Working on the big picture in math

[Andrius Kulikauskas]

Joe, Michel, Mike and all,

I'm interested in the big picture in math and how it unfolds. I am on a
quest to know everything and apply that knowledge usefully, and making
sense of math would be an important, satisfying and encouraging application.

A single person might be fortunate enough to discover a vantage point
from which to know everything, but applying that is certainly a group
activity, a science, a culture, a people. Similarly, math is a group
activity, and the existing culture reinforces certain expectations and
behaviors. So it takes some thoughtfulness to engage even just some
people and foster even a small space within that culture.

Math Future is, I find, an unusually supportive space for sharing one's
thoughts about the big picture in math. But I want to also engage
people who know more advanced math. So far at several communities I
have not succeeded: nLab https://ncatlab.org and nForum
https://nforum.ncatlab.org , the Azimuth Project
https://forum.azimuthproject.org and the Foundations of Mathematics
listserv http://www.cs.nyu.edu/mailman/listinfo/fom

I may have a chance, though, at MathOverflow.
http://mathoverflow.net
It seems to be the principal online community for research
mathematicians, including those at the highest level, such as Terry
Tao. However, to participate you have to either ask questions or
provide answer to other people's questions. You build up a reputation
as people upvote your questions or your answers. I have asked a couple
of questions:
http://mathoverflow.net/users/94182/andrius-kulikauskas
So if I'm able to break down my investigatory vision as a series of
precise questions, then that might be fruitful there.

I will write more about how I'm imagining the big picture in math and
also about two specific questions that I will ask for help here and at
MathOverflow to investigate. They follow up on my question about the
Mandelbrot set and the Catalan numbers:
http://mathoverflow.net/questions/247038/can-we-define-the-mandelbrot-set-in-terms-of-the-generating-function-of-the-cata
1) I'm curious where the power series P(c) for the Catalan numbers
converges, where it diverges, and where it is bounded. At this point I
am curious to study what happens if I (or you!) plug in various complex
numbers into P(c).
2) The Mandelbrot set is given by the behavior of the polynomials Pn(c)
as n approaches infinity. The coefficients of Pn(c) are always positive
integers that, for any term, ultimately equal those of P(c). Well, the
coefficients of the terms of P(c) are the Catalan numbers and they count
a variety of combinatorial objects, see:
https://en.wikipedia.org/wiki/Catalan_number
So this means that the Pn(c) should have a nice combinatorial
interpretation, for example, into strings of left and right
parentheses. A straightforward way to proceed is to calculate (expand)
Pn(c) for various n and compare their coefficients for c^k compare with
those of P(c) and thus investigate how each Catalan number gets built up.

I'm curious if SAGE or any other software would be helpful for this.

If somebody might be interested to collaborate, then I could explain
further and share more ideas on other questions to investigate. Our
research would improve our chances of getting help at MathOverflow.

The questions above connect one of the most fantastic objects in math
(the Mandelbrot set) with some of the most fundamental combinatorial
objects (as counted by the Catalan numbers). The latter objects are
very much related to push-down automata and context free grammars, which
are cognitively an important level in Chomsky's hierarchy of
computational complexity. That's the level that he thought human
language is defined at, if I recall and understand correctly.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Eiciunai, Lithuania