内容来源：https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/

内容总结：

新数学理论复兴几何学最古老难题

几何学中一个历史悠久的难题类型——枚举几何，正因一项名为“ motivic 同伦理论”的现代数学工具而重新焕发活力。该理论有望系统解决那些在不同数系中答案飘忽不定的经典计数问题。

枚举几何问题源自古希腊，其核心是计算满足特定几何条件的对象数量，例如“一个三次曲面包含多少条直线？”（答案为27条）。然而，当数学家试图在实数、整数或其他非复数数系中寻求答案时，解的数量会因几何构型的变化而改变，缺乏统一的求解框架，导致这一领域在20世纪中叶后逐渐沉寂。

转机出现在数学家杰西·卡斯和柯尔斯滕·维克尔格伦的合作中。他们发现，1977年的一项研究中隐含的“二次型”工具，其思想与前沿的motivic同伦理论不谋而合。基于此洞察，他们开创性地将枚举几何问题转化为对特定“空间”的研究，并最终导出一个二次型。

这一方法的强大之处在于其普适性：对于复数域，该二次型能直接给出精确解的数量；对于实数域，它能计算出解数量的下限；而对于更抽象的有限数系（如“钟表算术”），它也能揭示解集的某些结构性信息。2017年，他们成功应用该方法，统一处理了关于三次曲面27条线的著名定理在不同数系下的情况，首次实现了“一法解万题”。

目前，该理论已成为一个活跃的研究方向，吸引了众多年轻数学家的加入。它不仅复兴了经典的枚举几何，将其与代数、拓扑等领域紧密相连，更重要的是，它为探索不同数系本身的深层结构提供了一个前所未有的窗口。正如维克尔格伦所言，理解“一张纸上有多少有理曲线”这类问题，是触及数学现实根本的一部分。尽管理论输出的二次型中仍有许多信息有待破译，但这片未知领域正激励着新一代学者将计数问题带入21世纪。

中文翻译：

新数学方法复兴几何学最古老难题

引言
公元前三世纪，佩尔加的阿波罗尼奥斯提出：给定三个圆，能画出多少个圆与每个给定圆恰好相切于一点？这个问题花了1800年才得以证明答案：八个。

这类求解满足一组几何条件的解的数量问题，是古希腊人钟爱的课题。几千年来，它们也一直吸引着数学家们。一个三次曲面包含多少条直线？一个五次曲面包含多少条二次曲线？（答案分别是27条和609,250条。）"这些都是真正困难的问题，只不过表述起来容易理解，"伊利诺伊大学厄巴纳-香槟分校的数学家谢尔登·卡茨说。

随着数学的发展，数学家们想要计数的对象变得越来越复杂。这逐渐发展成为一个独立的研究领域，称为"计数几何"。

数学家们似乎能提出无穷无尽的计数几何问题。但到了20世纪中叶，数学家们开始对此失去兴趣。几何学家们超越了具体的计数问题，转而关注更一般的抽象概念和更深层的真理。除了在20世纪90年代有过短暂的复兴，计数几何似乎已被永久搁置。

现在，这种情况可能正开始改变。一小批数学家已经找到了如何将一项已有数十年历史的理论应用于计数问题。研究人员不仅为原始问题提供解答，还为这些问题在无穷多个奇异数系中的变体提供解答。"如果你做成一件事，那很了不起，"斯坦福大学的数学家拉维·瓦基尔说。"如果你能一而再、再而三地做成，那就成了一种理论。"

这一理论帮助复兴了计数几何领域，并将其与代数、拓扑和数论等其他几个研究领域联系起来——赋予了它新的深度和吸引力。这项工作也让数学家们对各种重要的数系获得了新的见解，远远超出了他们最熟悉的那些数系。

