Riemannian Geometry in Egison

v1 Oct.28.2017

Satoshi Egi

Rakuten Institute of Technology

Rakuten, Inc.

3

My aim is to create a language that can represent directly all algorithms that can be

discovered.

Currently, my biggest challenge is to improve Egison in order to represent directly

calculations that appear in mathematical physics.

4

Pattern-matching against the wider range of data types.

Customizable symbolic computation using Egison pattern-matching.

Tensor index notation in programming.

5

https://www.egison.org/math


6



7

The theory for investigating a curved space.

It is easy for us in the 3 dimensional space to recognize a 2 dimensional curved

surface is curved.

How about our world? Is it curved for who can recognize the higher dimensional

space?

https://commons.wikimedia.org/wiki/File:Trian

gles_(spherical_geometry).jpg

8

The theory explains the gravity as the curve of 4 dimensional time-space.

https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg

9

In our recognition, the space-time is flat.

10

But, according to the theory, the space-time is curved.

11

And, things move straight in this space-time.

12

In our recognition, it looks like things are falling.

13

Formulas in Riemannian geometry are represented with partial derivative

operator and tensor index notation.

We can represent both of them concisely in Egison!

14

Egison program that represents the above formula

Formula of Riemann curvature tensor~: superscript_: subscript

15




16




17

Pattern Body

Matcher

Target

18

are patterns.

19




21

We can apply Egison pattern-matching

against math expressions.

Math expressions are a multiset of terms.

Terms are a multiset of factors.

Therefore, Egison pattern-matching is

very useful to handle them.

22




23

Tensor index notation is a popular notation in the field of mathematics and physics.

24

We can control the way for multiplying vectors by indices.

25

We can use tensor index notation to multiply any order of tensors.

The tensor index notation is necessary to represent the multiplication of order

tensors higher than matrices.

26

In Egison method, we can apply directly both “∂/∂” and “.” functions to tensors.


Formula of Riemann curvature tensor~: superscript_: subscript

27

Wolfram program that represents the above formula


Formula of Riemann curvature tensor

28https://arxiv.org/abs/1702.06343

30

https://commons.wikimedia.org/wiki/File:T

riangles_(spherical_geometry).jpg

37

https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg

39

Currently, I am working to represent differential forms, exterior derivative, and

Hodge operator directly in Egison.

If we realize that, there are the wide range of application, e.g. mathematics,

physics, and computer simulation.

Riemannian Geometry in Egison

Technology

Transcript of Riemannian Geometry in Egison