S7) Mathematical Explorations of Geometric Algebras

Organizers: Hongbo Li, Dmitry Shirokov, Pierre-Philippe Dechant

Geometric Algebra is closely related to matrix algebra in that every real Clifford algebra is isomorphic to a matrix algebra over the reals, complex numbers, or quaternions according to Cartan-Bott periodicity. On the other hand, Geometric Algebra is usually represented in a matrix-independent way, no matter if in the basis form of the associated Grassmann space, or in a basis-free form. This proposes the long-lasting problems of converting algebraic manipulations done in the matrix form to the matrix-free form of Geometric Algebras, a typical one of which is finding a versor generator of an orthogonal transformation in any non-degenerate Geometric Algebra.

This session intends to bring to the audience new advances in or related to the following topics on Geometric Algebra:

1. Matrix-free algebraic manipulations in Geometric Algebra
2. The characteristic polynomial problem in Geometric Algebra
3. Elementary and transcendental functions defined on Geometric Algebra
4. Versor representation and computation in Geometric Algebra
5. Univariate versor polynomial factorization in Geometric Algebra
6. The Cartan-Dieudonné Theorem and its extensions
7. Grassmann algebra
8. Clifford modules
9. Quaternionic algebras, Octonions, Okubo algebra, and other hypercomplex algebras
10. Quaternionic equation solving and quaternionic matrix theory
11. Ternary Clifford algebras and Generalized Clifford algebras

Contact:

• Prof. Hongbo Li, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China. hli@mmrc.iss.ac.cn
• Prof. Dmitry Shirokov, National Research University Higher School of Economics, Russia. dm.shirokov2021@gmail.com
• Prof. Pierre-Philippe Dechant, University of Leeds, UK. ppd22@cantab.net