空間形狀知多少？ The Shapes of Space - 一位年輕俄羅斯數學家，證明了數學大師龐卡赫在一世紀前所提出的著名猜想，並且完成了三維空間的分類，還可能因而賺得100萬美元的獎金。 - A Russian mathematician has proved the century-old Poincaré conjecture and completed the catalogue of three-dimensional spaces. He might earn a $1-million prize

作者╱柯林斯 ( Graham P. Collins ) 譯者╱高湧泉

　　站起來，瞧瞧四周，走一圈，跳一下，甩甩手。你只不過是一組在三維流形（數學家對於三維空間的稱呼）中一小區域內運動的粒子，這三維流形在任何方向都延伸數十億光年之遠。
　　Stand up and look around. Walk in a circle. Jump in the air. Wave your arms. You are a collection of particles moving about within a small region of a 3-manifold—a three-dimensional space—that extends in all directions for many billions of light-years.

 　 　「流形」（manifold）是數學家建構出來的東西。自伽利略與克卜勒以來，物理上的成就其實就是各種數學（例如流形的數學）對於「實在」的成功描述。根據物理，所有的事件都發生在三維空間這個背景之中（我們先撇開弦論學家對於三維空間之外超小維度的臆想，參見延伸閱讀1）。所謂「三維」的意思是， 我們需要三個數字來標定一個粒子的位置，比如說，在地球附近這三個數字可能是經度、緯度與高度。
　 　Manifolds are mathematical constructs. The triumph of physics since the time of Galileo and Kepler has been the successful description of reality by mathematics of one flavor or another, such as the mathematics of manifolds. According to physics, everything that happens takes place against the backdrop of three-dimensional space (leaving aside the speculations of string theorists that there are tiny dimensions in addition to the three that are manifest) [see 「The Theory Formerly Known as Strings,」 by Michael J. Duff; Scientific American, February 1998]. Three dimensions means that three numbers are needed to specify the location of a particle. Near Earth, for instance, the three numbers could be latitude, longitude and altitude.

 　　根據牛頓力學和傳統量子物理學，所有事情皆發 生於三維空間，而這個空間是固定不可改變的。相反地，愛因斯坦的廣義相對論讓空間成為參與活動的演員之一：兩點之間的距離，會受到附近有多少物質、能量， 以及有多少重力波通過（參見延伸閱讀2）所影響。總之，無論是牛頓物理或是愛氏物理、不論空間是無限還是有限，空間都是個三維流形。因此如果想完全理解一 切物理以及一切科學的基礎，我們必得瞭解三維流形的性質（四維流形也很重要：空間與時間結合成一個四維流形）。
　　According to Newtonian physics and traditional quantum physics, the three-dimensional space where everything happens is fixed and immutable. Einstein's theory of general relativity, in contrast, makes space an active player: the distance from one point to another is influenced by how much matter and energy are nearby and by any gravitational waves that may be passing by [see 「Ripples in Spacetime,」 by W. Wayt Gibbs; Scientific American, April 2002]. But whether we are dealing with Newtonian or Einsteinian physics and whether space is infinite or finite, space is represented by a 3-manifold. Understanding the properties of 3-manifolds is therefore essential for fully comprehending the foundations on which almost all of physics—and all other sciences—are built. (The 4-manifolds are also important: space and time together form a 4-manifold.)

 　　數學家對於三維流形已經有很多認識，然而某些最基本的問題還是非常困難。數學中研究流形的領域是拓撲學。關於三維流形，拓撲學家可以研究的基本問題包括：什麼是三維流形最簡單的類型 （即何種三維流形有最不複雜的結構）？這個最簡單的流形有沒有同樣簡單的夥伴？或者是獨一無二的？有多少種三維流形？
　　Mathematicians know a lot about 3-manifolds, yet some of the most basic questions have proved to be the hardest. The branch of mathematics that studies manifolds is topology. Among the fundamental questions topologists can ask about 3-manifolds are: What is the simplest type of 3-manifold, the one with the least complicated structure? Does it have many cousins that are equally simple, or is it unique? What kinds of 3-manifolds are there?

　　對於第一個問題，數學家 早就知道答案了：一種稱為三維球的空間是最簡單的緊緻三維流形。（非緊緻流形可以想成是無限大的流形，我們爾後只去考慮緊緻流形。）至於另外兩個問題，百 年來都還沒有人找到答案。不過，一位俄羅斯數學家帕瑞爾曼（Grigory Perelman）或許已經在2002年將它解決了，他很可能已經證明出一個稱為「龐卡赫猜想」（Poincaré conjecture）的定理。
　 　The answer to the first of those questions has long been known: a space called the 3-sphere is the simplest compact 3-manifold. (Noncompact manifolds can be thought of as being infinite or having an edge. Hereafter I consider only compact manifolds.) The other two questions have been up for grabs for a century but may have been answered in 2002 by Grigori (「Grisha」) Perelman, a Russian mathematician who has most probably proved a theorem known as the Poincaré conjecture.

