Page 22 - Science Focus (Issue 19)

problem solving from a young age. While he was
extremely talented in mathematics, he was also honest
A Mysterious Maverick and careful to follow rules. Being a man of principle, 對於如何解決這道難題的見解分享給 Hamilton，可是最終

in Mathematics: mathematics had great appeal to him as one had to 並沒有收到任何的回覆。因此，他決定不與其他學者合作，
獨個兒解決這個問題，但其實他並不算孤身上路，因為他是
honestly follow established definitions and theorems to
Grigory Perelman arrive at a logically-sound conclusion. Graduating with 站在 Thurston 和 Hamilton 這些數學偉人的肩膀上，建
distinction from Leningrad State University and earning 基於他們的發現以架構自己完整的證明。在 Perelman 開
his PhD, he moved to the U.S. as he was invited to 始研究的五年後，克雷研究所才把龐加萊猜想列為千禧年難
तͭዹБٙग़।ᅰኪ࢕j numerous research positions in prestigious universities. 題，並懸賞 100 萬美元。換轉是其他人大概會對這筆可能飛
Grigory Perelman But, it wasn’t until he had become infatuated with 來的橫財大嘆好運，但 Perelman 卻心如止水，因為對他而
proving the Poincaré conjecture that he decided
言一切都沒有改變：由此至終他只是想找出真相。
to leave everything behind and return to his home
By Chun Ho Park 朴天浩 country to solely focus on this problem. 成功證明龐加萊猜想後，Perelman 並沒有把結果正式
發表到經同儕評審（peer review）的學術期刊，只是在網
Before his eight-year commitment to proving 上以預印本（preprint）的形式發表，就像他只想讓其他數
the conjecture, he met Richard Hamilton, a 學家了解到自己的發現一樣，而沒有想過要從自己的驚人發
mathematician at Cornell who had made significant 究竟一個數學家應該擁有怎樣的心態才算是達至最高
progress in solving the problem by formulating a 境界呢？當 Grigory Perelman 婉拒了因證明龐加萊猜想 現中獲取任何榮譽。為了闡明自己的發現，他欣然接受了所
framework called the Ricci Flow [2]. In the hope of （Poincaré conjecture）而得來的巨獎，他同時向世間證 有來自美國學術機構的邀請，包括普林斯頓大學、紐約州立
collaboration, Perelman wrote to Hamilton and shared 明了他並不為財富和名聲所動，這是在學術界鮮有的。單從 大學石溪分校和哥倫比亞大學，到場講解自己的發現並逐一
his idea on how to solve this problem, but did not 他拒絕接受百萬大獎，以及不久後在數學界消聲匿跡的兩件 回答在場人士的問題。不久之後，他贏得了菲爾茲獎和由克
receive any reply. So, instead of collaborating with 事可以看到他是特立獨行的一個。作為第一位，亦是唯一一 雷研究所所頒發的獎項，但兩者都被他拒絕接受。西班牙國
other scholars, he decided to tackle the problem in 位成功解開其中一條克雷研究所千禧年難題的數學家，他的 王也出席了頒獎典禮，卻未見 Perelman 的蹤影。「那個對
solitude, but built upon the works of mathematical 故事似乎比他對拒絕領獎一事的解釋更令人大惑不解。 我一點也不重要。」Perelman 評論道：「所有人都明白如果
giants like Thurston and Hamilton to solidify a complete 證明是正確就不需要額外的嘉許。」[3]
proof. It was after five years of him working on the 那甚麼是龐加萊猜想？解開了龐加萊猜想為甚麼會令世
problem that the Clay Institute announced it as one 界為之震驚？龐加萊猜想是拓撲學上其中一道最重要的問 智慧、勤勉和堅持絕對是使 Perelman 成功的因素，但
of the Millennium Problems with a million-dollar prize. 題，拓撲學是數學中一個關於特別幾何物件特性的分支。在 令他甘於奉獻八年時間去解答單一問題的原因要歸功於他
Although one could perceive this as a fortunate 拓撲學上，甜甜圈的形狀被視為與咖啡杯的一樣，因為兩者 對數學那樸素得稱得上為苦行的態度。這就是這個故事最
circumstance, nothing had changed for Perelman: he 都可以透過被延展和扭曲變成對方，而不需要切割或從中製 具啟發性的地方：他在數學上的堅決信念只是單純地出於
was only concerned with uncovering the truth. 造新的孔洞。雖然這是本質上是十分基礎的一個猜想，對於 對數學本身，而不是別的東西。如果要引述 Perelman 恩師
After successfully proving the conjecture, he did 理解我們身處的空間至關重要 [1]，可是它是一道著名的難 Mikhail Gromov 的一句話，那就是：「要完成偉大的工作，
not officially publish his results in a peer-reviewed 題，使得數學家奉上不少未如理想的證明，鎩羽而歸。 你要有一顆純潔的心。你想的一切只可以關於數學，其他都
journal, but merely posted it on the Internet as Perelman 與不少偉大數學家一樣出生於蘇聯時期的 是人性的弱點。接受獎項就是在暴露弱點。」[3]
What are the values that a mathematician a preprint. It was as if he just wanted to let other 俄羅斯，自小就熱衷於解決問題。在非常有數學天份的同時，
should hold at the highest standard? When Grigory mathematicians know of his discovery, rather than 他亦是一個誠實而且會小心翼翼遵循規矩行事的人。作為一
