Thouless and the final winner, Duncan Haldane, are both also using topology to pave the way for a new kind of material: topological insulators.

索利斯及最后一位的获奖者邓肯·霍尔丹都用拓学为一种新物质——拓绝缘开辟了道路。

Except, instead of rock and roll music, my concert was a lecture on black hole topology.

除了我排队的不是摇滚乐，我的演唱会是黑洞拓学的讲座。

This involved lots of weird quantum phenomena, plus a branch of math called topology, which looks at how objects can be arranged and manipulated.

这包括许多奇特的量子现象与数学中一个称为拓学的分支，拓学关注的是物可如何被排列与操纵。

You know, these concepts like topology and geometry, complicated, how do you do?

你知道，诸如拓学和几何学这样的概念，复杂难懂，该如何理解呢？

So that's the concept of retinotopy or the map in visual cortex.

这就是视网膜拓学或视觉皮层中的地图的概念。

This method can be super pretty, since there is a lot of color involved, and the core underlying idea applies to all sorts of math beyond this algorithm for solving equations, including a bit of topology, which I'll talk about afterwards.

这种方法可以非常漂亮， 因为涉及了大量的色彩，而且其核心理念不仅适用于这个解方程的算法， 还涵盖了各种数学领域，包括一些拓学的内容， 我稍后会进行讲解。

In topology since one block of buildings surrounded by roads is considered equivalent, settlements with a wide range of shapes and sizes, from the suburbs of Las Vegas to the slums of Mumbai, can be analyzed in the same way.

在拓学中，由于被道路包围的一栋建筑物被认为是等效的，从拉斯维加斯郊区孟买贫民窟的各种形状和大小的定居点，可以以相同的方式进行分析。

I'm not sure exactly what you might have had in mind there, maybe like topology or analysis or something.

我不确定你当时具了什么，可能是拓学或分析学之类的。

A donut and a coffee cup look the same to a topologist because they both have one hole.

对于拓学家来说，甜甜圈和咖啡杯看起来是一样的，因为它们都有一个洞。

So it's like the number theory, geometry, So like differential geometry, topology.

所以就像数论、几何，就像微分几何、拓学。

Topologists do this when the count the holes in a surface, physicists do this in defining the ideas of energy and momentum, or in special relativity when they define proper time.

拓学家在计算表面的孔洞数量时会这样做，物理学家在定义能量和动量的概念时也会这样做，或者在狭义相对论中定义固有时间时也是如此。

For a long time, it wasn't clear whether topology was useful for describing the behaviors of subatomic particles.

长期以来，人们并不清楚拓学是否适用于描述亚原子粒子的行为。

Now, scientists are trying to transform slums with a mathematical approach to urban design called " topology" .

现在，科学家们正试图通过一种称为“拓学”的数学方法来改造贫民窟。

For me, so what I'm saying is, I get a little bit of the joy of kind of learning something purely as a scientist, something completely outside of what I do, mathematics.

对我来说，我的意思是，我能感受一种纯粹作为科学家去学习某些完全超出我专业领域——数学之外的东西的乐趣。比如他感兴趣的规范理论，那种非常高级的几何拓学。

So that's the puzzle, remember it, and now let's take a seemingly unrelated sidestep to the total opposite side of the math universe: Topology.

所以这就是谜题， 记住它， 现在让我们来一个看似无关的转折， 转向数学宇宙的另一端：拓学。

First of all, topology is a branch of mathematics that focuses on fundamental properties of objects.

首先，拓学是数学的一个分支，专注于研究物的基础性质。

Gauge theory, the kind of very advanced geometry topology that he's interested in.

规范理论，即他感兴趣的那种高度复杂的几何拓学。

It also underlies the video on the Borsuk-Ulam theorem being used to solve What was basically a counting puzzle by using topological facts about spheres?

它还支撑了一个关于使用博苏克-乌拉姆定理来解决基本上是一个计数谜题的视频，这个谜题是通过球的拓学事实来解答的。

But when the topology gods close a door, they open a window, because the Borsuk-Ulam theorem guarantees that no matter what, there must be some other antipodal pair that now land on each other.

但当拓学的神关闭一扇门时，他们会打开一扇窗， 因为博苏克-乌拉姆定理保证无论怎样，总会有一对其他的对跖点现在会重合。

The more I talk to topologists, the more they seem to not operate at all in the visual space.

我越与拓学家交谈，越发现他们似乎完全不在视觉空间中操作。