Several followers expressed interest in my 2024 reviews of research papers on Geometric and Manifold Learning, Graph Neural Networks, and Topological Data Analysis.As a result, here's a summary!
I suggest the links to Slehar, the Cambridge GA group and John Denker. Also see Dorst's GA for Computer Science and his GAViewer software and tutorials, especially for the conformal model, though now more modern SW is available.
Substack ate my comment. The refs. I gave should be clear on the basics, though sometimes the binomial[n,m] (choose m basis bectors from an n-D set) / Pascal's triangle map of blade types gets buried. The connection between Lie algebras and bivectors is worth noting. e^Bivector allows rotating anything in n-D in the plane of B. This converts to a double-sided sandwich notation that allows arbitrarily chained rotations, e.g. exp(B/2)(X)exp(-B/2). If B is formed from a space and a time dimension (mixed signature) one gets a Lorentz boost. Adding a (+,-) pair of dims. and taking their sum and difference gives a pair of null-square dims. similar to a light-cone, called "origin" and "infinity", giving a conformal space that allows easily working with arbitrary circles, spheres, hypersperes which may be formed by multiplying 3, 4, or more arbitrary points on the figure together. If one of the points is at infinity, one gets lines, planes, etc. Conformal geometry allows unifying projective, affine, spherical and elliptic geometry: see "Recent Applications of Conformal Geometric Algebra", A. Lasenby 2004.
Also worth reading is David Hestenes' memoir: "The Genesis of Geometric Algebra: A Personal Retrospective" https://link.springer.com/article/10.1007/s00006-016-0664-z , contacts with Ed Jaynes John Wheeler, Kip Thorne, and Paul Dirac. Hestenes discovered that the GA del operator "reduces gradient, divergence and curl to a single operator in spaces of any dimension, and reduces the various integral theorems of Green, Gauss and Stokes to a single theorem"
GATr - Geometric Algebra Transformer / papers by Taco Cohen et al.
Very few people have a use for my post on classification of finite-dimensional Clifford algebras, but you might:
https://enonh.substack.com/p/classification-of-clifford-algebras
I suggest the links to Slehar, the Cambridge GA group and John Denker. Also see Dorst's GA for Computer Science and his GAViewer software and tutorials, especially for the conformal model, though now more modern SW is available.
Thanks for the reference. I have limited knowledge of the Clifford algebras.
Substack ate my comment. The refs. I gave should be clear on the basics, though sometimes the binomial[n,m] (choose m basis bectors from an n-D set) / Pascal's triangle map of blade types gets buried. The connection between Lie algebras and bivectors is worth noting. e^Bivector allows rotating anything in n-D in the plane of B. This converts to a double-sided sandwich notation that allows arbitrarily chained rotations, e.g. exp(B/2)(X)exp(-B/2). If B is formed from a space and a time dimension (mixed signature) one gets a Lorentz boost. Adding a (+,-) pair of dims. and taking their sum and difference gives a pair of null-square dims. similar to a light-cone, called "origin" and "infinity", giving a conformal space that allows easily working with arbitrary circles, spheres, hypersperes which may be formed by multiplying 3, 4, or more arbitrary points on the figure together. If one of the points is at infinity, one gets lines, planes, etc. Conformal geometry allows unifying projective, affine, spherical and elliptic geometry: see "Recent Applications of Conformal Geometric Algebra", A. Lasenby 2004.
Also worth reading is David Hestenes' memoir: "The Genesis of Geometric Algebra: A Personal Retrospective" https://link.springer.com/article/10.1007/s00006-016-0664-z , contacts with Ed Jaynes John Wheeler, Kip Thorne, and Paul Dirac. Hestenes discovered that the GA del operator "reduces gradient, divergence and curl to a single operator in spaces of any dimension, and reduces the various integral theorems of Green, Gauss and Stokes to a single theorem"