Perelman refused an enormous prize for proving the focusing on earning the credit for his astounding work. 個做事遵循原則的人，數學對他有一種特別的吸引力，因為
Poincaré conjecture, he also proved that he did not To explain his discovery, he accepted all the invitations
value wealth and fame like others in academia. Just from various institutes in the U.S., e.g. Princeton, Stony 那是需要從已確立的定義和理論，經嚴格邏輯推論後才能 References 參考資料：
considering the fact that he refused the million-dollar Brook and Columbia, to deliver lectures on his solution 達至結論的一門學科。Perelman 在列寧格勒國立大學以 [1] Veisdal, J. (2019). The Poincaré Conjecture: What is the
prize and vanished from the mathematics community and answer everyone’s questions about the proof. Not 優等成績畢業及取得博士學位後，因為受到不少著名大學邀 shape of the universe? Retrieved from https://medium.
shortly after, you can see that he is one of a kind. long after that, he won the Fields Medal and the prize 請其從事研究職位而旅居美國，一直到他著迷於證明龐加 com/cantors-paradise/the-poincaré-conjecture-
Being the first and only mathematician to solve one of from the Clay Institute, but refused both. The King of 萊猜想才決定放下在美國的一切回到老家，獨個兒專心鑽研 cb4ca7014cc5
Clay Institute’s Millennium Problems, his life story seems Spain attended the award ceremony, but Perelman 這道難題。 [2] Rubinstein, H. (2011). Millennium Prize: the
to be as mystifying as the million-dollar mystery he had was nowhere to be found. “It was completely Poincaré Conjecture. Retrieved from https://
deciphered. irrelevant for me,” Perelman remarked. “Everybody 在他開始花八年心力證明龐加萊猜想之前，他遇上過另 theconversation.com/millennium-prize-the-poincare-
conjecture-4245
So, what is the Poincaré conjecture, and why is it understood that if the proof is correct then no other 一位康奈爾大學的數學家 Richard Hamilton，Hamilton [3] Nasar, S., & Gruber, D. (2006). Manifold Destiny: A
以往曾構想出一 個叫「里奇流」（Ricci flow）的框 架而
worth all the fuss? The Poincaré conjecture was one of recognition is needed.” [3] 在這 個 問 題 上 推 進了一 大 步 [2]。Perelman 一 心 想 與 legendary problem and the battle over who solved
the most prominent questions in topology, a subfield Intelligence, diligence, and persistence were it. Retrieved from https://www.newyorker.com/
of mathematics concerned with the properties of certainly factors which allowed Perelman to succeed. Hamilton 合作，因此寫了一封很長的信給他，信中把自己 magazine/2006/08/28/manifold-destiny
special geometric objects. In topology, the shape of a But, his austere attitude towards mathematics was
doughnut is considered the same as that of a coffee probably what allowed him to dedicate eight years
cup, as one can be stretched and twisted to become of his life to solving one problem. This is what is most
the other, without cutting or making new holes in it. inspiring about his story: his firm belief on mathematics
While the conjecture was very fundamental in nature having to be purely about mathematics itself, nothing
and crucial in understanding the space we live in [1], else. To quote Mikhail Gromov, one of Perelman’s
it was notoriously difficult to prove, and many wrong teachers, “To do great work, you have to have a
proofs were submitted.
pure mind. You can think only about mathematics.
Born in Soviet Russia, home to many great Everything else is human weakness. Accepting prizes is
mathematicians, Perelman had great passion for showing weakness.” [3]