The Azimuth Project
Andrius Kulikauskas

Hello! I introduced myself in this post

At The Azimuth Project, I am currently interested in:

Project: Modeling the Composition of Perspectives.


I have divided the rest of my page here into the following sections:

  • Workspace A messy place where I’m working.
  • Projects My projects which someday I think will find their own pages.
  • Blog What I’m up to here and elsewhere.

Workspace

I’m currently collecting links for my map of the areas of mathematics which I write about below.

I realized that a main theme is the correspondence between the roots of a polynomial (which define a geometric construction in some vector space) and the polynomial in terms of its coefficients (in some ring). This comes up in:

  • The Fundamental Theorem of Algebra
  • Its generalization, Hilbert’s Nullstellensatz, and all of algebraic geometry.
  • Algebraic geometry can be nicely subdivided into “classical algebraic geometry” over the complex numbers or any algebraically closed field, and “modern algebraic geometry” (using schemes) over the real numbers or any non-algebraically-closed-field, per the NPTEL video lectures on algebraic geometry.
  • Galois theory links with the group of symmetries on those roots.
  • Lie theory has a similar intent.
  • Linear algebra is, apparently, just algebraic geometry for multilinear polynomials.
  • The Riemann hypothesis asks questions about zeros.

A related them is the different kinds of geometries. As I write in my page Discovery in Mathematics: A System of Deep StructureI think that there are four kinds geometries (affine, projective, conformal, symplectic) and six transformations between them. I’m finding it very helpful to watch Norman Wildberger’s videos on Universal Hyperbolic Geometry. I hope to collect all of the results he presents and express them in terms of symmetric functions. Perhaps the six natural bases of the symmetric functions are related to the six transformations that I’m looking for? I wrote my Ph.D. thesis on the values of these symmetric functions at the eigenvalues of an arbitrary matrix.

Perhaps the most important idea in math is the duality between 0 and infinity as mediated by the “lens” of 1. On the one hand, this is suggested by my talk on God’s Dance. On the other hand, I think this comes up with the field with one element. I think that element can be interpreted at different times as 0, 1 and infinity, for example: 0 = 1 / infinity. What I’m learning from the Universal Hyperbolic Geometry is that infinity and zero are often completely dual. For example, the idea that 0 is between tiny negative numbers and tiny positive numbers is in some sense not any more valid than the idea that infinity is “between” humungous negative numbers and humungous positive numbers, that is, more negative than the one and more positive than the other. Basically there is a sense of “symmetry breaking” by which 0 and infinity start to get treated differently. And zero is also related to the “zeros” discussed above.

I’m thinking of geometry as the way of embedding a lower dimensional space into a higher dimensional space. So that fits well with what I wrote about algebraic geometry as involving vector spaces for their geometry. And there is a duality between the bottom-up constructions and top-down deconstructions as with points and hyperplanes (or points and lines in 2 dimensions).

So I feel surprised to be making some progress. At some point I will need to try to again with my diagram of the areas of math.

Another idea I had for building this map was to study the top 100 or 200 top tags at Math Overflow and also, to ask them, for each pair of tags, the number of posts that use those tags.

  • Thank you to Kirby Urner for suggesting I include Synergetics
  • Synergetics - tetrahedron - icosahedron - geodesics - self-organizing systems
  • Field of one element - simplicial complex > simplex > empty set
  • simplex < simplicial homology < singular homology
  • homology group
  • Betti number

Project: Mapping the Areas of Mathematics

In trying to understand the big picture of math, we can make a map of how the various areas of math relate. In particular, what areas depend on what other areas?

I started by considering the areas of pure math listed in the Mathematics Subject Classification used for organizing journals and their articles. Here’s my first map, which I drew by hand with Dia.

I used Wikipedia to learn about the different areas and make my best guesses as to which areas depended on which. I was surprised to see that Geometry is evidently a very basic area in math. Which leads me to wonder, What is Geometry? For me, that kind of a question is a step forward.

I need to find a tool to make more complicated maps. I ended up making my maps with yEd, which is available for free. 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.

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)
  • I also want to add dualities that link various areas and structures.

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.

yEd provides a variety of layouts. Above, I used organic, which is most compact, but there are also hierarchical, orthogonal, circular, tree, radial, series parallel.

Here is a map based on the circular view. 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.


Blog

2016.05.22

Today, Sunday, I’ll be taking the train for a day trip to Kaunas. So I’ll have some time to collect some links for my map. I also need to research and think through my talk on beauty in mathematics. So that can inform my map as well. I’m also realizing which concepts in math are most relevant for my philosophy, so I will try to see what happens if I make them central. In particular, I am identifying God with “unmarked opposites” as in the case of the two imaginary square roots of -1. So that imaginary world is the “spiritual world”, whereas the “real world” is based on “marked opposites” 1 and -1. I also just recently learned of the “field of one element”, first because it appeared on the list of beautiful structures, and then yesterday from John Baez’s first lecture on Geometric Representation Theory. I’m imagining that the “field of one element” might model God, especially as he goes beyond himself. And the “roots of unity” might relate to my “divisions of everything”. So I want to find especially links from such areas in Math that I think are key. My hypothesis is that if I elaborate and highlight the concepts in Math that might express my philosophical concepts, then my map of areas of mathematics will organize itself in revealing ways. As things stand, it’s just a spaghetti diagram. :(

2016.05.16

I wrote a letter about the little known fact that the generating function of the Catalan numbers is the limit of the polynomials which generate the Mandelbrot set. I find this highly interesting because the Catalan numbers count well ordered sets of parentheses, precisely the kinds of expressions that are processed by context-free grammars. And a complex number can be interpreted as two infinite sequences of 1s and 0s, perhaps an input tape and an output tape. So my hunch is that:

  • the Catalan function converges when the processing of the parentheses is successful
  • it is bounded when there are only finitely many “mistakes”
  • it goes off to infinity if there are an unbounded number of “mistakes” where a “mistake” may be that the number of open parentheses is slightly more than the number of closed parentheses.

I found the main hand book on (especially complex) analytic combinatorics which describes the connection but in terms of the height of binary trees. It has lots of links from the Catalan numbers (and other key numbers) to other areas of math such as the Chebyshev polynomials (and other orthogonal polynomials), and the Bernoulli distribution (and other probability distributions). I also found a book that relates the q-t-Catalan numbers to the Macdonald polynomials which are relevant to root systems and everything Lie representational, I think. So I have lots of new connections to make in my map of mathematical areas.

2016.05.14

Hello, David, John and all! I learned of the Azimuth Project through John Baez’s website, blogs and videos, which I’m very grateful for as currently I’m trying to learn enough advanced math to get a sense of the big picture. I would like to share some of my projects that I imagine could be relevant here. I am thinking of writing about them on this page and then somebody may suggest on what pages I might pursue them.

David Tanzer, I very much enjoyed your post about Blog - Zero to the X. These are the kinds of math ideas that I’m keen to think about. I read on your page:

My plan is to study math and science and then teach it to colleagues in software development. We need more scientists to solve the myriad of problems that beset the human race, and the world of programmers looks like a good recruitment base for the sciences. In the process I hope to develop myself as a scientist!

This sentiment has come up recently in the Math Future google group, which has some very creative educators for math, computers, engineering and science, mostly as relates to high school, though.

I have a Ph.D. in Mathematics and a B.A. in Physics but my main interest has always been “to know everything and apply that knowledge usefully”, for example, by saving the planet. I’m writing a book about my philosophy and I want to show its implications for math and physics. So there is a lot of math that I am trying to learn now. As one project, I am making a map of the areas of math.

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