　 　這個猜測是由法國數學家龐卡赫於距今100年前提出來的，它的意思是，三維球在三維流形之中的確是獨一無二的，再也沒有其他三維流形，具備三維球那種簡 單的性質。比三維球更複雜的其他三維流形，要不是具有邊界（就好像你會碰上的一堵磚牆），不然就是各區域之間有著多重連結（就好像樹林裡的路徑岔開後又會 連結在一起）；龐卡赫猜想要說的，正是「三維球是唯一沒有以上這些較複雜結構的緊緻三維流形」。因為任何三維物體，只要和球具有同樣的性質，都可以變形成 三維球的模樣；因此就拓撲學的角度而言，這種物體只是另一個三維球而已。此外，帕瑞爾曼的證明也回答了上面的第三個問題：他完成了三維流形的分類工作。
　　First postulated by French mathematician Henri Poincaré exactly 100 years ago, the conjecture holds that the 3-sphere is unique among 3-manifolds; no other 3-manifold shares the properties that make it so simple. The 3-manifolds that are more complicated than the 3-sphere have boundaries that you can run up against like a brick wall, or multiple connections from one region to another, like a path through the woods that splits and later rejoins. The Poincaré conjecture states that the 3-sphere is the only compact 3-manifold that lacks all those complications. Any three-dimensional object that shares those properties with the sphere can therefore be morphed into the same shape as a 3-sphere; so far as topologists are concerned, the object is just another copy of the 3-sphere. Perelman's proof also answers the third of our questions: it completes work that classifies all the types of 3-manifolds that exist.

　 　我們需要用些腦筋才能想像三維球的樣子，它不是日常意義中的球（見58~59頁〈球的多重樂音〉），但是和我們熟悉的二維球有很多共同性質：拿一個圓形 氣球來說，球的那一層橡膠皮就是二維球。二維球是二維流形，因為只要緯度與經度兩個座標，就足以標定球上的位置。而且你如果取一小片球皮，用放大鏡檢視， 這一小片球皮和一小片平坦的二維橡膠面，看起來會很類似，它只是稍微有些曲率而已。對於在氣球上爬行的小蟲而言，球看起來就像是平面，但是小蟲如果沿著一 條「直線」（對小蟲而言）爬行得夠遠，牠終究會回到原先的出發點。
　　It takes some mental gymnastics to imagine what a 3-sphere is like—it is not simply a sphere in the everyday sense of the word [see box on pages 98 and 99]. But it has many properties in common with the 2-sphere, which we are all familiar with: If you take a spherical balloon, the rubber of the balloon forms a 2-sphere. The 2-sphere is two-dimensional because only two coordinates—latitude and longitude—are needed to specify a point on it. Also, if you take a very small disk of the balloon and examine it with a magnifying glass, the disk looks a lot like one cut from a flat two-dimensional plane of rubber. It just has a slight curvature. To a tiny insect crawling on the balloon, it would seem like a flat plane. Yet if the insect traveled far enough in what would seem to it to be a straight line, eventually it would arrive back at its starting point.

　　同樣地，三維球中的蚊子（或者是人，如果球和我們所處的宇宙一樣大！）會感覺自己處於「平常」的三維空間，但是，牠如果沿著任何方向的直線飛行，只要飛得夠遠，終究會環繞三維球一週而回到出發點，就好像氣球上的小蟲或是某個繞地球一圈的人。
　 　Similarly, a gnat in a 3-sphere—or a person in one as big as our universe!—perceives itself to be in 「ordinary」 three-dimensional space. But if it flies far enough in a straight line in any direction, eventually it will circumnavigate the 3-sphere and find itself back where it started, just like the insect on the balloon or someone taking a trip around the world.

　　球也存在於其他維度的空間之中。一維球你很熟悉：它就是圓（圓盤的邊，不包括圓盤內部）。n維球就是n維空間中的球。
　　Spheres exist for dimensions other than three as well. The 1-sphere is also familiar to you: it is just a circle (the rim of a disk, not the disk itself). The n-dimensional sphere is called an n-sphere.


證明龐卡赫猜想
Proving Conjectures

　　龐卡赫提出關於三維球的猜想之後，半個世紀過去了，卻一直沒有人能夠真正證明這個猜想。到了1960年 代，數學家證明了適用於五維或是更高維球的類似假想。在每個維度，n維球的確是唯一的最簡單流形。奇怪的是，高維比四維或是三維的情形還容易證明。 1982年，特別困難的四維猜想證明出來了，只剩下龐卡赫原始的三維假設還沒得到解決。
　　AFTER POINCARÉ proposed his conjecture about the 3-sphere, half a century went by before any real progress was made in proving it. In the 1960s mathematicians proved analogues of the conjecture for spheres of five dimensions or more. In each case, the n-sphere is the unique, simplest manifold of that dimensionality. Paradoxically, this result was easier to prove for higher-dimensional spheres than for those of four or three dimensions. The proof for the particularly difficult case of four dimensions came in 1982. Only the original three-dimensional case involving Poincaré's 3-sphere remained open.