与此同时，这些结果引发的新问题与它们解答的问题一样多。该理论能输出数学家们所寻求的数字，但也提供了他们难以解读的额外信息。

这种神秘感激励着新一代人才投身其中。他们正共同将计数问题带入21世纪。

向前计数
所有计数几何问题本质上都归结为计算空间中的对象数量。但即使是最简单的例子也可能很快变得复杂。

设想纸上两个相距一定距离的圆。你能画出多少条与每个圆恰好相切一次的直线？答案是四条：

你可以将这些圆分得更开，或者将一个圆缩小一半，答案不会改变。但如果移动一个圆使其像维恩图那样与另一个圆相交，答案就会突然改变——从四条变为两条。将较小的那个圆完全滑入较大的圆内部，答案就变成了零：你无法画出任何与每个圆只相切一次的直线。

这种不一致性确实令人头疼。在这个例子中，只需考虑三种不同的构型，但通常问题过于复杂，研究人员无法穷尽所有可能的情况。你可能得到了某种情况下的答案，但当移动物体时，你完全不知道答案会如何变化。

实践中，数学家们试图将问题的几何约束写成一方程组，然后找出有多少个解能同时满足所有这些方程。但尽管他们知道解的数量并非总是一致，他们所写出的方程本身却无法表明是否偶然遇到了会产生不同答案的新构型。

有一个例外——当问题是在复数域中定义时。复数由两部分组成："实部"（一个普通数字）和"虚部"（一个普通数字乘以-1的平方根，数学家称之为i）。

在上面圆和直线的例子中，如果你询问方程的复数解的数量，无论你观察何种圆的位置排列，答案总是四个。

到1900年左右，数学家们已经发展出技术来解决复数域中的任何计数几何问题。这些技术无需考虑不同的构型：无论数学家得到什么答案，他们都知道该答案对每种构型都必然成立。

但是，当数学家们只想找到计数几何问题中方程的实数解数量，或整数解数量时，这些方法就不再有效了。如果他们在复数以外的任何数系中提出计数几何问题，不一致性会再次出现。在这些其他数系中，数学家无法系统地处理计数问题。

与此同时，当数学家们将自身局限于整数或实数时，所遇到的这种神秘且变化的答案，使得计数问题成为探究这些其他数系的绝佳途径——以便更好地理解它们之间的差异以及存在于其中的对象。数学家们认为，开发处理这些场景的方法将开辟出新的、更深入的数学领域。

数学巨匠大卫·希尔伯特就是其中之一。当他列出他认为20世纪最重要的未解决问题时，其中就包括一个关于使解决计数几何问题的技术更加严谨的问题。

20世纪60年代和70年代，亚历山大·格罗滕迪克及其后继者发展了一套新颖的概念工具，帮助解决了希尔伯特的问题，并为现代代数几何领域奠定了基础。随着数学家们追求理解这些抽象到非专业人士仍难以企及的概念时，他们最终将计数几何抛在了后面。与此同时，对于其他数系中的计数几何问题，"我们的技术碰壁了，"卡茨说。计数几何从未成为希尔伯特所设想的那座灯塔；反而是其他的研究线索照亮了数学家们的道路。

计数几何不再让人觉得是一个核心、活跃的研究领域。卡茨回忆说，在20世纪80年代，作为一名年轻教授，他曾被警告不要涉足这个领域，"因为这对我职业生涯没好处。"

但几年后，弦理论的发展暂时让计数几何获得了第二次生命。弦理论中的许多问题都可以用计数来表述：弦理论家希望找到某种特定类型的不同曲线的数量，这些曲线代表弦的运动——弦是他们认为构成宇宙基本组成部分的一维物体，存在于十维空间中。计数几何"再次变得非常流行，"卡茨说。

但这只是昙花一现。一旦物理学家回答了他们的疑问，他们就转向了其他问题。数学家们仍然缺乏一个适用于其他数系中计数几何问题的通用框架，并且对此兴趣寥寥。其他领域似乎更具吸引力。

