内容来源：https://www.quantamagazine.org/string-theory-inspires-a-brilliant-baffling-new-math-proof-20251212/

内容总结：

今年8月，一篇数学论文在学界引发震动。由菲尔兹奖得主马克西姆·孔采维奇领衔的研究团队宣称，他们解决了代数几何领域一个悬置半个世纪的核心难题——多项式方程的分类问题。令人惊讶的是，证明过程中使用的并非该领域的传统方法，而是源自弦理论的陌生工具。

多项式方程是数学中最基础且应用广泛的对象，其解集常表现为曲线、曲面等高维几何形状。数学家长期致力于区分两类多项式：一类解集结构简单，可通过“有理参数化”映射到简单空间；另一类则结构复杂，无法实现此类映射，却蕴含更丰富的数学内涵。自20世纪70年代以来，对于涉及五个变量的三次多项式（其解集构成四维流形），学界一直无法判定其是否可参数化。

孔采维奇团队此次突破的关键，在于引入了他数十年来倡导的“同调镜像对称”纲领中的思想。该纲领源于弦理论，旨在建立代数几何与物理学的深刻联系。团队通过计算四维流形上特定曲线的数量，将其相关的霍奇结构分解为更基本的“原子”，并借助京都大学数学家入谷浩史提供的关键公式，最终证明这类四维流形无法被参数化——这意味着它们的解集具有内在的复杂结构。

然而，这项成果在带来兴奋的同时也引发了广泛质疑。由于证明大量依赖弦理论的概念，许多专攻多项式分类的数学家坦言“看不懂”。全球多个数学中心已自发组织研讨班，试图解读这篇“天书般”的论文。米兰大学的斯特拉里教授评价道：“我们或许正在见证未来数学的雏形。”

尽管验证过程可能长达数年，但这项研究已为沉寂已久的领域注入了新的活力。它不仅是多项式分类问题的重大进展，更可能为数学与物理学的交叉开辟前所未有的道路。正如论文合著者卡茨尔科夫所言：“我们完成了这项工作，且确信它是正确的。我愿用余生继续发展这套理论。”

中文翻译：

弦论催生了一项卓越而令人困惑的新数学证明

引言

今年八月，一个数学家团队发表了一篇论文，声称用完全陌生的方法解决了代数几何领域的一个重大问题。这一成果瞬间点燃了整个数学界，既激起了部分数学家的热情，也引发了另一些人的怀疑。

这项成果涉及多项式方程——即由变量的幂次组合构成的方程（如 y = x 或 x² − 3xy = z²）。这些方程是数学中最简单、最普遍的存在之一，如今已成为众多不同研究领域的基础。因此，数学家希望研究它们的解，这些解可以表示为曲线、曲面以及被称为流形的高维物体等几何形状。

数学家想要驾驭的多项式方程类型无穷无尽，但它们都可以归入两个基本类别之一：一类是其解可以通过简单步骤计算得出的方程，另一类是其解具有更丰富、更复杂结构的方程。第二类才是数学的精华所在，也是数学家希望集中精力以取得重大突破的领域。

然而，在将少数几种多项式归入“简单”和“困难”两类后，数学家们便陷入了困境。过去半个世纪以来，即使是看起来相对简单的多项式也一直难以分类。

直到今年夏天，新的证明出现了。它声称要打破僵局，为如何对许多迄今为止似乎完全无法触及的多项式类型进行分类，提供了一个诱人的前景。

问题是，代数几何领域没有人能理解这个证明。至少目前还不能。该证明依赖于从弦论世界引入的思想。对于那些毕生致力于多项式分类的数学家来说，其技术手法是完全陌生的。

一些研究者信任论文作者之一、菲尔兹奖得主马克西姆·孔采维奇的声誉。但孔采维奇也以提出大胆主张而闻名，这让其他人有所保留。世界各地的数学系纷纷成立了阅读小组，试图解读这一突破性成果，以缓解紧张情绪。

这项审查可能需要数年时间。但它也为一个停滞的研究领域重新点燃了希望。同时，它标志着孔采维奇倡导了数十年的一个更宏大的数学计划取得了初步胜利——他希望通过这个计划在代数、几何和物理学之间架起桥梁。

“普遍的观感是，”未参与这项工作的米兰大学数学家保罗·斯特拉里说，“我们可能正在见证未来数学的一角。”