　　2002年11月，解決這個三維問題關鍵性的一步終於跨出。那時俄國聖彼得堡斯特克洛夫數學研究所的帕瑞爾曼在www.arxiv.org（物 理學家與數學家公開最新研究結果的網站）上放了一篇文章，這篇文章並沒有提到龐卡赫猜想這個名稱，但是拓撲學專家一眼就識出兩者之間的關係。帕瑞爾曼隨後 在2003年3月公開了第二篇論文，然後在同年4~5月訪問美國，於麻省理工學院與紐約州立大學石溪分校做了一系列關於這些結果的演講。十幾所頂尖大學的 數學團隊開始研讀他的論文、檢查每個細節，並尋找錯誤。
　　A major step in closing the three-dimensional problem came in November 2002, when Perelman, a mathematician at the Steklov Institute of Mathematics at St. Petersburg, posted a paper on the www.arxiv.org Web server that is widely used by physicists and mathematicians as a clearinghouse of new research. The paper did not mention the Poincaré conjecture by name, but topology experts who looked at it immediately realized the paper's relevance to that theorem. Perelman followed up with a second paper in March 2003, and from April to May that year he visited the U.S. to give a series of seminars on his results at the Massachusetts Institute of Technology and Stony Brook University. Teams of mathematicians at nearly a dozen leading institutes began poring over his papers, verifying their every detail and looking for errors.

　　在石溪，帕瑞爾曼每天正式或非正式地演講三到六個小時，前後總共兩個星期。石溪數學家 安德森（Michael Anderson）說：「他回答了每個問題，也解釋得非常清楚，還沒有人可以提出真正讓他頭痛的問題。」他又說，只差一小步的證明就可以完成整個結果， 「但是沒有人懷疑這最後一步會出問題。」帕瑞爾曼在第一篇文章解釋了整個基本想法，可以算是已經通過了檢驗而廣為大家接受；第二篇文章則包含應用與更技術 性的論證，但是人們對它的信心尚不如對第一篇文章那麼高。
　 　At Stony Brook, Perelman gave two weeks of formal and informal lectures, talking from three to six hours a day. 「He answered every question that arose, and he was very clear,」 says mathematician Michael Anderson of Stony Brook. 「No one has yet raised any serious doubts.」 One more comparatively minor step has to be proved to complete the result, Anderson says, 「but there are no real doubts about the validity of this final piece.」 The first paper contains the fundamental ideas and is pretty well accepted as being verified. The second paper contains applications and more technical arguments; its verification has not reached the level of confidence achieved for the first paper.

　　龐卡赫猜想的證明有100萬美元的獎賞，這是美國麻州劍橋的克萊數學研究所，在 2000年所提出的七個「千禧數學題」（Millennium Problem）之一。帕瑞爾曼的證明必須正式出版，而且經得起隨後兩年的檢驗，才有資格領獎。（克萊數學研究所應該會同意在網路上公開便等同於「出 版」，因為它也受到嚴格的同儕審查，就好像一般論文一樣。）
　 　The Poincaré conjecture has a $1-million reward on offer for its proof: it is one of seven such 「Millennium Problems」 singled out in 2000 by the Clay Mathematics Institute in Cambridge, Mass. Perelman's proof has to be published and withstand two years of scrutiny before he becomes eligible for the prize. (The institute might well decide that its posting on the Web server qualifies as 「published」 because the result is undergoing as rigorous a peer review as any paper gets.)

　　帕瑞爾曼的工作其實是推廣並完成了美國哥倫比亞大學的漢密爾頓（Richard S. Hamilton）在1990年代所探討的研究方向。克萊數學研究所在2003年年底頒了一個研究獎給漢密爾頓，而帕瑞爾曼的計算與分析則進一步掃除了好 幾個漢密爾頓不能克服的路障。
　　 Perelman's work extends and completes a program of research that Richard S. Hamilton of Columbia University explored in the 1990s. The Clay Institute recognized Hamilton's work with a research award in late 2003. Perelman's calculations and analysis blow away several roadblocks that Hamilton ran into and could not overcome.

　　如果帕瑞爾曼的證明是正確的，一如大家所預期的那樣，那麼，他所完成的事實上是一個比龐卡赫猜想更為龐大的架 構。現在任教於美國康乃爾大學的佘斯頓（William P. Thurston），就率先提出了「佘斯頓幾何化猜想」（Thurston geometrization conjecture），這個猜想提供了所有三維流形的完整分類。三維球──唯一具有崇高簡單性質的流形──便是這個偉大分類的基礎。如果龐卡赫猜想是錯 的，也就是說存在著很多和球一樣「簡單」的空間，那麼三維流形的分類就會遠比佘斯頓的提議來得複雜。反之，有了帕瑞爾曼與佘斯頓的結果，一切三維空間可能 具有的形狀，也就是所有數學允許的宇宙形狀（只考慮空間，不包括時間），就有了完整的分類。
　　If, as everyone expects, Perelman's proof is correct, it actually completes a much larger body of work than the Poincaré conjecture. Launched by William P. Thurston—now at Cornell University—the Thurston geometrization conjecture provides a full classification of all possible 3-manifolds. The 3-sphere, unique in its sublime simplicity, anchors the foundation of this magnificent classification. Had the Poincaré conjecture been false—that is, if there were many spaces as 「simple」 as a sphere—the classification of 3-manifolds would have exploded into something infinitely more complicated than that proposed by Thurston. Instead, with Perelman's and Thurston's results, we now have a complete catalogue of all the possible shapes that a three-dimensional space can take on—all the shapes allowed by mathematics that our universe (considering just space and not time) could have.