这种情况一直持续到数学家柯尔斯滕·威克尔格伦和杰西·卡斯突然意识到：计数几何或许能提供希尔伯特所期望的那种深刻见解。

鸟瞰视角
卡斯和威克尔格伦在2000年代末相遇，很快成为固定的合作者。他们的性格在许多方面截然不同。威克尔格伦热情但克制、审慎。每当我请她确认是否理解了某个陈述时，她会停顿片刻，然后坚定地回答"Yes, please"——这是她表示"完全正确，你理解了！"的方式。而卡斯则充满紧张的热情。他容易兴奋，语速极快。

但卡斯和威克尔格伦合作得很好，并且有许多共同的兴趣——包括热爱将几何学的范围扩展到其他领域。

2015年，卡斯途经威克尔格伦居住的亚特兰大，决定与她探讨自己最新的着迷点：他想重新审视受限数系中的计数问题，这个长期被放弃的尝试。

他带去了许多松散的想法和似乎相关的旧论文。"我意识到这是一个有点异想天开的项目，"卡斯说。"她非常有礼貌地向我解释，我所有的答案都是无稽之谈。"然后他提到了1977年的一项成果，突然"灵光一现"。

在那篇1977年的论文中，数学家哈罗德·莱文和大卫·艾森巴德正在研究一个涉及计数的证明。他们最终得到了一种特殊的表达式，称为"二次型"——一种简单的多项式，其中每一项的指数总和总是2，例如x² + y²，或z² - x² + 3yz。

艾森巴德和莱文意识到，他们感兴趣的计数结果就明摆在那里。答案在于该二次型的"符号差"：正项的数量减去负项的数量。（例如，二次型z² - x² + 3yz有两个正项z²和3yz，一个负项x²，因此其符号差是2 - 1 = 1。）

这就是威克尔格伦的灵光时刻。在艾森巴德和莱文发表他们的证明之后的几十年里，数学家们设计了一个看似不相关的框架，称为" motive 同伦理论"。该框架将方程的解视为特殊的数学空间，并研究它们之间的关系，既精密又强大。除其他用途外，它还让数学家能够使用特定类型的二次型来描述这些关系。

听着卡斯的讲述，威克尔格伦立刻意识到艾森巴德和莱文得到的正是这样一种形式。这些数学家在不自知的情况下运用了 motive 同伦理论——而该理论给了他们一直在寻求的答案。

尽管艾森巴德和莱文当时研究的并非计数几何问题，但其风格足够相似——毕竟涉及计数——这促使卡斯和威克尔格伦思考。或许他们也可以利用 motive 同伦理论的框架来解决他们自己的计数问题。而且，由于 motive 同伦理论可以广泛应用于任何数系，或许它能解开那些困扰数学家许久的、在其他数系背景下的计数几何问题。

更深层的视角
请记住，通常一个计数几何问题涉及寻找满足一组方程的解的数量。卡斯和威克尔格伦的洞见不在于试图直接求解那些方程——在复数以外的数系中，这很少奏效。相反，两人将给定的计数几何问题（设定在某个特定数系中）改写为关于方程空间和描述这些空间之间关系的函数空间的问题。

通过这种方式重新表述问题后，他们就可以对其应用 motive 同伦理论。这使他们能够计算出一个二次型。现在他们必须弄清楚这个二次型包含了关于他们原始问题的哪些信息。

他们意识到，当在复数域中工作时，他们只需计算所得二次型中不同变量的个数。这个数字就给出了他们计数几何问题的解的数量。当然，这对他们来说并不特别有趣：数学家们已经有了很好的技术来得到这个答案。

于是他们转向其他数系。对于实数域，情况变得稍微复杂一些。一旦他们在实数背景下计算出二次型，他们就必须查看其符号差。而符号差并不给出精确答案：它给出的是答案可能的最小值。也就是说，对于任何涉及实数的计数几何问题，他们有了计算下界的方法——这是一个好的起点。

