將這個問題歸結為對系統能量的估計。

我沒有去選擇問題；是問題選擇了我。問題會勾住我，讓我理所應當地為之思索幾年或幾十年。有時我得到錯誤的想法。錯誤的想法好比是鍋里的原料。加了充分多的原料以後就可以熬湯了。如果運氣好，味道就不錯。

在普林斯頓，我通常教一門研究生課（通常是論述我本人的工作）和一門本科生課（通常是初等微積分）。當研究停滯不前時，一想到我正在做一些有用的事情可以陪伴我的大一新生度過一段不那麼痛苦的時光，我就很知足。

附：原文

Charles Louis Fefferman

Fourier analysis, partial differential equations

Fields Medal

Herbert E. Jones, Jr., University Professor of Mathematics, Princeton University

As a little kid, I wanted to know how rockets work and borrowed a physics book from a library. I didn’t understand one word. My wise father explained to me that the book was full of math, which I hadn’t studied.

I set about reading math textbooks, starting with fourthgrade arithmetic. After I got through calculus, my father took me to our local university, the University of Maryland, for tutoring.That was the start of my relationship with the University of Maryland. They were wonderful. It’s a big state school, but I had the feeling that their whole math department was giving me private tutoring. I attended Maryland as an undergraduate. It was illegal since I was only fourteen years old, but the chairman of the math department threatened to resign if the university didn’t admit me. You wouldn’t be interviewing me today if it hadn’t been for them.

I went to Princeton as a grad student. It was a great stroke of luck for me to study with Eli Stein, a great mathematician and perhaps the best teacher of math I’ve ever known. Eli’s teaching and example are still a major influence on my work.

I would like to describe two of my contributions. The first is a connection between Kakeya sets and Fourier analysis. Kakeya sets are strange shapes in the plane. One can turn a 1-inch-long needle through a full 360 degrees, keeping the needle entirely inside a Kakeya set; yet the area of a Kakeya set is as small as you please. Fourier analysis is the study of how complicated vibrations break up into simple ones. For example, the complicated motion of a violin string is made up of a fundamental note, a first overtone, a second overtone, and so on. The sound of the violin string is degraded if the high frequencies are removed.  In part, that’s because the violin string is one-dimensional. A photograph is a two-dimensional image, also built up from simple pieces analogous to the fundamental note and overtones of a string. Because a photo is two-dimensional, it may be out of focus yet come into sharp focus when its high frequencies are removed. That’s because of the existence of Kakeya sets. I discovered this in the 1970s. Kakeya sets in dimension higher than two continue to present challenging problems. The photos in this book are in perfect focus.

Secondly, I’ve spent many years on mathematical problems about atoms. Any quantum mechanics textbook explains why one electron and one proton combine to make one hydrogen atom. The textbook won’t tell you why a billion, billion, billion electrons and a billion, billion, billion protons combine to make lots of hydrogen atoms. That’s a much harder problem, which entails a lot of math; the full solution isn’t yet known. I made a contribution by reducing the problem to an estimate for the energy of the system.

I don’t choose problems; they choose me. A question will grab hold of me and I feel compelled to think about it for years or decades. On a typical day, I get no ideas, but on a good day, I get a wrong idea. Wrong ideas are ingredients in the pot. Add enough ingredients and the stew cooks. With luck, it tastes

good.

At Princeton, I usually teach a graduate class (often on my own work) and an undergraduate class (often elementary calculus). When research is going badly, it’s pleasing to think that I’m doing something useful by not giving my freshmen a hard time.

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