alekseevskii attempts numerical classification of unimodular Einstein Lie groups towards resolving the long-standing generalized Alekseevskii conjecture. This repository was updated in 2020 for modern Python/OpenCL.

License

This project is distributed under the MIT license.

Note

• In the 1970s, it was conjectured by D. Alekseevskii that any (non-compact) homogeneous Einstein space of negative scalar curvature is diffeomorphic to R^n. In other words, the Classical Alekseevskii Conjecture states: Given a homogeneous Einstein space G/K with negative scalar curvature, K must be a maximal compact subgroup of G [Jablonski].
• blocking contains an implementation of a numerical search algorithm in python and cython
  • blocks are sampled from the unit cube of a specified dimension in R^n
  • each is checked for a number of necessary conditions given by our classification goal and the conjecture: sphere admissibility, jacobi condition, and possibility of Einstein metrics
  • for a few related ideas, please see Arroyo and Lafuente, Jablonski and Petersen, or Prof. Jablonski's research statement
• tests include a simple benchmark of computing spatial array index
• opencl_helper.py provides a nice wrapper around pyopencl for loading simple kernels and running sequences over sets of numpy arrays