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爱因斯坦将引力和几何紧密地联系起来了，引力是建立在仿射联络这样一个几何对象的基础上，后者由矢量的平行移动来定义。所以仿射联络决定了局域时空坐标系的相对定向。外尔力图将几何概念应用到电磁力，也就是包括当时所知的全部基本力

爱因斯坦指出，如果外尔理论正确，从原点出发的两条路径，由于标度的连续变化，一般将会有不同大小，因而两座钟快慢会不同，时间将依赖于每个人的历史，那就没有客观规律，也就没有物理学了

坐标不变性保持能动量守恒一样，规范不变性保持了电荷守恒。

大数巧合的猜测和研究的一个副产品，是现代规范场论和标准粒子模型的诞生。

弗里德曼将他的结果寄给爱因斯坦，后者很快认识到自己错了，他的静态宇宙是不切实际的一类宇宙模型。只要对它作扰动，它就会开始膨胀或收缩

哈勃发现，星系越远，它离我们而去的速度也越快，正比于距离。哈勃发现的乃是宇宙的膨胀。这是20世纪最伟大的科学发现之一，它证实了爱因斯坦广义相对论对宇宙所作的预言：宇宙不可能是静态的

在上半世纪，不仅建立了一般拓扑学的基础，还创立了拓扑学中相互关联的四大领域

前7位主要研究高维拓扑，而后5位则研究低维拓扑

首先攻克球面同伦群计算的大难题，证明有限性定理，证明除了两个无穷系列之外，其他同伦群都是有限阿贝尔群

托姆的获奖工作主要是1954年发表的配边理论。配边是流形间的一个等价关系，两个n维流形称为配边，如果它们共同构成一个n＋1维流形的边。流形按配边关系划分成等价类，这些等价类构成一个阿贝尔群Nn

这时他的兴趣转向生物学、语言学和哲学，并建立“语义物理学”

1956年他证明7维球面上存在多种微分结构而引起轰动，由此开创微分拓扑学的新纪元

K理论是第一个重要的广义上同调理论，有广泛应用

到1970年代阿蒂亚启动新一轮研究，即规范理论和拓扑与几何关系，进而导致20世纪最后25年低维拓扑及几何和理论物理如量子场论与弦论的奇妙关系的发现

斯梅尔早期工作是关于流形的浸入问题，特别是他发现了不弄破球面而把里面翻到外面的方法。他最大的成就是证明5维及5维以上的庞加莱猜想

瑟斯顿的主要贡献是闭3维流形的分类。他把3维流形分解为“素”流形的连通和，然后提出一个分类纲领，即每一种素流形都具有8种几何结构的一种，他完成了这个纲领的大部分

他要找出刻画流形的全组拓扑不变量

拓扑不变量由弱到强可分为 三个层次

2维定向闭流形很简单，不变量完全组只有一个整数，即欧拉-庞加莱示性数?字， ?字=2-2 g，其中g代表曲面孔洞数

广义庞加莱猜想有一个更为简单易懂的说法：任何n-1连通的闭流形一定与n维球面同胚。这里n-1连通是单连通的推广

他的工具包括莫尔斯（Morse）理论以及吴文俊的示嵌类理论

他们的工具也不同凡响，唐纳森的工具使用了杨-米尔斯的规范场理论。这一下子把几何拓扑与物理学拉近，并影响其后20多年的历史

迄今与庞加莱猜想有关的成就已有3人获菲尔兹奖。正确地讲，还有第4位获奖者，他就是瑟斯顿（W.Thurston，1946— ）。瑟斯顿在1977年提出一个3维流形分类的几何化方案，一旦方案完成，庞加莱猜想只不过是其中一个推论而已。

声称克服了汉密尔顿纲领中的所有几何的、分析的技术困难。他面对这些有限时间奇点并加以分类，而且用他的手术（surgery）改造过的里奇流代替原来的里奇流，并把汉密尔顿对原有里奇流的估计以及极限情形的处理推广到手术改造过的里奇流上，这样就一举证明瑟斯顿的几何化猜想，自然也证明了椭圆化猜想以及其推论——庞加莱猜想

Although Okounkov's work is very varied indeed, there are two main themes that
occur repeatedly: the
notion of randomness and the idea of permuting, in other words
rearranging, the set of numbers from 1 to n.

He analysed the resulting random surfaces using ideas from permutation
theory and came to a surprising conclusion: when you project the melted part
of the crystal onto two dimensions, you will always see the same distinctive shape which can be encircled by an algebraic curve.

it provides the maths, in particular the geometry, necessary to understand
what physicists call critical phenomena

it "represents one of the
most exciting and fruitful interactions between mathematics and physics in
recent times.

"To rise above oneself and
grasp the world"

This was the first known result on a topological invariant

He tried to describe the 'one-sided' property of the Möbius band in terms of non-orientability.

he first to examine connectivity of surfaces

Riemann introduced Riemann surfaces, determined by the function f(w, z), so that the function w(z) defined by the equation f(w, z) = 0 is single valued on the surfaces.

He called a simple closed curve on a surface which does not intersect itself an irreducible circuit if it cannot be continuously transformed into a point.

Jordan proved that the number of circuits in a complete independent set is a topological invariant of the surface.

Further in the distance, you could see the
same scene again: yourself standing in the middle of the more distant but
otherwise identical copy of the street. Like a hall of mirrors, the
pattern would go on infinitely and in all directions.

This
property is known as non-orientability

These four
surfaces are the only connected surfaces that can be constructed from
a flat rectangle.

We can get rid of the third dimension entirely and visualize a compact
surface using a tiling picture.

Alternatively, we can
visualize the the compact space by gluing together identical copies of the
fundamental cell edge-to-edge according to our simple rules

The octagon cannot tile a flat plane, but it can tile a
surface which is
everywhere negatively curved.

By gluing opposite sides of the octagonal cell we can build the double
torus of fig. 11. The double torus can be deformed into a mug with
two handles

But we can begin first with a volume of fixed
curvature, select a fundamental tile, and apply the rules for gluing
the edges

With the // operator, dividing by zero anywhere in the computation will result in the entire computation returning a special value of the Maybe monad called "Nothing", which indicates a failure to compute a value.

In Haskell, this function is called return due to the way it is used in the do-notation described later. The unit function has the polymorphic type t→M t

a monad, a normal function result, stores function results and side-effect representations.

just the underlying type (represented by wrapping with "Just")

For the safe division example, "(/)" is the underlying function, "(//)" is the safe monadic version.

maps any value x to the expression that follows

A monad can optionally define a "zero" value for every type.

Intuitively, the zero represents a value in the monad that has only monad-related structure and no values from the underlying type. In the Maybe monad, "Nothing" is a zero. In the List monad, "[]" (the empty list) is a zero

concatMap maps a function over a list and concatenates (flattens) the resulting lists

The following definitions for safe division for values in the Maybe monad are also equivalent

In Haskell this is called liftM2

The type signature says: If m is a monad, we can "lift" any binary function into it

The parser can maintain a set that represents the possible syntaxes, and scan until the set has one item (meaning that a syntax was recognized) or no items (meaning that the input is unacceptable)

So the list monad is a simple way to implement a backtracking algorithm in a lazy language

it is also possible to define a monad in terms of "return" and two other operations, "join" and "map". This formulation fits more closely with the definition of monads in category theory.