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David, John and all, Hello!
I became interested in the Azimuth Project because of John Baez's mathematical expositions. I'm truly grateful for his videos and blog posts. I'm currently interested to get a grip on the big picture in mathematics, and he makes me feel that he is too, and that it's within reach, however outrageous that may seem. I especially liked his videos of his favorite numbers because he is able to toss a line from the most concrete to the most abstract. Coincidentally, the numbers 8 and 24 are at the heart of my own explorations. And he knows so much about subjects that I think I should, too. Subjects I didn't even knew existed! Finally, he wants to save the planet. Well, I'm part of the planet, so maybe he will care about me, too.
But I was definitely encouraged by David Tanzer's wiki page. http://www.azimuthproject.org/azimuth/show/David+Tanzer He writes: "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!" Kirby Urner at the Math Future google group expressed a very similar strategy. And my own wish is to learn more math. So I thought I might have a chance here.
For the sake of being myself, I will present my own goals here, as I imagine them, starting with the most reasonable and continuing towards the ever more unreasonable.
My starting point is that "advanced math literacy" might be a very practical endeavor for Azimuth Project. Mostly because John excels at fostering that and also at attracting people like David and me who care about that. But also because a lack of such literacy is arguably a major reason why global warming is not being taken seriously and why we have a shortage of scientists who might address it.
Personally, in learning advanced math, I am making a map of the areas in math. I have set up a page Andrius Kulikauskas at the Azimuth Project wiki where I have started writing about that: http://www.azimuthproject.org/azimuth/show/Andrius+Kulikauskas Indeed, I link to a map of some 200 nodes that I drew with yEd software. I would very much appreciate discussion to make my map as comprehensive and profound as possible. And also to consider how such a map could be helpful for math literacy and for saving the planet. So that seems a most reasonable goal here. I have no idea where I should pursue that at the wiki so for now I'm active at my own page there.
I would also love help to learn particular areas of math, many of which John writes and talks about. My big goal in life since childhood has been "to know everything and apply that knowledge usefully". Along the way I got a B.A. in Physics at the University of Chicago (1986) and a Ph.D. in Math at UCSD (1993). In 2014, I wrote an illustrated summary: http://www.selflearners.net/wiki/Truth/Book I'm now thinking through a detailed book which I plan to write next year. As part of that, I want to show that my philosophy can say useful things about Math and Physics, especially the big picture. So there is a lot for me to learn, even though I need to be very selective. That's where I appreciate help, fool that I am, and that's where I find John and Urs's writings so helpful.
My own background is in algebraic combinatorics. I thought of combinatorics as the "basement" of math from which arose the concrete objects of math. I became interested in the question, Why is it that the symmetric functions have infinitely many bases, but precisely six bases seem interesting to the human mind? (elementary, homogeneous, power, monomial, Schur, forgotten). I realized an interesting fact that we can calculate the symmetric functions of the eigenvalues of an arbitrary matrix (the determinant being the product and the trace being the sum, in particular). We thus generate collections of all manner of cycles, walks, words, Lyndon words. And if we set the off-diagonal matrix elements to zero, then the eigenvalues are on the diagonal, and so we recover the original symmetric functions. So this is a strange case where specializing (to the eigenvalues) makes a function more general. And since the symmetric functions are the basis for all of combinatorics (everything that has labels which can be permuted) I was able to show how that could be taken to arise from facts about arbitrary matrices. http://www.ms.lt/derlius/AndriusKulikauskasThesis.pdf
One area that I would like to learn the key ideas of is category theory. In my philosophy, perspectives play an important role. I imagine there must be an "algebra of perspectives". For example, if a lost child is smart, then they realize that they are the child and not the parent, and so they should not look for their parent, but should rather go where their parents think they would be. We thus have "the child's view of the parent's view of the child's view of the parent's view of the child's view". In which case, the child's view and the parent's view coincide, even though they are not in communication! So this is the kind of thing that I think I could model with category theory if I was fluent in it.
Another area to learn is Lie groups and Lie algebra because that seems central to Math but certainly for Physics. For my philosophy, I studied about 200 different ways that I had figured things out. I came up with a system of 24 ways, a "house of knowledge". Then I realized that I could come up with such a system for discovery in Math, which I did. Now I'd like to do that for Physics, for which I need to learn more Math, starting with tensors.
In Math, I read George Polya's book "How to Solve It". He has a "pattern of two loci" which is at work in trying to draw an equilateral triangle given one side AB. You draw circles centered at A and B and see where they intersect. I realized that when we solve this problem, what is happening in our mind is that we're working with a lattice of conditions given by the plane (no conditions), circle A (one condition), circle B (another condition), and the two intersections (both conditions). Thus the crux of the discovery has nothing to do with triangles but is given by a math structure. That structure is implicit in our mind. So this is a way to see what kind of math is natural, what kind we actually leverage in our minds as we do math. I studied Paul Zeitz book and found 24 such structures/methods. I wrote them up here: http://www.ms.lt/sodas/Mintys/MatematikosRūmai and I should work on that further.
I also want to do that for Physics. I'm studying and sorting physics experiments listed at Wikipedia. I also need to learn a lot of related Math to appreciate how things are figured out in theoretical physics.
Another of Math/Physics that I want to learn key ideas from include Entropy because I want to sharpen my concepts of grace and justice which I picture entropically as the distinction between an open system (fed by an infinitely loving sun) and a closed system (where everything tends to fall apart). Real life is quite ambiguous and I'd like to model that ambiguity and its implications.
I'm intrigued to learn some Geometric Representation Theory because I want to know how different areas in Math are linked. And I realized from my map that geometry must be quite fundamental. But who can tell me, What is geometry? I'm guessing that it's the way that a lower dimensional space is embedded in a higher dimensional space. And tensors must be crucial in that they are the "trivial" answer to that, but especially as they model the duality between the top-down and bottom-up views. So I need to learn tensors well. I'm not there yet.