Natural Numbers **(N) = {1,2,3, 4…}**

1-dimension: a Line

2-dimension: a plane

n-dimensional flat space: a Vector Space

Now imagine in a world where we replace every natural number by vector space:

1 by a Line

2 by a Plane

n by a *flat space* Vector Space

Sum of numbers = Direct sum of vector space.

E.g. Add a 1-D Line to a 2-D Plane = 3-D Space

Product of numbers = **Tensor Product** (of two vector spaces of respective dimension **m** & **n**) with dimension **m.n**

This new world would be much interesting and richer than the Natural Number world: vector spaces have symmetries, whereas numbers are just numbers with no symmetries.

**(Interesting):** we can add 1 to 2 in only ONE way (1+2), but there are many ways to embed (add) a Line in a Plane (perpendicular, slanting in any angle, etc).

**(Richer):** the Lie Group SO(3) is the rotation of a 3-dimensional space. The number 3 is just an attribute of its dimensionality.

Numbers forms a **Set**

Vector Space forms a **Category**

Category not only contains *‘objects’* the vector spaces, also **‘morphisms’** from any object to any other object.

Note 1: Morphism of an object to itself = symmetries

Note 2: The Functional Language: **Haskel** is the next generation of computer languages based on Category, rather than on Set theory.

**Categorification**: The paradigm shift in modern math, by creating a new world in which the familiar concepts are *elevated* to a higher level — **欲穷千里路, 更上一层楼**。( 6th CE Tang Dynasty Poem: “To see further thousand miles away, climb up one level higher”)

Examples: from sets to categories:

Categorification (1)

Function (sine) to 30 degrees: eg.

The **function** ‘sine’ assigns to a point of the circle (manifold) a **number 1/2.**

A **Sheaf** assigns to a point in the manifold a vector space.

Categorification (2)

Categorification (3)

**Functions** were the concepts of *archaic* math, and **sheaves** are the concepts of **modern math.** Sheaf can have symmetries. By elevating a function to a sheaf, we can explore these symmetries to learn more than we could ever learn using functions.

Sheaves applied in the right column of Langrands Program:

**Number Theory ** | **Curves over Finite Fields ** | **Riemann Surfaces **

The 2 rows (L & R) below each column contains the names of the objects on the 2 sides of the Langrands relation specific to that column.

**Love and Math by Edward Frenkel** http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY

Reblogged this on Singapore Maths Tuition.

Alexander Grothendieck 法国犹太人。父亲参加西班牙革命牺牲, 童年时母亲和他住监牢。读Strasbourg大学的(博士)导师是中国数学家 吴文俊 (百岁老人, 邵逸夫数学奖, 发明Machine-Proving Geometry)。

Grothendieck 得Field Medal, 发明 Algebraic Topology 的Sheaf (束)。成名后不满法国高等研究院有军方背景, 离职隐居在法/西边界的Pyranée(牛庇山)中20多年, 高龄86岁于2014 去世。