Audio Edition: New Proofs Probe Soap-Film Singularities

欢迎收听Quanta音频版。

Welcome to the Quanta audio edition.

在每两周一次的节目中，我们会为您带来来自Quanta网站的故事，讲述基础科学和数学领域的最新进展。

In each of these biweekly episodes, we bring you a story direct from the Quanta website about developments in basic science and mathematics.

我是苏珊·瓦拉特。

I'm Susan Vallat.

数学家们突破了研究所谓最小曲面这一长期存在的障碍。

Mathematicians have broken through a long standing barrier in the study of so called minimizing surfaces.

这类曲面可能出现在肥皂膜、细胞、黑洞中。

The sort of thing you might find in soap films, cells, black holes.

接下来请收听。

That's next.

如果你发现了一种天然的基因编辑机制，并由此引发了一场革命，会怎样？

What if your discovery of a natural gene editing mechanism launched a revolution?

如果你的抽象数学证明与现实世界有着意想不到的联系，会怎样？

What if your abstract math proofs had unexpected connections to the real world?

如果你的诺贝尔奖获奖工作关于宇宙膨胀只是故事的开端呢？

What if your Nobel Prize winning work on the expanding universe was just the beginning of the story?

如果你加入我们，史蒂夫·斯特罗加茨和贾娜·莱文，在一个播客里了解所有这些以及更多内容呢？

And what if you joined us, Steve Strogatz And Jana Levin to learn about all of this and much more on one podcast?

请收听《Quanta Magazine》第五季的《The Joy of Why》。

Listen to season five of Quanta Magazine's The Joy of Why.

新剧集从6月11日起每隔一个周四更新。

New episodes drop every other Thursday starting June 11.

在19世纪中期，比利时物理学家约瑟夫·普拉托，他从小就开始设计和进行科学实验，将金属丝环浸入肥皂溶液中，并研究形成的薄膜。

In the mid-nineteenth century, Belgian physicist Joseph Plateau, who had been designing and conducting scientific experiments since he was a child, submerged loops of wire in a soapy solution and studied the films that formed.

当他把金属丝弯成一个圆环时，肥皂膜覆盖在上面，形成了一个扁平的圆盘。

When he bent his wire into a circular ring, a soap film stretched across it, creating a flat disc.

但是当他将两个平行的金属丝环浸入溶液中时，肥皂在它们之间拉伸，形成了一个沙漏形状，数学家称之为悬链面。

But when he dipped two parallel wire rings into the solution, the soap stretched between them to form an hourglass shape instead, what mathematicians call a catenoid.

不同的金属丝框架产生了各种不同的薄膜，有的像马鞍或螺旋坡道，有的则复杂到难以描述。

Different wireframes produced all sorts of different films, some shaped like saddles or spiraling ramps, others so complicated they defied description.

普拉托提出，这些肥皂膜应该总是占据尽可能小的面积。

Plateau posited that these soap films should always take up the smallest area possible.

它们就是数学家所说的面积最小曲面。

They're what mathematicians call area minimizing surfaces.

数学家花了将近一个世纪才证明他是对的。

It would take nearly a century for mathematicians to prove him right.

在二十世纪三十年代初，杰西·道格拉斯和T.

In the early nineteen thirties, Jesse Douglas and T.

博拉多独立证明了普拉托问题的答案是肯定的。

Borrado independently showed that the answer to the plateau problem is yes.

对于三维空间中的任何闭合曲线或你的线框，你总能找到一个以它为边界的面积最小的二维曲面，也就是你的肥皂膜。

For any closed curve or your wireframe in three-dimensional space, you can always find a minimizing two dimensional surface or your soap film that has the same boundary.

这个证明后来为道格拉斯赢得了首届菲尔兹奖，这是数学界最负盛名的奖项。

The proof later earned Douglas the first ever Fields Medal, the most prestigious award in mathematics.

从那以后，数学家们不断扩展普拉托问题，希望能更深入地了解最小曲面。

Since then, mathematicians have expanded on the plateau problem in hopes of learning more about minimizing surfaces.