但最令人兴奋的是，当他们为其他更奇特的数系计算二次型时，也能获取重要信息。以包含七个数字、基于所谓"钟算术"的循环系统为例：在这种系统中，7 + 1 等于 1 而不是 8。在这种系统中，他们将二次型改写为一个称为矩阵的数字阵列。然后他们计算一个称为行列式的量，并证明虽然它不能告诉他们解的总数，但它确实告诉了他们关于具有某些几何特性的解所占比例的信息。

2017年，卡斯和威克尔格伦将这种方法应用于计数几何中最著名的定理之一：一个三次曲面最多包含27条直线。使用他们的新方法，他们证明了在复数域中答案确实是27。他们复制了实数域中已知的下界——并为每个有限数系提供了新的数值信息。所有这些都包含在一个统一的框架中。

这是数学家们首次能够对复数域和实数域之外数系中的计数几何问题说出点实质内容。此外，尽管问题的答案可能因数系和其中形状的构型而异，但数学家们首次找到了一个能够涵盖所有那些潜在不同答案的理论。

"这不仅仅是关于实数或复数，"威克尔格伦说。"它们只是一个适用于任何数系的结果的特例。"

而这仅仅是个开始。

新的开端
自那以后的几年里，威克尔格伦、卡斯和其他人利用 motive 同伦理论重新框架化了大量其他计数问题，在各种数系中推导出相关的二次型。

"所有过去给人整数答案的几何构造，"杜伊斯堡-埃森大学的数学家马克·莱文说（他一直在独立探索相同的想法），"现在你可以把问题输入进去，得到某个东西，它会给你一个二次型作为答案。"

自卡斯和威克尔格伦的原始工作以来，在理解二次型在不同数系中能提供什么信息方面，数学家们已经取得了很大进展。但有时，他们不确定该在二次型中寻找什么。"我们仍然有点搞不清它到底告诉了你什么，"莱文说。还有很多需要解读的地方。

"目前，"南加州大学的阿拉温德·阿索克说，试图从二次型中获取关于计数几何问题的信息"已经成了一个完整的产业。"他补充说，这项工作具体且易于理解，吸引了年轻数学家的关注。"这很令人兴奋，因为学生们可以较快地接触到有实质内容的东西。"

这种具体性在当今抽象的数学图景中是不寻常的。"数学不断向更高层次的抽象发展，有时我感觉自己都不知道在说什么了，"扎布丽娜·保利说，她是威克尔格伦的第一位研究生，现在是德国达姆施塔特工业大学的教授。但这个新的研究领域给了她一种方法，将那种高层次的抽象拉回现实。

威克尔格伦、卡斯、莱文等人最近利用他们的技术重新审视了与弦理论相关的计数问题——但是在新的数系和背景下。

在所有这些案例中，数学家们找到了一种新的方式来探索点、线、圆以及更复杂的对象在不同数值背景下的不同行为。卡斯和威克尔格伦复兴的计数几何版本，为深入理解数本身的结构提供了一个意想不到的窗口。"我很难不被'一张纸上存在多少条有理曲线'这样的问题所吸引，"威克尔格伦说。"那是一张纸的数学现实的一个基本组成部分。"

