See: Classical Lie Groups
I'm applying for the Summer School on the Foundations of Geometry in Historical Perspective.
Research goals
- I am seeking the cognitive foundations of everything, including all of mathematics, but especially geometry.
- I am currently investigating why, intuitively, there are four classical infinite families of Lie groups and algebras, and how they ground different geometries, which I expect are affine, projective, conformal and symplectic.
- I am looking out for the various kinds of dualities which appear in mathematics.
Summer school goals
- I wish to gain an intuitive understanding of the nature of affine, projective, conformal and symplectic geometries, as well as other potentially fundamental geometries.
- I wish to understand how the intuitive characteristics of various geometries are grounded in different structures such as the classical Lie groups and algebras.
- I would like to acquire a general understanding of the fundamental theorems, structures and concepts of modern geometry.
- I would like to understand the basics of algebraic geometry, including Grothendieck's constructs, especially topoi.
- I would like to understand the relationship between geometry and logic.
A historical approach may help me personally understand the bigger picture, or at least, how various ideas have unfolded so far.