理性方法

对所有多项式进行分类的努力，涉及最古老的数学类型：解方程。例如，要解简单的多项式 y = 2x，你只需要找到满足方程的 x 和 y 的值。这个方程有无穷多解，比如 x = 1, y = 2。当你在坐标平面上绘制所有解时，就得到一条直线。

其他多项式则更难直接求解，它们的解在空间中切割出更复杂、更高维的形状。

但事实证明，对于其中一些方程，存在一种非常简单的方法来找到所有可能的解。你无需将不同的数字分别代入每个变量，而是可以通过用新变量 t 来重写变量，从而一次性获得所有解。

考虑多项式 x² + y² = 1，它定义了一个圆。现在令 x 等于 2t/(1 + t²)，y 等于 (1 − t²)/(1 + t²)。当你将这些新公式代回原方程时，得到 1 = 1，这是一个无论 t 取何值都成立的陈述。这意味着，通过为 t 选择任意实数值，你都能立即得到原多项式的一个解。例如，令 t 等于 1，则得到 x = 2(1)/(1 + (1)²) = 1，y = 0。确实，x = 1, y = 0 是原方程的一个解：(1)² + (0)² = 1。

这种直接表示所有解的方法称为有理参数化。它等价于将原多项式图像（本例中是一个圆）上的每个点映射到一条直线上的唯一点。

任何一次多项式方程（即各项次数最高为 1 的多项式）都可以这样参数化。方程有多少个变量并不重要：它可能有两个变量，也可能有 200 个。一旦变量超过两个，多项式方程的解就会形成复杂的高维形状。但由于多项式仍然可以参数化，因此存在一种方法，可以将高维形状中的每个点映射到具有相同维数的某个特别简单的空间（比如直线）中的点。这反过来又提供了一种直接计算多项式所有解的方法。

类似地，任何二次多项式（各项次数最高为 2）也有有理参数化。

但是，如果方程的次数是 3 或更高，它就不一定能被参数化。这取决于方程有多少个变量。

以典型的三次多项式为例：椭圆曲线，如 y² = x³ + 1，它只有两个变量。“椭圆曲线是辉煌的，是美妙的，但你不可能参数化它们，”布朗大学的布伦丹·哈塞特说。没有简单的 x 和 y 公式能给出椭圆曲线的所有解，因此无法将曲线映射到一条直线上。“如果能做到，它们就不会这么有趣了，”哈塞特说。

相反，椭圆曲线的解具有丰富得多的结构——这种结构几个世纪以来在数论中发挥了至关重要的作用，密码学家也利用它来编码秘密信息。

那么，具有更多变量的三次方程呢？它们是可以参数化的，还是像椭圆曲线那样，其解的结构更有趣？

1866 年，德国数学家阿尔弗雷德·克莱布施证明，具有三个变量（其解形成二维曲面）的三次方程通常是可以参数化的。一个多世纪后，赫伯特·克莱门斯和菲利普·格里菲斯发表了一项里程碑式的证明，他们证明对于大多数具有四个变量的三次方程，情况正好相反。这些方程形成被称为三维流形（three-folds）的三维流形，它们是不可参数化的：它们的解无法映射到一个简单的三维空间。

许多数学家猜测，下一个待分类的多项式——具有五个变量（形成被称为四维流形（four-folds）的四维流形）的三次方程——通常也应该不可参数化。事实上，他们认为多项式在超过某个点之后永远不应该可参数化。但克莱门斯和格里菲斯的技术对四维流形无效。

因此，分类工作沉寂了数十年。

转变一位先知

2019 年夏天，在莫斯科的一次会议上，当马克西姆·孔采维奇起身谈论四维流形的分类时，数学家们感到惊讶。

一方面，孔采维奇以高层次的数学方法而闻名，他更喜欢提出雄心勃勃的猜想并勾勒宏大计划，而将更微妙的细节和正式的证明写作留给他人。他形容自己介于先知和白日梦者之间。

过去三十年来，他一直专注于发展一个名为同调镜像对称的计划，该计划源于弦论。在 20 世纪 80 年代，弦理论家希望计算高维流形上的曲线数量，以回答关于宇宙基本组成部分可能如何行为的问题。为了计算给定流形上的曲线，他们考虑了它的“镜像”——另一个与原始流形非常不同但具有相关性质的流形。特别是，他们发现与镜像相关的代数对象（称为霍奇结构）可以揭示原始流形上的曲线数量。反之亦然：如果你计算镜像上的曲线，你将获得关于原始流形霍奇结构的信息。