英文来源：

New Math Revives Geometry’s Oldest Problems
Introduction
In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.
Such questions, which ask for the number of solutions that satisfy a set of geometric conditions, were a favorite of the ancient Greeks. And they’ve continued to entrance mathematicians for millennia. How many lines lie on a cubic surface? How many quadratic curves lie on a quintic surface? (Twenty-seven and 609,250, respectively.) “These are really hard questions that are only easy to understand,” said Sheldon Katz, a mathematician at the University of Illinois, Urbana-Champaign.
As mathematics advanced, the objects that mathematicians wanted to count got more complicated. It became a field of study in its own right, known as enumerative geometry.
There seemed to be no end to the enumerative geometry problems that mathematicians could come up with. But by the middle of the 20th century, mathematicians had started to lose interest. Geometers moved beyond concrete problems about counting, and focused instead on more general abstractions and deeper truths. With the exception of a brief resurgence in the 1990s, enumerative geometry seemed to have been set aside for good.
That may now be starting to change. A small cadre of mathematicians has figured out how to apply a decades-old theory to enumerative questions. The researchers are providing solutions not just to the original problems, but to versions of those problems in infinitely many exotic number systems. “If you do something once, it’s impressive,” said Ravi Vakil, a mathematician at Stanford University. “If you do it again and again, it’s a theory.”
That theory has helped to revive the field of enumerative geometry and to connect it to several other areas of study, including algebra, topology and number theory — imbuing it with fresh depth and allure. The work has also given mathematicians new insights into all sorts of important number systems, far beyond the ones they’re most familiar with.
At the same time, these results are raising just as many questions as they answer. The theory spits out the numbers that mathematicians are seeking, but it also gives additional information that they’re struggling to interpret.
That mystery has inspired a new generation of talent to get involved. Together, they’re bringing counting into the 21st century.
Counting Forward
All enumerative geometry problems essentially come down to counting objects in space. But even the simplest examples can quickly get complicated.
Consider two circles some distance apart on a piece of paper. How many lines can you draw that touch each circle exactly once? The answer is four:
You can slide these circles further apart, or shrink one to half its size, and the answer won’t change. But move one circle so that it intersects the other like a Venn diagram, and suddenly the answer does change — from four to two. Slide whichever circle is smaller entirely inside the bigger one, and now the answer is zero: You can’t draw any lines that touch each circle only once.
Such inconsistencies are a real pain. In this example, there were only three different configurations to consider, but often the problem is too complicated for researchers to work through every possible case. You might find the answer for one case, but you’ll have no idea how it will change when you move things around.
In practice, mathematicians try to write the problem’s geometric constraints as a collection of equations, then figure out how many solutions satisfy all those equations simultaneously. But even though they know that the number of solutions won’t always stay consistent, there’s nothing in the nature of the equations they write down that indicates whether they’ve stumbled on a new configuration that will yield a different answer.
There’s one exception — when the problem is defined in terms of complex numbers. A complex number has two parts: a “real” part, which is an ordinary number, and an “imaginary” part, which is an ordinary number multiplied by the square root of −1 (what mathematicians call i).
In the example above with the circles and lines, if you ask for the number of complex solutions to your equations, you always get four as your answer, no matter what arrangement you look at.
By around 1900, mathematicians had developed techniques to solve any enumerative geometry problem in the complex realm. These techniques didn’t have to take different configurations into account: No matter what answer mathematicians got, they knew it had to be true for every configuration.
But the methods were no longer effective when mathematicians only wanted to find, say, the number of real solutions to the equations in an enumerative geometry problem, or the number of integer solutions. If they asked an enumerative geometry problem in any number system other than the complex one, inconsistencies cropped up again. In these other number systems, mathematicians couldn’t address enumerative questions systematically.
At the same time, the mysterious, shifting answers that mathematicians encountered when they limited themselves to the integers, or to the real numbers, made enumerative questions a great way to probe those other number systems — to better understand the differences between them, and the objects that live inside them. Mathematicians thought that developing methods to deal with these settings would open up new, deeper areas of mathematics.