橡皮甜甜圈
Rubber Doughnuts

　　如果要更深入理解龐卡赫猜想與帕瑞爾曼的證明，你必須懂一點拓撲學。在這個數學領域中，我們不關心物體 的精確形狀，就把它們當做勞作黏土所做出來的，你可以極盡所能地將它拉長、擠壓、扭曲。但是我們為什麼得在乎用想像的勞作黏土所做出來的東西（或是空間） 呢？原因是：物體的精確形狀（從其中一點到另一點的距離）是另外一層的數學結構，我們稱這結構為物體的幾何。拓撲學家藉由研究黏土物體，發現它們有一些非 常基本的性質，這些性質極為基本，以至於和其幾何結構毫不相干。研究拓撲學就好像藉由研究「黏土人」（它們可以變形成任何特定的人）的性質來發現人類共同 的性質。
　　TO UNDERSTAND the Poincaré conjecture and Perelman's proof in greater depth, you have to know something about topology. In that branch of mathematics the exact shape of an object is irrelevant, as if it were made of play dough that you could stretch, compress and bend to any extent. But why should we care about objects or spaces made of imaginary play dough? The reason relates to the fact that the exact shape of an object—the distance from one point on it to another—is a level of structure, which is called the geometry of the object. By considering a play-dough object, topologists discover which properties of the object are so fundamental that they exist independently of its geometric structure. Studying topology is like discovering which properties humans have in general by considering the properties of a 「play-dough person」 who can be morphed into any particular human being.

　　如果你讀過任何介紹拓撲學的科普文章，你或許已經知道一個老掉牙的講法：對拓撲學家而言，杯子和甜甜圈是沒有差別的（這裡指的是環圈 狀的甜甜圈，而不是實心裡頭有果醬的那種）。重點是，你可以在不把黏土穿個洞或把任何兩部份黏在一起的條件下，將黏土杯子捏成甜甜圈的樣子（參見下頁 圖）。反之，如果想把一個球變形成甜甜圈，你就得在球的中心穿個洞，或是將它拉成圓柱形然後把兩端黏在一起。因為必須穿個洞或是黏起來，所以對拓撲學家來說，球和甜甜圈是不同的東西。
　　If you have read any popular account of topology, you have probably encountered the hoary old truism that a cup and a doughnut are indistinguishable to a topologist. (The saying refers to a ring-shaped doughnut, not the solid, jam-filled kind.) The point is that you can morph the play-dough cup into a doughnut shape simply by smushing the clay around, without having to cut out any holes or glue any patches together [see illustration on page 100]. A ball, on the other hand, can be turned into a doughnut only by either boring a hole through its middle or stretching it into a cylinder and gluing the ends together. Because such cutting or gluing is needed, a ball is not the same as a doughnut to a topologist.

　　拓撲學家最感興趣的是球和甜甜圈的表面，我們不把它們看成是實心的物體，而是像氣球的東西。這兩者的拓撲還是不 一樣：圓形氣球不能變形成環面（torus）的氣球。所以就拓撲學而言，球和環面是不同的東西。早期的拓撲學家就已開始探討究竟有多少拓撲上相異的物體， 以及如何將它們分類。對於也稱為「曲面」（surface）的二維物體來說，答案很單純：一切取決於一個面有多少「把手」。
　　What interests topologists most are the surfaces of the ball and the doughnut, so instead of imagining a solid we should imagine a balloon in both cases. The topologies are still distinct—the spherical balloon cannot be morphed into the ring-shaped balloon, which is called a torus. Topologically, then, a sphere and a torus are distinct entities. Early topologists set out to discover how many other topologically distinct entities exist and how they could be characterized. For two-dimensional objects, which are also called surfaces, the answer is neat and tidy: it all boils down to how many 「handles」 a surface has.

　　到了19世紀末， 數學家已經瞭解如何分類「曲面」。他們知道在所有的曲面當中，球的簡單性是獨一無二的。接下來，他們很自然會想知道在三維流形中的情況。首先，三維球是否 也和二維球一樣，在其簡單性上是獨一無二的？在這個基本問題提出後的一世紀中，出現了太多錯誤的嘗試與證明。
　　By the end of the 19th century, mathematicians understood how to classify surfaces. Out of all the surfaces, the sphere, they knew, had a unique simplicity. Naturally they started wondering about three-dimensional manifolds. To start with, was the 3-sphere unique in its simplicity, analogous to the 2-sphere? The century-long history that follows from that elementary question is littered with false steps and false proofs.

　　龐卡赫迎頭就處理起這個問題，他 是20世紀初兩位最傑出的數學家之一（另一位是希爾伯特）。人們稱龐卡赫為最後一位數學通才，他精通每個數學領域，無論是純數學或是應用數學。除了推進眾 多數學領域的進展，他還對天體力學、電磁學與科學哲學（他寫了好幾本廣為閱讀的科哲普及書）有重要貢獻。
　　Henri Poincaré tackled this question head-on. He was one of the two foremost mathematicians who were active at the turn of the 20th century (the other being David Hilbert). Poincaré has been called the last universalist—he was at ease in all branches of mathematics, both pure and applied. In addition to advancing numerous areas of mathematics, he contributed to the theories of celestial mechanics and electromagnetism as well as to the philosophy of science (about which he wrote several widely read popular books).