1994 年，孔采维奇勾勒了一个计划来解释这种对应关系的根本原因。他的计划还预测，这种对应关系会扩展到与弦论无关的所有类型的流形。

目前，没有人知道如何证明孔采维奇的镜像对称计划。“这将是下个世纪的数学，”他说。但多年来，他在证明方面取得了部分进展，同时也探索了该计划的潜在后果。

2002 年，孔采维奇的一位朋友、迈阿密大学的卢德米尔·卡扎尔科夫假设了这样一个后果：该计划可能与多项式方程的分类有关。

卡扎尔科夫熟悉克莱门斯和格里菲斯 1972 年关于三维流形不可参数化的证明。在那项工作中，两人直接考察了给定三维流形的霍奇结构，并利用它证明了该三维流形无法映射到一个简单的三维空间。但与四维流形相关的霍奇结构过于复杂，无法使用相同的工具进行分析。

卡扎尔科夫的想法是间接地获取四维流形的霍奇结构——通过计算其镜像上特定类型曲线的数量。通常，研究四维流形霍奇结构的数学家不会考虑这类曲线计数：它们只出现在看似无关的数学领域，如弦论。但如果镜像对称计划成立，那么镜像上的曲线数量应该能阐明原始四维流形霍奇结构的特征。

具体来说，卡扎尔科夫希望将镜像的曲线计数分解成若干部分，然后利用镜像对称计划证明，存在一种相应的方法来分解四维流形的霍奇结构。接着，他可以处理这些霍奇结构的“碎片”，而不是整个结构，以证明四维流形无法被参数化。如果其中任何一个“碎片”无法映射到一个简单的四维空间，他就能完成证明。

但这一推理思路依赖于一个假设：孔采维奇的镜像对称计划对于四维流形是成立的。“很明显它应该是成立的，但我没有技术能力去实现它，”卡扎尔科夫说。

不过，他知道有人具备这种能力：孔采维奇本人。

但他的朋友对此不感兴趣。

深入钻研

多年来，卡扎尔科夫试图说服孔采维奇将他在镜像对称方面的研究应用于多项式分类，但都徒劳无功。孔采维奇希望专注于整个计划，而不是这个具体问题。直到 2018 年，两人与宾夕法尼亚大学的托尼·潘特夫合作研究了另一个涉及将霍奇结构和曲线计数分解成部分的问题。这终于说服孔采维奇听取卡扎尔科夫的想法。

卡扎尔科夫再次向他详细阐述了自己的想法。孔采维奇立刻发现了一条卡扎尔科夫长期寻求但从未找到的替代路径：一种从镜像对称中汲取灵感，而无需实际依赖它的方法。“在你思考了多年之后，你看到它在几秒钟内发生，”卡扎尔科夫说，“那是一个壮观的时刻。”

孔采维奇认为，应该有可能利用四维流形自身的曲线计数（而非其镜像的）来分解霍奇结构。他们只需要找出一种方法，将两者以某种方式联系起来，从而得到他们需要的“碎片”。然后，他们就能分别专注于霍奇结构的每个“碎片”（他们称之为“原子”）。

这就是孔采维奇在 2019 年莫斯科会议上向听众阐述的计划。对一些数学家来说，这听起来好像一个严谨的证明即将到来。数学家们通常比较保守，往往要等到绝对确定才会提出新想法。但孔采维奇总是更大胆一些。“他非常开放地分享他的想法，非常有前瞻性，”研究镜像对称的波士顿马萨诸塞大学数学家丹尼尔·庞勒阿诺说。

孔采维奇警告说，还有一个关键要素他们完全不知道如何解决：一个描述当数学家试图将四维流形映射到新空间时，每个“原子”将如何变化的公式。只有掌握了这样的公式，他们才能证明某些“原子”永远无法达到与“简化”后的四维流形相对应的状态。这将意味着四维流形不可参数化，且其解是丰富而复杂的。“但人们不知怎么地以为他说已经完成了，”庞勒阿诺说，他们期待很快能看到证明。