Among them was the mathematical great David Hilbert. When he penned a list of what he considered the most important open problems of the 20th century, he included one about making the techniques for solving enumerative geometry questions more rigorous.
In the 1960s and ’70s, Alexander Grothendieck and his successors developed novel conceptual tools that helped resolve Hilbert’s problem and set the foundation for the field of modern algebraic geometry. As mathematicians pursued an understanding of those concepts, which are so abstract that they remain impenetrable to nonspecialists, they ended up leaving enumerative geometry behind. Meanwhile, when it came to enumerative geometry problems in other number systems, “our techniques hit a brick wall,” Katz said. Enumerative geometry never became the beacon that Hilbert had imagined; other threads of research illuminated mathematicians’ way instead.
Enumerative geometry no longer felt like a central, lively area of study. Katz recalled that as a young professor in the 1980s, he was warned away from the subject “because it was not going to be good for my career.”
But a few years later, the development of string theory temporarily gave enumerative geometry a second wind. Many problems in string theory could be framed in terms of counting: String theorists wanted to find the number of distinct curves of a certain type, which represented the motion of strings — one-dimensional objects in 10-dimensional space that they believe form the building blocks of the universe. Enumerative geometry “became very much in fashion again,” Katz said.
But it was short-lived. Once physicists answered their questions, they moved on. Mathematicians still lacked a general framework for enumerative geometry problems in other number systems and had little interest in pursuing one. Other fields seemed more approachable.
That was the case until the mathematicians Kirsten Wickelgren and Jesse Kass came to a sudden realization: that enumerative geometry might provide the exact kind of deep insights that Hilbert had hoped for.
A Bird’s-Eye View
Kass and Wickelgren met in the late 2000s and soon became regular collaborators. In many ways their demeanors couldn’t be more different. Wickelgren is warm, but restrained and deliberate. Whenever I asked her to confirm that I’d understood a given statement correctly, she’d pause for a moment, then answer with a firm “Yes, please” — her way of saying “Exactly, you’ve got it!” Kass, on the other hand, is nervously enthusiastic. He’s easily excited and talks at a rapid-fire pace.
But Kass and Wickelgren worked well together and shared many interests — including a love for extending geometry’s reach into other fields.
In 2015, Kass was passing through Atlanta, where Wickelgren lived, and decided to approach her with his latest obsession: He wanted to revisit enumerative questions in restricted number systems, that long-abandoned endeavor.
He brought along a bunch of loose ideas and old papers that seemed relevant. “I realized this was a kind of pie-in-the-sky project,” Kass said. “She very politely explained to me that all my answers were nonsense.” Then he mentioned a result from 1977, and suddenly “a light bulb went off.”
In that 1977 paper, the mathematicians Harold Levine and David Eisenbud were working out a proof that involved counting. They ended up with a special type of expression called a quadratic form — a simple polynomial where each term’s exponents always add up to 2, such as x2 + y2, or z2 − x2 + 3yz.
Eisenbud and Levine realized that the count they were interested in was hidden in plain sight. The answer lay in the form’s “signature”: the number of positive terms minus the number of negative terms. (For example, the quadratic form z2 − x2 + 3yz has two positive terms, z2 and 3yz, and one negative term, x2, so its signature is 2 − 1, or 1.)
This was Wickelgren’s light bulb. In the decades since Eisenbud and Levine had published their proof, mathematicians had devised a seemingly unrelated framework called motivic homotopy theory. That framework, which treated solutions to equations as special mathematical spaces and studied the relationships between them, was both sophisticated and powerful. Among other things, it gave mathematicians a way to describe those relationships using particular kinds of quadratic forms.
Listening to Kass, Wickelgren immediately recognized that Eisenbud and Levine had come up with one of these forms. The mathematicians had been doing motivic homotopy theory without realizing it — and it had given them the answer they’d been seeking.
And while Eisenbud and Levine weren’t working on an enumerative geometry problem, it was similar enough in flavor — it involved counting, after all — that it got Kass and Wickelgren thinking. Perhaps they could solve their own counting problems using the framework of motivic homotopy theory, too. And since motivic homotopy theory could be broadly applied to any number system, perhaps it would unlock the enumerative geometry questions in those settings that had eluded mathematicians for so long.
A Deeper View