　　龐卡赫是「代數拓撲」這門學問的主要 創始者，他在1900年前後利用代數拓撲的技巧，建立了一種量度物體拓撲的方法，稱為「同倫」（homotopy）。如何決定一個流形的同倫？方法如下： 想像你將一個封閉迴圈嵌入流形之中（參見左頁），這個迴圈可以用任何方式環繞著流形。我們接著問，是不是只要將迴圈上的點移一移，不必將其中某一部份拿到 流形之外，就一定能夠將迴圈收縮成一個點？對於環面來說，答案是「不」。如果迴圈環繞了環面的圓週一圈，它就無法收縮成一個點，它會受阻於環面的內圈。利 用同倫，我們便能量度究竟有多少種方式可以阻礙一個迴圈的收縮。
　　Poincaré largely created the branch of mathematics called algebraic topology. Around 1900, using techniques from that field, he formulated a measure of an object's topology, called homotopy. To determine a manifold's homotopy, imagine that you embed a closed loop in the manifold [see box on next page]. The loop can be wound around the manifold in any possible fashion. We then ask, Can the loop always be shrunk down to a point, just by moving it around, without ever lifting a piece of it out of the manifold? On a torus the answer is no. If the loop runs around the circumference of the torus, it cannot be shrunk to a point—it gets caught on the inner ring of the doughnut. Homotopy is a measure of all the different ways a loop can get caught.

　　在一個n維球上面，無論迴圈纏繞的情形有多複雜，它永遠可以解開來然後收縮到 一個點。（在移動的時候，迴圈可以穿過自己。）龐卡赫推測，在三維流形中，唯有三維球能夠讓其中每個迴圈都收縮成一個點，但是他自己無法證明，後來這個想 法就稱為龐卡赫猜想。數十年來，很多人宣稱找到了證明，但其實都是錯的。
　　On an n-sphere, no matter how convoluted a path the loop takes, it can always be untangled and shrunk to a point. (The loop is allowed to pass through itself during these manipulations.) Poincaré speculated that the only 3-manifold on which every possible loop can be shrunk to a point was the 3-sphere itself, but he could not prove it. In due course this proposal became known as the Poincaré conjecture. Over the decades, many people have announced proofs of the conjecture, only to be proved wrong.

　　為了明確起見，我在這裡不考慮兩個複雜的狀況：所謂「不可定向的流 形」（nonorientable manifold）以及有邊界的流形。例如，莫比斯帶（Möbius band）這種扭了一圈再接起來的帶子就是不可定向的流形。在一個球上切下一個圓盤，這個球就有了邊界；莫比斯帶也是有邊界的流形。
　　For clarity, here and throughout I ignore two complications: so-called nonorientable manifolds and manifolds with edges. For example, the Möbius band, a ribbon that is twisted and joined in a loop, is nonorientable. A sphere with a disk cut out from it has an edge. The Möbius band also has an edge.


幾何化
Geometrization

　　帕瑞爾曼的證明是頭一個禁得起考驗的證明。他分析三維流形的方法和一種稱為「幾何化」的步驟有關。幾何 是流形或物體的真實形狀：就幾何的觀點而論，一個東西不是黏土做的，而是由陶瓷材料做出來的。例如一個杯子的幾何就和甜甜圈的幾何不同，因為兩種曲面彎曲 的情形不一樣。我們說，只有一個握把的杯子和甜甜圈是拓撲環面的兩個例子，但是它們被賦予不同的幾何。
　　PERELMAN'S PROOF is the first to withstand close scrutiny. His approach to analyzing three-dimensional manifolds is related to a procedure called geometrization. Geometry relates to the actual shape of an object or manifold: for geometry, an object is made not of play dough but of ceramic. A cup, for example, has a different geometry than a doughnut; its surface curves in different ways. It is said that the cup and the doughnut are two examples of a topological torus (provided the cup has one handle) to which different geometries have been assigned.

　　為瞭解「幾何化」如何幫助帕瑞爾曼，我 們得先想一想如何用幾何來分類二維流形（曲面）。我們賦予每個拓撲曲面一個特殊、唯一的幾何：曲面曲率均勻地分佈在流形上的幾何。對於球來說，這個唯一的 幾何就是正圓球面。蛋殼形是另一種拓撲球可能有的幾何，但是它並沒有均勻分佈的曲率：蛋的尖端比鈍端更為彎曲。
　　To gain a sense of how geometrization served to help Perelman, consider how geometry can be used to classify 2-manifolds, or surfaces. Each topological surface is assigned a special, unique geometry: the one for which the curvature of the surface is spread completely evenly about the manifold. For the sphere, that unique geometry is the perfectly spherical sphere. An eggshell shape is another possible geometry for a topological sphere, but it does not have curvature evenly spread throughout: the small end of the egg is more curved than the big end.