这些曲面在数学和科学中随处可见，出现在几何和拓扑学重要猜想的证明中，也出现在细胞和黑洞的研究中，甚至用于生物分子的设计。

These surfaces appear throughout math and science, in proofs of important conjectures in geometry and topology, in the study of cells and black holes, and even in the design of biomolecules.

斯坦福大学的奥蒂斯·乔多什说，这些是非常漂亮的研究对象，非常自然、吸引人，又令人着迷。

Otis Chodosh of Stanford University says they're very beautiful objects to study, very natural, appealing, and intriguing.

数学家们现在知道，普拉托的预测在七维及以下都是完全成立的。

Mathematicians now know that Plateau's prediction is categorically true up through dimension seven.

但在更高维度中，有一个需要注意的地方。

But in higher dimensions, there's a caveat.

形成的极小曲面可能并不总是像圆盘或沙漏那样光滑。

The minimizing surfaces that form might not always be nice and smooth, like the disk or hourglass.

相反，它们可能会在某些地方折叠、挤压或自我交叉，形成所谓的奇点。

Instead, they might fold, pinch, or intersect themselves in places, forming what are known as singularities.

当极小曲面出现奇点时，理解和处理它们就变得困难得多。

When minimizing surfaces have singularities, it becomes much harder to understand and work with them.

因此，数学家们想知道这种不光滑的极小曲面有多常见，以及它们可能具有哪些性质。

Mathematicians consequently want to know how common such nonsmooth minimizing surfaces are and what properties they might have.

如果奇点在某个维度中很罕见，只在人为构造的情况下出现，那么只要稍微调整一下你的线框，它们就会消失。

If singularities are rare in a given dimension, appearing only under contrived circumstances, then they'll disappear if you wiggle your wireframe just right.

这样你就会得到一个光滑的极小曲面，更容易研究，从而有机会深入理解该维度中的这类曲面。

You'll be left with a smooth, minimizing surface that you can study more easily, which will give you the chance to develop a thorough understanding of such surfaces in that dimension.

1985年，数学家证明了在八维空间中，奇点确实可以通过微调来消除。

In 1985, mathematicians proved that in eight dimensional space, singularities can indeed be wiggled away.

但在更高维度中，乔多什说情况就完全失控了。

But in higher dimensions, Chodosh says all hell breaks loose.

奇点变得难以分析得多。

The singularities become much more difficult to analyze.

近四十年来，没有人能在这个问题上取得太大进展。

For nearly forty years, no one can make much progress on the problem.

这个障碍终于被打破了。

That barrier has finally been broken.

2023年，乔多什与莱斯大学的克里斯托斯·曼图利迪斯以及华威大学的费利克斯·舒尔策合作，证明了在九维和十维空间中，光滑极小曲面是常态。

In 2023, Chodosz, along with Christos Mantoulidis of Rice University and Felix Schultze of the University of Warwick, showed that in dimensions nine and ten, smooth minimizing surfaces are the norm.

去年，这个团队又加入了康奈尔大学的王继汉，证明了在11维空间中也是如此。

And last year, the team, joined by Jihan Wang of Cornell University, showed that the same is true in dimension 11.

这项工作标志着向理解更高维度中可能出现的奇异最小曲面迈出了重要一步，数学家们现在可以利用这一结果来解决许多长期以来仅限于八维或以下维度的数学问题，使这些定理更加强大。

The work marks a major advance toward understanding the strange kinds of minimizing surfaces that can arise in higher and higher dimensions, and mathematicians can now use the result to resolve a host of other math problems that have long been limited in scope to dimension eight or below, making those theorems even more powerful.

1962年，数学家温德尔·弗莱明证明了所有二维最小曲面，也就是普拉托可能试图研究的任何可能的肥皂膜，都必须是光滑的。

In 1962, mathematician Wendell Fleming proved that all minimizing two dimensional surfaces, any possible soap film that Plateau might have tried to study, must be smooth.

