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Summer School on the Foundations of Geometry in Historical Perspective at the Max Planck Institute, July 2-7, 2018

April 2, 2018

Letter of Intent

Dear Summer School Organizers,

I wish to be exposed to the history of the foundations of geometry so that I might gain intuition for my investigation into how the four classical Lie groups/algebras {$A_{n},B_{n},C_{n}$} and {$D_{n}$} may ground different geometries, but in particular, affine, projective, conformal and symplectic geometries. Indeed, I would greatly benefit from interaction with others to make sense of these four geometries as well as others that are important from a historical perspective.

In 1993, I received my Ph.D. in Mathematics from UCSD. My thesis was in algebraic combinatorics. However, I studied math as a tool for my truly comprehensive interest to know everything and apply that knowledge usefully. You could say that is philosophy, and indeed, in 2016, I started my academic career as a philosopher at VGTU, teaching Philosophy, Ethics and Creative Writing. Basically, I've worked all of my life independently on hundreds of investigations. In particular, as you can read in my CV, I'm documenting a system of cognitive frameworks which structure the perspectives that our minds may have. Now I'm writing up my work. As part of that, I'm trying to show how my approach can be fruitful for mathematical research.

I would like to show that it is possible to understand the big picture in math, how the various branches of math unfold with all of their many concepts. In one investigation, I simply organized the subjects listed in the Mathematics Subject Classification by considering how some subjects depended on other subjects. Of course, algebra and analysis are two major "hemispheres". But I came to appreciate that geometry is evidently very central. In particular, the Lie groups/algebras seem to be very important in bridging analysis and algebra.

In an earlier investigation, I documented and systematized the ways of figuring things out in math. I studied problem solving books by Polya and Zeitz and noted the problem solving "patterns" that they taught. Polya teaches the pattern of two loci, which is how Euclid has us construct an equilateral triangle given one side AB: You draw one circle centered at A, and another at B, and you see where they intersect. In contemplating this example, I realized that the crux of solving this problem is to manipulate in our minds a tiny lattice of conditions: "None", "A", "B", "A and B". As in linguistics, we can say the surface structure is the equilateral triangle but the deep structure is this lattice. And so we can collect such patterns, and the related structures, which are natural to our mind, and not simply contrived. Which is to say, this approach suggests that we can distinguish implicit math which we intuit and do in our minds without any explicit system or axioms, and explicit math, which we don't intuit directly, but which we can express and study on paper and express axiomatically.

In the system that I observed there seemed to be slots for four geometries. In a very different investigation of emotions and moods, I discovered the importance of the boundary between self and world, between what we know and what we don't know. In studying how poems evoke moods, I noted six different transformations: reflection, sheer, rotation, dilation, squeeze and translation. I argued that these transformations arise as enrichments of geometries. I noted that we can think of a triangle in progressively richer ways: first, in terms of paths forward from A to B to C and back to A; second, in terms of intersecting lines which may go back and forth; third, in terms of angles; fourth, in terms of an oriented area which is swept out. I will be giving a talk about this at the 24th World Congress of Philosophy in Beijing, China, this August: A Geometry of Moods: Evoked by Wujue Poems of the Tang Dynasty.

Those four geometries seemed to me to be related to affine, projective, conformal and symplectic geometries. And John Baez and others have written that these geometries are related to the different Lie groups/algebras. "Whenever we pick a Dynkin diagram and a field we get a geometry". But I have yet to understand this. At Math Stack Exchange, I have asked the question: Intuitively, why are there 4 classical Lie groups/algebras? Gaining such intuition, and relating that to different geometries, is a major goal for me at the summer school.

I have made some progress by studying Victor Kac's notes and learning how to write out Lie algebras using Serre's relations. As I write in my own answer to my question at Math Stack Exchange, I have noticed that if the simple roots for An are written out as: {$x_1-x_2,\dots,x_{n-1}-x_n$} then the options for the next root are as follows:

Attach:simpleroots.png Δ

As I write in my answer, it is as if there are different kinds of "mirrors" that we can engage.

I am also relating this to different kinds of infinite families of polytopes, namely, the simplexes, cross-polytopes, hypercubes, and demihypercubes. This has led me to a novel interpretation of the -1 simplex as the unique center of the simplex which keeps getting reinterpreted as a vertex. The simplex also has a unique totality, which is to say, a pseudoscalar. In the case of the cross-polytopes, the center keeps getting reinterpreted as a new pair of vertices on opposite sides.

Ultimately, I am thinking that at the heart of logic and math is a taxonomy of duality.

I am also interested in understanding how Norman Wildberger's Universal hyperbolic geometry relates to other kinds of geometry.

Also, I would be interested to get some help in learning about the basic tools of modern algebraic geometry: sheaves, stacks, sites, and ultimately, topoi, as well as Grothendieck's six operations.

Here in Lithuania, I meet with algebraic geometrist Rimvydas Krasauskas and we talk for a few hours at a time. He's especially interested in Clifford algebras. I would benefit from a week of such learning with a variety of teachers and fellow students. I am especially interested in Jürgen Jost's research.

I hope you might grant me this opportunity to learn from you!

Andrius Kulikauskas, Vilnius, Lithuania