　　二維流形有三種幾何類型（參見 上頁〈幾何化〉）。球有所謂的「正曲率」，就像山丘頂的形狀；幾何化後的環面則是平坦的，曲率為零，像是平面。其他的流形則有兩個以上的把手，有「負曲 率」，好像山谷通道或馬鞍的形狀：從前到後，馬鞍是向上彎的，從左到右，馬鞍是向下彎的。龐卡赫（不然還有誰？）以及寇貝（Paul Koebe）與克萊恩（Felix Klein，克萊恩瓶就是以他為名）對於這個二維流形的幾何分類，或者說幾何化，有很大貢獻。
　　The 2-manifolds form three geometric types [see box at right]. The sphere has what is called positive curvature, the shape of a hilltop. The geometrized torus is flat; it has zero curvature, like a plain. All the other manifolds, with two or more handles, have negative curvature. Negative curvature is like the shape of a mountain pass or a saddle: going from front to back, a saddle curves up; from left to right, it curves down. Poincaré (who else?), along with Paul Koebe and Felix Klein (after whom the Klein bottle is named), contributed to this geometric classification, or geometrization, of 2-manifolds.

　　數學家很自然就想將類似的方法用於三維流形。我們能夠替每個拓撲三維流形找到曲率均勻分佈於流形之上的唯一幾何嗎？
　　It is natural to try to apply similar methods to 3-manifolds. Is it possible to find unique geometries for each topological 3-manifold, for which curvature is spread evenly throughout the manifold?

　 　事實上，三維流形遠比二維流形複雜太多了；多數三維流形無法被賦予均勻的幾何。反之，它們必須切成許多小部份，每個部份有不同的標準幾何。此外，和二維 流形的三種基本幾何不同，三維流形的每一部份可以是八種標準幾何之一。把三維幾何切開來，就類似於把一個數字分解成質數的乘積。
　　It turns out that 3-manifolds are far more complicated than 2-manifolds. Most 3-manifolds cannot be assigned a uniform geometry. Instead they have to be cut up into pieces, each piece having a different canonical geometry. Furthermore, instead of three basic geometries, as with 2-manifolds, the 3-manifold pieces can take any of eight canonical geometries. The cutting up of each 3-manifold is somewhat analogous to the factorization of a number into a unique product of prime factors.

　　佘斯頓在1970年代末期首先設想出以上三維流形的分類方法。他和合作者證明了這個假設的一大部份，但是決定整個系統的關鍵，包括涵蓋龐卡赫猜想的那個部份，還是無法處理。三維球是獨特的嗎？這個問題的答案以及佘斯頓方案，只有在帕瑞爾曼的文章出現後才得以確立。
　　 This classification scheme was first conjectured by Thurston in the late 1970s. He and his colleagues proved large swaths of the conjecture, but crucial points that the entire system depended on remained beyond their grasp, including the part that embodied the Poincaré conjecture. Was the 3-sphere unique? An answer to that question and completion of the Thurston program have come only with Perelman's papers.

　 　我們如何幾何化一個流形，也就是說給它一個均勻的曲率呢？一個方法是從某個任意的幾何出發，例如上頭有種種凹凹凸凸的蛋殼形狀，然後再設法撫平這些凹 凸。漢密爾頓在1990年初期開始利用這種方法分析三維流形，他的方案使用了一個稱為「瑞奇流」（Ricci flow）的方程式（瑞奇是一位數學家Gregorio Ricci-Curbastro），它有些類似於控制熱流的方程式。
　　How might we try to geometrize a manifold—that is, give it a uniform curvature throughout? One way is to start with some arbitrary geometry, perhaps like an eggshell shape with various lumps and indentations, and then smooth out all the irregularities. Hamilton began such a program of analysis for 3-manifolds in the early 1990s, using an equation called the Ricci flow (named after mathematician Gregorio Ricci-Curbastro), which has some similarities to the equation that governs the flow of heat.

　　物體中如果有熱點與冷點，熱當然是從溫度較高的地方流到較低的地方，直到物體的溫度到處都相同。瑞奇流方程式對於曲率有類似的效應，能把流形上的凹凸之處撫平。如果流形一開始是蛋殼狀，它會慢慢地變成完美的球形。
　　In a body with hot and cold spots, heat naturally flows from the warmer regions to the cooler ones, until the temperature is uniform everywhere. The Ricci flow equation has a similar effect on curvature, morphing a manifold to even out all the bumps and hollows. If you began with an egg, it would gradually become perfectly spherical.

　 　漢密爾頓的分析遇到了一些困難：在某些情況下，瑞奇流會讓流形緊束成一個點。（這是瑞奇流和熱流不一樣的地方之一，緊束的地方就好像是溫度變成無窮大的 點。）其中一個例子是，當流形呈現啞鈴狀的時候，亦即兩個圓球用一條細頸子連起來，球會從頸子吸取材料而逐漸變大，使得頸子漸漸在中央變成一個點（參見左 頁〈如何對付奇異點？〉）。
　　Hamilton's analysis ran into a stumbling block: in certain situations the Ricci flow would cause a manifold to pinch down to a point. (This is one way that the Ricci flow differs from heat flow. The places that are pinched are like points that manage to acquire infinite temperature.) One example was when the manifold had a dumbbell shape, like two spheres connected by a thin neck. The spheres would grow, in effect drawing material from the neck, which would taper to a point in the middle [see box above].