带有奇异点的最小曲面根本不存在。

Minimizing surfaces with singularities simply don't exist.

这两种曲面存在于我们熟悉的三维空间中。

These two surfaces exist in our familiar three-dimensional space.

但是当我们进入更高维度，问题变得更难可视化时，会发生什么呢？

But what happens when we move to higher dimensions where the problem gets harder to visualize?

例如，在四维空间中，我们线框的类比物是一个二维曲面，而普拉托问题要求我们找到填充该曲面的最小体积的三维形状。

For instance, in four dimensions, the analog of our wireframe is a two d surface, and the plateau problem asks us to find the three d shape that fills that surface with the smallest possible volume.

那个形状可能是什么样子呢？

What might that shape look like?

斯坦福大学的布莱恩·怀特说，就他们当时所知，它可能非常糟糕，像分形一样，或者极其不规则。

Brian White of Stanford says for all they knew, it could be very horrible, fractal like, or extremely irregular.

在弗莱明证明之后的几年里，数学家们证明了这种情况在四维、五维、六维或七维中从未发生过。

In the years that followed Fleming's proof, mathematicians showed that this never happens in four, five, six, or seven dimensions.

最小曲面总是光滑的。

Minimizing surfaces are always smooth.

但在1968年，数学家吉姆·西蒙斯在八维空间中构造了一个七维形状，它在一个点上有一个奇点。

But in 1968, mathematician Jim Simons constructed a seven dimensional shape in eight dimensions that had a singularity at just one point.

第二年，数学家们证明了这个形状是一个最小曲面，从而确立了八维空间中的最小曲面确实可能存在奇点。

The following year, mathematicians proved that this shape was a minimizing surface, establishing that minimizing surfaces in eight dimensional space could in fact be singular.

那么问题就变成了，这些奇点到底有多糟糕？

The question then became, just how bad are these singularities, really?

它们是罕见还是常见？

Are they rare or common?

你能通过稍微改变线框的形状，以恰到好处的方式消除它们吗？

And can you get rid of them by changing the shape of your wireframe just a bit in just the right way?

怀特说，如果你想了解一个曲面的性质，奇点会让分析变得困难得多。

White says if you want to figure out things about a surface, singularities make it much harder to analyze.

但如果奇点很少出现，而且你可以轻松地调整它们，得到一个光滑的曲面，那事情就简单多了。

But if singularities arise only rarely and you can easily nudge them away to get a smooth surface, life becomes much easier.

比如，你可以使用微积分的工具。

You can use the tools of calculus, for example.

1985年，罗伯特·哈特和莱昂·西蒙证明了八维空间中的极小曲面具有这个很好的性质，数学家称之为“一般正则性”。

In 1985, Robert Hart and Leon Simon proved that minimizing surfaces in eight dimensions have this nice property, which mathematicians call generic regularity.

但没人能想出如何调整他们的方法，来证明在更高维度中是否也存在这种性质。

But no one could figure out how to adapt their techniques to show whether it exists in higher dimensions.

几十年来，情况一直如此，直到乔多什、曼图利德斯和舒尔茨介入。

That's where things stood for decades until Czodosz, Mantoulides, and Schultze stepped in.

这三位数学家想要探索未知的高维领域，理解其中极小曲面的性质，就像生物学家可能想了解一个新发现岛屿上的动植物一样。

The three mathematicians wanted to explore uncharted higher dimensional realms and understand the nature of their minimizing surfaces, the way a biologist might seek to understand the flora and fauna of a newly discovered island.

于是他们开始尝试，看看能否把这些奇点“挤”走。

And so they set out to see whether they could wiggle these singularities away.

他们一开始重新证明了哈特和西蒙在八维空间中的那个几十年前的结果，这次用的是他们想测试的一种新方法。

They started by reproving Hart and Simon's decades old result in eight dimensions, this time using a different method they hoped to test out.