Remember that typically, an enumerative geometry problem involves finding the number of solutions that satisfy a collection of equations. Kass and Wickelgren’s insight was not to try to solve those equations directly — it rarely worked in settings other than the complex numbers. Instead, the pair rewrote a given enumerative geometry question (set in a given number system) in terms of spaces of equations and functions that described the relationship between those spaces.
With the problem reformulated in this way, they could apply motivic homotopy theory to it. This allowed them to compute a quadratic form. Now they had to figure out what information that quadratic form contained about their original problem.
When they were working in the complex numbers, they realized, all they had to do was count up the number of different variables in the quadratic form they’d computed. That number gave them the number of solutions to their enumerative geometry problem. Of course, this wasn’t particularly interesting to them: Mathematicians already had good techniques for getting this answer.
So they moved on to other number systems. For the real numbers, it got a little trickier. Once they computed the quadratic form in this setting, they had to look at its signature instead. And the signature didn’t give the precise answer: It gave a minimum for what the answer could be. That is, for any enumerative geometry problem involving real numbers, they had a way to calculate a lower bound — a good starting place.
But most exciting of all was that when they computed a quadratic form for other, stranger number systems, they could also glean important information. Take a looping system of seven numbers that operates on what’s called clock arithmetic: In such a system, 7 + 1 equals 1 instead of 8. In this system, they rewrote their quadratic form as an array of numbers called a matrix. They then calculated a quantity called the determinant and proved that while it didn’t tell them the total number of solutions, it did tell them something about what proportions of those solutions had certain geometric properties.
In 2017, Kass and Wickelgren showcased this for one of enumerative geometry’s most famous theorems: that a cubic surface can contain at most 27 lines. Using their new methods, they showed that indeed, the answer is 27 in the complex numbers. They replicated a known lower bound for the real numbers — and provided new numerical information for every finite number system. It all came in one package.
It was one of the first times mathematicians had been able to say anything significant about enumerative geometry problems for systems outside the complex and real numbers. Moreover, while the problem’s answer might change depending on the number system and the configuration of shapes within it, for the first time mathematicians had found one theory that could encompass all those potential different answers.
“It’s not just about the real numbers or the complex numbers,” Wickelgren said. “They’re just special cases of a result that holds in any number system.”
And that was only the beginning.
A New Start
In the years since, Wickelgren, Kass and others have reframed a host of other enumerative problems using motivic homotopy theory, deriving the relevant quadratic forms in various number systems.
“All the geometric constructions used to give people integer answers,” said Marc Levine, a mathematician at the University of Duisburg-Essen who has been independently exploring the same ideas. “Now you can feed [the problem] in and get something which will give you a quadratic form as an answer.”
Mathematicians have made a lot of progress since Kass and Wickelgren’s original work when it comes to understanding what information a quadratic form can give them in different number systems. Sometimes, though, they’re not sure what to look for in the quadratic form. “We’re still kind of mystified about what exactly it tells you,” Levine said. There’s a lot left to interpret.
“At this point,” said Aravind Asok of the University of Southern California, trying to glean information about enumerative geometry problems from quadratic forms “is an entire industry.” It’s also concrete and accessible, which has attracted the attention of young mathematicians, he added. “It’s exciting because students can get into something with meat sort of quickly.”
Such concreteness is unusual in today’s abstract mathematical landscape. “The math keeps going one level higher in abstraction, and then sometimes I feel like I don’t know what I’m talking about anymore,” said Sabrina Pauli, who was Wickelgren’s first graduate student and is now a professor at the Technical University of Darmstadt in Germany. But this new area of research gives her a way to bring that high level of abstraction back down to earth.
Wickelgren, Kass, Levine and others have recently used their techniques to revisit enumerative questions related to string theory — but in new number systems and settings.
In all these cases, mathematicians have found a new way to explore how points, lines, circles and far more complicated objects act differently in different numerical contexts. Kass and Wickelgren’s revived version of enumerative geometry provides an unlikely window into the very structure of numbers. “It would be hard for me not to be drawn to the question that asks how many rational curves are there on a sheet of paper,” Wickelgren said. “That’s a fundamental part of the mathematical reality of a sheet of paper.”