当证明并未如期而至时，一些数学家开始怀疑他是否真的有解决方案。与此同时，当时在法国国家科学研究中心工作的托尼·岳宇加入了团队。孔采维奇说，宇的新鲜见解和严谨的证明风格，被证明对这个项目至关重要。

新冠疫情期间封锁开始时，宇前往法国附近的高等科学研究所拜访孔采维奇。宇回忆说，他们享受着空旷研究所的宁静，在拥有更多黑板的演讲厅里度过了许多时光。

他们通过 Zoom 定期与潘特夫和卡扎尔科夫会面，很快完成了证明的第一部分，精确地弄清了如何利用给定四维流形上的曲线数量将其霍奇结构分解成“原子”。但他们苦苦寻找一个公式来描述这些“原子”随后如何转化。

他们不知道的是，一位曾出席孔采维奇莫斯科讲座的数学家——京都大学的入谷浩——也已经开始追寻这样一个公式。“他被我的猜想迷住了，”孔采维奇说，“我不知道，但他开始研究它了。”

2023 年 7 月，入谷浩证明了一个公式，描述当四维流形被映射到新空间时，“原子”将如何变化。它没有提供孔采维奇及其同事所需的全部信息，但在接下来的两年里，他们找到了如何完善它的方法。然后，他们使用新公式证明，四维流形总是至少有一个“原子”无法被转化以匹配简单的四维空间。四维流形不可参数化。

仍在消化中

当团队在八月发布他们的证明时，许多数学家感到兴奋。这是几十年来分类项目取得的最大进展，并暗示了一种超越四维流形、处理多项式方程分类的新方法。

但其他数学家并不那么确定。距离莫斯科讲座已经过去了六年。孔采维奇最终兑现了他的承诺，还是仍有细节需要填补？

而且，当证明的技术如此完全陌生——属于弦论，而非多项式分类——他们如何能打消疑虑呢？“他们说，‘这是黑魔法，这是什么工具？’”孔采维奇说。

“他们突然带着这种全新的方法出现，使用的工具以前被广泛认为与这个主题无关，”麻省理工学院的柏绍云说，“了解这个问题的人不理解这些工具。”

柏绍云是现在试图弥合这种理解差距的几位数学家之一。过去几个月，他共同组织了一个“阅读研讨会”，由研究生、博士后研究员和教授组成，希望能理解这篇新论文。每周，一位不同的数学家深入研究证明的某个方面，并向小组其他成员展示。

但即使现在，在进行了 11 次每次 90 分钟的会议之后，参与者对于证明的主要细节仍然感到困惑。“这篇论文包含了卓越的原创思想，”柏绍云说，这些思想“需要大量时间来消化”。

类似的阅读小组已在巴黎、北京、韩国等地聚集。“全世界的人现在都在研究同一篇论文，”斯特拉里说，“这是一件特别的事情。”

哈塞特将其比作格里戈里·佩雷尔曼 2003 年对庞加莱猜想的证明，该证明也使用了全新的技术来解决一个著名问题。直到其他数学家用更传统的工具重现了佩雷尔曼的证明后，数学界才真正接受了它。

“会有阻力，”卡扎尔科夫说，“但我们完成了这项工作，我确信它是正确的。”他和孔采维奇也将其视为镜像对称计划的一次重大胜利：虽然他们离证明它并没有更近，但这一结果为它的真实性提供了进一步的证据。

“我很老了，也很累了，”卡扎尔科夫说，“但只要我活着，我就愿意发展这个理论。”