首先，他们假设了与想要证明的结论相反的情况，即当你用定义曲面的线框稍微扰动时，奇点或单个点总是会存在。

First, they assumed the opposite of what they wanted to show, that when you slightly perturb with a wireframe that defines your surface, a singularity or a single point always persists.

每次扰动都会产生一个新的极小曲面，而这个曲面仍然带有奇点。

Each time you make a perturbation, you get a new minimizing surface that still has a singularity.

然后你可以把这些极小曲面一个个叠起来，这样奇点出现的位置就形成了一条线。

You can then stack all of these minimal surfaces on top of each other so that the points where the singularities occur form a line.

但这是不可能的。

But that's impossible.

1970年，数学家赫伯特·费德勒发现，在n维空间中，极小曲面上的任何奇点的维度最多只能是n减八。

In 1970, mathematician Herbert Federer found that any singularity on a minimizing surface in n dimensional space can have a dimension of at most n minus eight.

这意味着在八维空间中，任何奇点都必须是零维的。

That means that in eight dimensions, any singularity must be zero dimensional.

换句话说，就是一个孤立的点。

In other words, an isolated point.

线是不允许存在的。

Lines aren't allowed.

Czodosz、Mantelides和Schulze将Federer的论证扩展到了八维空间中的曲面堆叠情况。

Czodosz, Mantelides, and Schulze extended Federer's argument to apply to stacks of surfaces in eight dimensions as well.

然而，在他们的证明中，他们却构造出了带有这样一条线的曲面堆叠。

Yet in their proof, they'd produced a stack of surfaces with just such a line.

这个矛盾表明他们最初的假设是错误的，也就是说，你确实可以通过扰动线框来消除奇点。

The contradiction showed that their original assumption was false, meaning that you can perturb the wireframe to get rid of the singularity after all.

现在他们觉得有信心解决九维空间中的问题了。

They now felt ready to tackle the problem in nine dimensions.

他们以同样的方式开始了证明。

They started their proof in the same way.

他们假设了最坏的情况，进行了一系列扰动，最终得到了一个无限堆叠的极小曲面，每个曲面都有奇点。

They assumed the worst, made a series of perturbations, and ended up with an infinite stack of minimizing surfaces that all had singularities.

然后他们引入了一个新工具，叫做分离函数，用来测量这些奇点之间的距离。

They then introduced a new tool called a separation function, which measures the distance between these singularities.

如果任何扰动都无法干扰奇点，那么分离函数应该始终保持很小。

If no perturbation can interfere with the singularity, then the separation function should always stay small.

但这三人能够证明，有时这个函数会变得很大。

But the trio was able to show that sometimes the function could get large.

某些扰动可以使奇点消失。

Some perturbations could make the singularity disappear.

数学家们证明了在九维空间中极小曲面的通用正则性。

The mathematicians had proved generic regularity for minimizing surfaces in dimension nine.

他们能够将同样的论证应用于十维空间。

They were able to use the same argument in dimension 10.

但在十一维空间中，奇点变得更加难以处理。

But in 11 dimensions, the singularities get even harder to deal with.

他们的方法对一种特定的三维奇点不适用。

Their techniques didn't work for a particular kind of three-dimensional singularity.

曼特利德斯说，奇点类型多种多样，任何成功的论证都必须足够广泛，以涵盖所有类型。

Mantelides says there's a zoo of singularity types, and any successful argument must be broad enough to handle all of them.

团队决定与王继汉合作，他对这类奇点有深入的研究。

The team decided to collaborate with Jihan Wang, who had studied this kind of singularity extensively.

他们一起改进了分离函数，使其在这种情况下也能发挥作用。

Together, they honed their separation function to work in this case too.

他们解决了11维中的问题。

They'd solved the problem in dimension 11.

斯坦福大学的布莱恩·怀特表示，他们将我们的理解拓展了几个维度，这真的很了不起，但处理更高维度时，他们可能需要找到不同的方法。

Brian White of Stanford says the fact that they extended our understanding by a few dimensions is really fantastic, but they'll likely have to find a different approach to handle higher dimensions.

