Math, Areas of math, Math concepts, Theorems

I want to overview and understand all of math. I'm taking two approaches.

- I'm documenting and organizing the ways of figuring things out in mathematics.
- I'm creating a map that show how the branches of math unfold to yield its key concepts and results.

- Collect and organize mathematical results.

I'm working on this at my page on the Azimuth project.

Areas of math

- Mathematics Subject Classification revision from 2010 to 2020, no changes foreseen in the 63 areas of 2 digit classification.

Ieškau matematikos pagrindų. Apžvelgiu matematikos sritis ir jas išdėstau pagal tai, kaip viena nuo kitos priklauso.

**Matematikos apžvalga**

- Add time to the diagram
- catalan numbers are related to semantics and to the generating function of the mandelbtot set
- Consider Mathematics, Form and Function by Aleksandrov, Kolmogorov, Lavrentev
- Mathematics: Its Content, Methods and Meaning
- Wikipedia: The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces. The power series defined below is generalized by the idea of a germ. The general theory of analytic continuation and its generalizations are known as sheaf theory.
- Wikipedia: Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.
- Characteristic class? of different kinds are related to the classical linear groups.

Kirby, Christian,

Thank you for your advice!

I ended up making my maps with yEd, which is available for free at http://www.yworks.com It's a well rounded tool.

- There are a variety of ways to import data from Excel spreadsheets. I imported a list of 100 nodes that way.
- But the graphical user interface is also just right for creating notes and edges. I quickly created another 100 nodes and 300 edges.
- And there is a variety of export options including SVG, HTML Image Map and HTML Flash Viewer. I will show some examples of the latter.

Here's a flash viewer of one map of mathematical areas: http://www.ms.lt/derlius/Math/MathWays2/mathways2.html At the bottom I've placed the math areas which are starting points for math such as "logic" and "geometry" seem to be. Then each arrow leads to a type of math that requires a bit more structure or knowledge. At the very top is "number theory" which seems to pull together absolutely every kind of math. You can zoom into the image using the "zoom" scale at top. Then you can move around the image using your browser's scroll bars on the right and on the bottom.

I've colored coded:

- yellow nodes are areas of theoretical math
- orange nodes are areas of applied math
- blue nodes are math structures known for their beauty
- green nodes are math structures that I think would be helpful to be familiar with
- purple nodes are for ways of figuring things out which I'm systematizing (they appear in the second map)

So this is a further development of this earlier map...

The new map has twice as many nodes but it hasn't made things clearer for me. However, the last map was not scalable, which is to say, I couldn't make it any bigger. Whereas this new map I could probably grow to include 10,000 nodes, or simply a node for every math page in Wikipedia. So I can play around with this new map and I think within a year I will find helpful ways of organizing the big picture in math.

Kirby, yes it could be advantageous to put it on a sphere or tetrahedron, etc. However, if there is some deep pattern lurking here, it may be 6 dimensional or 600 dimensional for all we know. So on the one hand a 2-dimensional spherical surface isn't that much of an advance over a 2-dimensional flat surface. In any event, what really matters to me is to come up with a good mental model and then visualize accordingly. For example, it seems there may be a single "starting point" (foundations) and a single "ending point" (number theory). In that case I have a globe with a specified south pole and north pole. Then it's not a sphere where any point could be the axis. Instead, there is only one axis for the globe, and so basically it is a cylinder like the usual map of the world, where, as usual, the right hand side is adjacent to the left hand side. There may be a handful of "starting points", it's still not clear.

I also made a second map based on the system of ways of figuring out which I've uncovered: http://www.ms.lt/sodas/Mintys/MatematikosR%C5%ABmai Here's that system linked together with the areas in math: http://www.ms.lt/derlius/Math/MathWays/math.html

These maps use "organic" layout in yEd, which is most compact. Other possible layouts include: hierarchical, orthogonal, circular, tree, radial, series parallel.

Here is a third map based on the circular view: http://www.ms.lt/derlius/Math/MathWays3/mathways3.html You can zoom in. This view was very helpful for seeing how the nodes group together by subject. There do seem to be some general patterns in terms of content. I tried to pick a node from each group and make a large node so that it would stand out. The groups are I think more arbitrary than they may seem, however. Anyways, this was helpful.

Everybody is welcome to download the data and try it out in yEd. http://www.ms.lt/derlius/Math/MathWays3/mathways3.graphml Perhaps somebody can put it on a 3-D surface.

This circular view also reminds me (superficially?) of circle folding. Bradford, I look forward to folding more circles and sharing how that goes.

Hi Kirby, Joseph and all,

Kirby, thank you for mentioning Synergetics. I will look into that and add it to my map of math areas. http://www.azimuthproject.org/azimuth/show/Andrius+Kulikauskas I hope to work on that tomorrow. Also, I want to highlight some areas that I think relate to my philosophy.

Andrius

Kirby, thank you for encouraging me regarding my map of the big picture. I will keep working on that. I like your idea of adding a time of discovery, thank you! I now made number theory more central...

It's getting messy. I wonder if anybody knows of a diagramming/visualizaing tool that I might try to use. I'm currently using DIA. I'm thinking of trying out TouchGraph which I've used before. There's a free version: https://sourceforge.net/projects/touchgraph/

Algebra and analysis

- Express the link between algebra and analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.

Parsiųstas iš http://www.ms.lt/sodas/Book/MapOfMath

Puslapis paskutinį kartą pakeistas 2018 lapkričio 11 d., 16:13