英文来源：

String Theory Inspires a Brilliant, Baffling New Math Proof
Introduction
In August, a team of mathematicians posted a paper claiming to solve a major problem in algebraic geometry — using entirely alien techniques. It instantly captivated the field, stoking excitement in some mathematicians and skepticism in others.
The result deals with polynomial equations, which combine variables raised to powers (like y = x or x2 − 3xy = z2). These equations are some of the simplest and most ubiquitous in mathematics, and today, they’re fundamental to lots of different areas of study. As a result, mathematicians want to study their solutions, which can be represented as geometric shapes like curves, surfaces and higher-dimensional objects called manifolds.
There are infinitely many types of polynomial equations that mathematicians want to tame. But they all fall into one of two basic categories — equations whose solutions can be computed by following a simple recipe, and equations whose solutions have a richer, more complicated structure. The second category is where the mathematical juice is: It’s where mathematicians want to focus their attention to make major advances.
But after sorting just a few types of polynomials into the “easy” and “hard” piles, mathematicians got stuck. For the past half-century, even relatively simple-looking polynomials have resisted classification.
Then this summer, the new proof appeared. It claimed to end the stalemate, offering up a tantalizing vision for how to classify lots of other types of polynomials that have until now seemed completely out of reach.
The problem is that no one in the world of algebraic geometry understands it. At least, not yet. The proof relies on ideas imported from the world of string theory. Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials.
Some researchers trust the reputation of one of the paper’s authors, a Fields medalist named Maxim Kontsevich. But Kontsevich also has a penchant for making audacious claims, giving others pause. Reading groups have sprung up in math departments across the world to decipher the groundbreaking result and relieve the tension.
This review may take years. But it’s also revived hope for an area of study that had stalled. And it marks an early victory for a broader mathematical program that Kontsevich has championed for decades — one that he hopes will build bridges between algebra, geometry and physics.
“The general perception,” said Paolo Stellari, a mathematician at the University of Milan who was not involved in the work, “is that we might be looking at a piece of the mathematics of the future.”
The Rational Approach
The effort to classify all polynomials deals with the oldest kind of math: solving equations. To solve the simple polynomial y = 2x, for instance, you just need to find values of x and y that satisfy the equation. There are infinitely many solutions to this equation, such as x = 1, y = 2. When you graph all the solutions in the coordinate plane, you get a line.
Other polynomials are harder to solve directly, and their solutions cut out more complicated, higher-dimensional shapes in space.
But for some of these equations, it turns out, there’s a really simple way to find every possible solution. Instead of separately plugging different numbers into each variable, you can get all the solutions at once by rewriting the variables in terms of a new variable, t.
Consider the polynomial x2 + y2 = 1, which defines a circle. Now set x equal to 2t/(1 + t2), and y equal to (1 − t2)/(1 + t2). When you plug these new formulas back into your original equation, you get 1 = 1, a statement that’s always true, no matter what t is. This means that by choosing any real-number value for t, you’ll instantly get a solution to the original polynomial. For instance, when you set t equal to 1, you get x = 2(1)/(1 + (1)2) = 1, and y = 0. And indeed, x = 1, y = 0 is a solution to the original equation: (1)2 + (0)2 = 1.
This straightforward way of framing all your solutions is called a rational parameterization. It’s equivalent to mapping every point on the graph of your original polynomial — in this case, a circle — to a unique point on a straight line.
Any degree-1 polynomial equation — that is, any polynomial whose terms are raised to a power of at most 1 — can be parameterized like this. It doesn’t matter how many variables the equation has: It might have two variables, or 200. Once you go beyond two variables, the solutions to your polynomial equation will form complicated higher-dimensional shapes. But because the polynomial can still be parameterized, there’s a way to map every point in your high-dimensional shape to points on a particularly simple space in the same number of dimensions (like the line). This, in turn, gives you a straightforward way to compute all the polynomial’s solutions.
Similarly, any degree-2 polynomial (whose terms are raised to a power of at most 2) has a rational parameterization.
But if an equation’s degree is 3 or more, it can’t always be parameterized. It depends on how many variables the equation has.
Take a typical kind of degree-3 polynomial: elliptic curves, like y2 = x3 + 1, which have only two variables. “Elliptic curves are glorious, they’re wonderful, but you can’t possibly parameterize them,” said Brendan Hassett of Brown University. There’s no simple formula for x and y that gives you all of an elliptic curve’s solutions, so there’s no way to map the curve to a straight line. “If you could, they would not be so much fun,” Hassett said.