舒尔茨说，他们需要新的思路。

Schultz says they need a new ingredient.

与此同时，数学家们期待这一新成果能帮助他们在数学和物理的其他问题上取得进展。

In the meantime, mathematicians expect the new result to help them make progress on other problems in math and physics.

例如，几何和拓扑学中许多关于具有特定曲率性质的形状的存在性和行为的猜想，其证明都依赖于最小曲面的光滑性。

The proofs of many conjectures in geometry and topology about the existence and behavior of shapes with certain curvature properties, for instance, rely on the smoothness of minimizing surfaces.

因此，这些猜想目前只证明了到八维的情况。

As a result, these conjectures have only been proved up to dimension eight.

现在，其中很多都可以推广到九维、十维和十一维。

Now, many of them can be extended to dimensions nine, ten, and 11.

广义相对论中一个重要的结论——正质量定理也是如此。这个定理大致是说，宇宙的总能量必须是正的。

The same is true for an important statement in general relativity called the positive mass theorem, which claims, loosely speaking, that the total energy of the universe must be positive.

在七十年代，Richard Shane 和 Shingtong Yao 利用极小曲面证明了七维及以下的情况。

In the nineteen seventies, Richard Shane and Shingtong Yao used minimizing surfaces to prove this statement in dimension seven and below.

2017年，他们把这个结果推广到了所有维度。

In 2017, they extended their results to all dimensions.

现在，关于普拉托问题的最新进展为在九维、十维和十一维中验证正质量定理提供了新的途径。

Now the latest progress on Plateau's problem offers a new way to confirm the positive mass theorem in dimensions nine, ten, and 11.

White 说，他们提供了另一种更直观的推广方式，不同的证明方法带来了不同的见解。

White says they provide another more intuitive way to do the extension with different proofs giving different insights.

这项工作可能还会带来很多意想不到的后果。

The work could also have plenty of unforeseen consequences.

普拉托问题已经被用来研究各种其他问题，包括一个与冰融化相关的问题。

The plateau problem has been used to study all sorts of other questions, including one related to how ice melts.

数学家们希望这个团队的新方法能帮助他们更深入地理解这些联系。

Mathematicians hope the team's new methods will help deepen their understanding of these connections.

至于普拉托问题本身，现在有两条路可以走。

As for the plateau problem itself, there are now two paths forward.

要么数学家们继续在更高维度上证明一般正则性，要么他们会发现，在维度11以上，奇点就无法再被消除掉了。

Either mathematicians will continue to prove generic regularity in higher and higher dimensions, or they'll discover that beyond dimension 11, it's no longer possible to wiggle singularities away.

舒尔茨说，那也会是一个小小的奇迹，又一个待解之谜。

Schultze says that would be a bit of a miracle too, another mystery to unravel.

迈克尔·肯永戈洛协助制作了本期节目。

Michael Kenyongolo helped with this episode.

我是苏珊·瓦莱特。

I'm Susan Vallet.

想了解更多这个故事，请阅读史蒂夫·纳塔斯的完整文章《新证明探索肥皂膜奇点》，在我们的网站quantamagazine.org上。

For more on this story, read Steve Natas' full article, New Proofs Probe Soap Film Singularities, on our website, quantamagazine.org.

顺便提一下：吉姆·西蒙斯创立了西蒙斯基金会，该基金会也资助了编辑独立的《量子杂志》和这个播客。

Just a quick note: Jim Simons founded the Simons Foundation, which also funds the editorially independent Quanta Magazine and this podcast.

西蒙基金会的活动不会影响我们的报道。

Simon's foundation activities have no influence on our coverage.

每周二收听这个频道，获取Quanta播客。

Check out this feed every Tuesday for the Quanta Podcast.

在那里，主编萨米尔·帕特尔会与我们的作者和编辑讨论更多Quanta最受欢迎、有趣且引人深思的故事。

That's where editor in chief Samir Patel talks to our writers and editors about more of Quanta's most popular, interesting, and thought provoking stories.

来自PRX。

From PRX.