　　另一個可能的情況是，一根細棍子從流形中凸出來，這時瑞奇流可能會製造一個稱為「雪茄奇異點」（cigar singularity）的麻煩。當流形以這種方式緊束起來的時候，它就是奇異的，而不再是真正的三維流形了。在真正的三維流形中，任何一點附近的小區 域，看起來都像是普通的三維流形，但是這個性質在緊束的點就失效了。帕瑞爾曼想出了解決這個障礙的辦法。
　　Another possible example arose when a thin rod stuck out from the manifold; the Ricci flow might produce a trouble called a cigar singularity. When a manifold is pinched in this way, it is called singular—it is no longer a true three-dimensional manifold. In a true three-dimensional manifold, a small region around any point looks like a small region of ordinary three-dimensional space, but this property fails at pinched points. A way around this stumbling block had to wait for Perelman.


躲起來苦幹的帕瑞爾曼

　　帕瑞爾曼在1992年以博士後研究員的身份到了美國，分別在紐約大學與石溪從事研究，然後再到加州大學 柏克萊分校待了兩年。他很快就闖出名號，成為一顆閃亮新星，在某一幾何領域中，證明出很多重要且深刻的結果。歐洲數學學會頒了個獎給他（他婉拒了），國際 數學家大會邀請他發表一場演講（他接受了），能獲得這項邀請是很高的榮譽。1995年春天，一些頂尖的數學系邀請他去任教，但是他都拒絕了，選擇回到家鄉 聖彼得堡。「文化上，他是非常俄羅斯的，」一位美國同事如此評論，「他非常不注重物質生活。」
　　Perelman came to the U.S. as a postdoctoral student in 1992, spending semesters at New York University and Stony Brook, followed by two years at the University of California at Berkeley. He quickly made a name for himself as a brilliant young star, proving many important and deep results in a particular branch of geometry. He was awarded a prize from the European Mathematical Society (which he declined) and received a prestigious invitation to address the International Congress of Mathematicians (which he accepted). In spring 1995 he was offered positions at a number of outstanding mathematics departments, but he turned them all down to return to his home in St. Petersburg. 「Culturally, he is very Russian,」 commented one American colleague. 「He's very unmaterialistic.」

　　回到聖彼得堡，他基本上就從數學家的雷達螢幕消失。多年後，他還在活動的唯一跡像是，他偶爾會傳送電子信件給以前的同事，例如說指出他們公佈於網路上論文中的錯誤；至於詢問他在做什麼的電子信件，則都得不到答覆。
　　Back in St. Petersburg, he essentially disappeared from mathematicians' radar screens. The only signs of activity, after many years, were rare occasions when he e-mailed former colleagues, for example, to point out errors in papers they had posted on the Internet. E-mails inquiring about his pursuits went unanswered.

　 　最後在2002年底，幾個人收到了他的電子信件，提醒他們他公佈於數學網路上的論文；這些信和往常一樣簡短，只說他們可能會對這篇文章有點興趣。這個輕 描淡寫的聲明，預告了他第一階段對於龐卡赫猜想的研究。在這篇文章之中，他除了聲明其工作單位為斯特克洛夫數學研究所，還說明他受益於之前從美國博士後研 究員薪水所省下來的積蓄。
　　Finally, in late 2002 several people received an e-mail from him alerting them to the paper he had posted on the mathematics server—just a characteristically brief note saying they might find it of interest. That understatement heralded the first stage of his attack on the Poincaré conjecture. In the preprint, along with his Steklov Institute affiliation, Perelman acknowledged support in the form of money he had saved from his U.S. postdoctoral positions.

　　帕瑞爾曼在論文中多補了一項到瑞奇流方程式裡。修改後的方程式並沒有除去奇異點的麻煩，但是卻能讓帕瑞爾曼的分析進 展得更遠。對於啞鈴的奇異點而言，他證明可以施展一項「手術」：把緊束點兩邊的細管子剪斷，然後在每個啞鈴細管子的開口上蓋一頂小圓帽。如此一來，對於手 術後的流形來說，瑞奇流就可以繼續進行，直到遇上下一次緊束，那時就再次重複同樣的步驟。他也證明了雪茄奇異點不可能發生。用這種方法，任何三維流形都可 以化約成許多小部份的組合，每一部份都有均勻的幾何。
　　In his paper, Perelman added a new term to the Ricci flow equation. The modified equation did not eliminate the troubles with singularities, but it enabled Perelman to carry the analysis much further. With the dumbbell singularities he showed that 「surgery」 could be performed: Snip the thin tube on each side of the incipient pinch and seal off the open tube on each dumbbell ball with a spherical cap. Then the Ricci flow could be continued with the surgically altered manifold until the next pinch, for which the same procedure could be applied. He also showed that cigar singularities could not occur. In this way, any 3-manifold could be reduced to a collection of pieces, each having a uniform geometry.