Instead, the solutions to an elliptic curve have a far richer structure — one that’s played a vital role in number theory for centuries, and that cryptographers have taken advantage of to encode secret messages.
What about degree-3 equations with more variables, then? Are they parameterizable, or is the structure of their solutions more fun, the way it is for elliptic curves?
In 1866, the German mathematician Alfred Clebsch showed that degree-3 equations with three variables — whose solutions form two-dimensional surfaces — are usually parameterizable. More than a century later, Herbert Clemens and Phillip Griffiths published a monumental proof in which they showed that the opposite is true for most degree-3 equations with four variables. These equations, which form three-dimensional manifolds called three-folds, are not parameterizable: Their solutions can’t be mapped to a simple 3D space.
Many mathematicians suspected that the next polynomial to be classified — degree-3 equations with five variables (forming four-dimensional manifolds known as four-folds) — wouldn’t usually be parameterizable either. In fact, they figured that polynomials should never be parameterizable past a certain point. But Clemens and Griffiths’ techniques didn’t work for four-folds.
And so for decades, the classification effort lay dormant.
Converting a Prophet
Mathematicians were surprised when, at a conference in Moscow in the summer of 2019, Maxim Kontsevich got up to speak about classifying four-folds.
For one thing, Kontsevich is known for taking a high-level approach to mathematics, preferring to pose ambitious conjectures and sketch out broad programs, often leaving the subtler details and formal proof-writing to others. He’s described himself as something between a prophet and a daydreamer.
©IHES/Flann Mérer
For the past three decades, he’s been focused on developing a program called homological mirror symmetry, which has its roots in string theory. In the 1980s, string theorists wanted to count the number of curves on high-dimensional manifolds to answer questions about how the building blocks of the universe might behave. To count the curves on a given manifold, they considered its “mirror image” — another manifold that, though very different from the original, had related properties. In particular, they found that an algebraic object associated to the mirror image, called a Hodge structure, could reveal the number of curves on the original manifold. The reverse was also true: If you count the curves on the mirror image, you’ll get information about the original manifold’s Hodge structure.
In 1994, Kontsevich sketched out a program to explain the underlying reason for this correspondence. His program also predicted that the correspondence extended to all kinds of manifolds beyond those relevant to string theory.
For now, no one knows how to prove Kontsevich’s mirror symmetry program. “It will be next-century mathematics,” he said. But over the years, he’s made partial progress toward a proof — while also exploring the program’s potential consequences.
In 2002, one of Kontsevich’s friends, Ludmil Katzarkov of the University of Miami, hypothesized one such consequence: that the program might be relevant to the classification of polynomial equations.
Katzarkov was familiar with Clemens and Griffiths’ 1972 proof that three-folds aren’t parameterizable. In that work, the pair looked at a given three-fold’s Hodge structure directly. They then used it to show that the three-fold couldn’t be mapped to a simple 3D space. But the Hodge structures associated with four-folds were too complicated to analyze using the same tools.
Katzarkov’s idea was to access the four-fold’s Hodge structure indirectly — by counting how many curves of a particular type lived on its mirror image. Typically, mathematicians studying the Hodge structures of four-folds don’t think about curve counts like these: They only come up in seemingly unrelated areas of math, like string theory. But if the mirror symmetry program is true, then the number of curves on the mirror image should illuminate features of the original four-fold’s Hodge structure.
Natalia Leal
In particular, Katzarkov wanted to break the mirror image’s curve count into pieces, then use the mirror symmetry program to show that there was a corresponding way to break up the four-fold’s Hodge structure. He could then work with these pieces of the Hodge structure, rather than the whole thing, to show that four-folds can’t be parameterized. If any one of the pieces couldn’t be mapped to a simple 4D space, he’d have his proof.
But this line of reasoning depended on the assumption that Kontsevich’s mirror symmetry program was true for four-folds. “It was clear that it should be true, but I didn’t have the technical ability to see how to do it,” Katzarkov said.
He knew someone who did have that ability, though: Kontsevich himself.
But his friend wasn’t interested.
Digging In
For years, Katzarkov tried to convince Kontsevich to apply his research on mirror symmetry to the classification of polynomials — to no avail. Kontsevich wanted to focus on the whole program, not this particular problem. Then in 2018, the pair, along with Tony Pantev of the University of Pennsylvania, worked on another problem that involved breaking Hodge structures and curve counts into pieces. It convinced Kontsevich to hear Katzarkov out.
Katzarkov walked him through his idea again. Immediately, Kontsevich discovered an alternative path that Katzarkov had long sought but never found: a way to draw inspiration from mirror symmetry without actually relying on it. “After you’ve spent years thinking about this, you see it happening in seconds,” Katzarkov said. “That’s a spectacular moment.”
Kontsevich argued that it should be possible to use the four-fold’s own curve counts — rather than those of its mirror image — to break up the Hodge structure. They just had to figure out how to relate the two in a way that gave them the pieces they needed. Then they’d be able to focus on each piece (or “atom,” as they called it) of the Hodge structure separately.
This was the plan Kontsevich laid out for his audience at the 2019 conference in Moscow. To some mathematicians, it sounded as though a rigorous proof was just around the corner. Mathematicians are a conservative bunch and often wait for absolute certainty to present new ideas. But Kontsevich has always been a little bolder. “He’s very open with his ideas, and very forward-thinking,” said Daniel Pomerleano, a mathematician at the University of Massachusetts, Boston, who studies mirror symmetry.
There was a major ingredient they still had no idea how to address, Kontsevich warned: a formula for how each atom would change as mathematicians tried to map the four-fold to new spaces. Only with such a formula in hand could they prove that some atom would never reach a state corresponding to a properly “simplified” four-fold. This would imply that four-folds weren’t parameterizable, and that their solutions were rich and complicated. “But people somehow got the impression that he said it was done,” Pomerleano said, and they expected a proof soon.
When that didn’t come to pass, some mathematicians began to doubt that he had a real solution. In the meantime, Tony Yue Yu, then at the French National Center for Scientific Research, joined the team. Yu’s fresh insights and meticulous style of proof, Kontsevich said, turned out to be crucial to the project.
When lockdowns began during the Covid pandemic, Yu visited Kontsevich at France’s nearby Institute for Advanced Scientific Studies. They relished the quiet of the deserted institute, spending hours in lecture halls where there were more blackboards, Yu recalled.
Meeting regularly with Pantev and Katzarkov over Zoom, they quickly completed the first part of their proof, figuring out precisely how to use the number of curves on a given four-fold to break its Hodge structure into atoms. But they struggled to find a formula to describe how the atoms could then be transformed.
What they didn’t know was that a mathematician who had attended Kontsevich’s lecture in Moscow — Hiroshi Iritani of Kyoto University — had also started pursuing such a formula. “He was enchanted by my conjecture,” Kontsevich said. “I didn’t know, but he started to work on it.”
In July 2023, Iritani proved a formula for how the atoms would change as four-folds were mapped to new spaces. It didn’t give quite as much information as Kontsevich and his colleagues needed, but over the next two years, they figured out how to hone it. They then used their new formula to show that four-folds would always have at least one atom that couldn’t be transformed to match simple 4D space. Four-folds weren’t parameterizable.
Still Processing
When the team posted their proof in August, many mathematicians were excited. It was the biggest advance in the classification project in decades, and hinted at a new way to tackle the classification of polynomial equations well beyond four-folds.
But other mathematicians weren’t so sure. Six years had passed since the lecture in Moscow. Had Kontsevich finally made good on his promise, or were there still details to fill in?
And how could they assuage their doubts, when the proof’s techniques were so completely foreign — the stuff of string theory, not polynomial classification? “They say, ‘This is black magic, what is this machinery?’” Kontsevich said.
“Suddenly they come with this completely new approach, using tools that were previously widely believed to have nothing to do with this subject,” said Shaoyun Bai of the Massachusetts Institute of Technology. “The people who know the problem don’t understand the tools.”
Bai is one of several mathematicians now trying to bridge this gap in understanding. Over the past few months, he has co-organized a “reading seminar” made up of graduate students, postdoctoral researchers and professors who hope to make sense of the new paper. Each week, a different mathematician digs into some aspect of the proof and presents it to the rest of the group.
But even now, after 11 of these 90-minute sessions, the participants still feel lost when it comes to major details of the proof. “The paper contains brilliant original ideas,” Bai said, which “require substantial time to absorb.”
Similar reading groups have been congregating in Paris, Beijing, South Korea and elsewhere. “People all over the globe are working on the same paper right now,” Stellari said. “That’s a special thing.”
Hassett likens it to Grigori Perelman’s 2003 proof of the Poincaré conjecture, which also used entirely new techniques to solve a famous problem. It was only after other mathematicians reproduced Perelman’s proof using more traditional tools that the community truly accepted it.
“There will be resistance,” Katzarkov said, “but we did the work, and I’m sure it’s correct.” He and Kontsevich also see it as a major win for the mirror symmetry program: While they’re not closer to proving it, the result provides further evidence that it’s true.
“I’m very old, and very tired,” Katzarkov said. “But I’m willing to develop this theory as long as I’m alive.”