　　當我們把瑞奇流和手術應用到所有三維流形上時，任何與三維球一樣「簡單」（精準點說，就是和三維球有相同的同倫）的流形，最終都必須和三維球有一樣的均勻幾何。這個結果意味著就拓撲而言，所研究的流形就是三維球。換句話說，三維球是獨一無二的。
　　When the Ricci flow and the surgery are applied to all possible 3-manifolds, any manifold that is as 「simple」 as a 3-sphere (technically, that has the same homotopy as a 3-sphere) necessarily ends up with the same uniform geometry as a 3-sphere. That result means that topologically, the manifold in question is a 3-sphere. Rephrasing that, the 3-sphere is unique.

　 　除了證明了龐卡赫猜想，帕瑞爾曼的研究還有技術上的重要性，因為他引入了新的分析技巧；數學家已經在發表奠基於他的結果、或是將他這些新技巧應用於其他 問題的論文。除此之外，數學和物理還有奇怪的關聯──漢密爾頓與帕瑞爾曼所使用的瑞奇流，和某種稱為「重整化群」的物理概念有關；重整化群告訴我們，交互 作用的強度如何隨著碰撞的能量改變而有所不同。舉例來說，電磁交互作用的強度在低能量時，大約是0.0073（約1/137），然而兩個電子如果以近乎光 速迎頭對撞，則強度就比較接近0.0078。
　　 Beyond proving Poincaré's conjecture, Perelman's research is important for the innovative techniques of analysis it has introduced. Already mathematicians are posting papers that build on his work or apply his techniques to other problems. In addition, the mathematics has curious connections to physics. The Ricci flow used by Hamilton and Perelman is related to something called the renormalization group, which specifies how interactions change in strength depending on the energy of a collision. For example, at low energies the electromagnetic interaction has a strength characterized by the number 0.0073 (about 1/137). If two electrons collide head-on at nearly the speed of light, however, the strength is closer to 0.0078.

　　增加碰撞能量等於在研究更小尺度下的作用力。因此重整化群就好像顯微鏡，其放大倍率可以調大或調 小，以研究各種尺度下的物理過程。同樣地，瑞奇流就像是顯微鏡，可以用所選擇的放大倍率去觀看流形。在某個尺度看得到的山丘或凹坑，在另一個尺度之下就不 見了。物理學家預期在大約10^-35公尺，也就是普朗克長度（Planck length）的尺度下，我們所在的空間看起來會很不一樣，就好像有很多迴圈與把手的「泡沫」，以及其他拓撲結構（參見2004年2月號〈時空原子〉）。描述物理交互作用強度如何改變的數學，很類似於描述流形幾何化的數學。
　　Increasing the collision energy is equivalent to studying the force at a shorter distance scale. The renormalization group is therefore like a microscope with a magnification that can be turned up or down to examine a process at finer or coarser detail. Similarly, the Ricci flow is like a microscope for looking at a manifold at a chosen magnification. Bumps and hollows visible at one magnification disappear at another. Physicists expect that on a scale of about 10^-35 meter, or the Planck length, the space in which we live will look very different—like a 「foam」 with many loops and handles and other topological structures [see 「Atoms of Space and Time,」 by Lee Smolin; Scientific American, January]. The mathematics that describes how the physical forces change is very similar to that which describes geometrization of a manifold.

　 　另外，還有一個和物理的關聯：廣義相對論方程式（這是描述重力現象與宇宙大尺度結構的方程式）和瑞奇流方程式密切相關。除此之外，帕瑞爾曼所加到漢密爾 頓瑞奇流方程式中的項，也出現在弦論（一種重力的量子理論）之中。我們還不知道，他的技巧是否會對於廣義相對論或弦論有所幫助；如果有幫助，帕瑞爾曼所教 導我們的，就不僅和抽象三維空間的形狀有關，而且和我們所處的這個空間形狀有關。  
 
　　Another connection to physics is that the equations of general relativity, which describe the workings of gravity and the large-scale structure of the universe, are closely related to the Ricci flow equation. Furthermore, the term that Perelman added to the basic flow used by Hamilton arises in string theory, which is a quantum theory of gravity. It remains to be seen if his techniques will reveal interesting new information about general relativity or string theory. If that is the case, Perelman will have taught us not only about the shapes of abstract 3-spaces but also about the shape of the particular space in which we live.

	1.The Theory Formerly Known as Strings. Michael J. Duff in Scientific American; February 1998.
	2.Ripples in Spacetime. W. Wayt Gibbs in Scientific American; April 2002
	3.The Poincaré Conjecture 99 Years Later: A Progress Report. John W. Milnor. February 2003. 線上閱讀：www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf
	4.Jules Henri Poincaré (biography). October 2003. 線上閱讀：www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html Millennium Problems. 克萊數學研究所：www.claymath.org/millennium/
	5.Notes and commentary on Perelman's Ricci flow papers. Compiled by Bruce Kleiner and John Lott. 線上閱讀：www.math.lsa.umich.edu/research/ricciflow/perelman.html
	6.Topology. Eric W. Weisstein in Mathworld—A Wolfram Web Resource. 線上閱讀：mathworld.wolfram.com/Topology.html