The aim of the project is to study the projective and affine geometry of matrices and the related questions in the
research field of linear preserver problems.
In the geometry of matrices, there are four kinds of matrices first studied by L. K. Hua: the symmetric, Hermitian,
alternate, and rectangular matrices. The aim of the study is to characterize the group of motions by as few
geometric invariants as possible. For example, Hua found that the invariant adjacency is sufficient to characterize
the basic group. This statement is called the fundamental theorem of the geometry of matrices.
The fundamental theorem of the geometry of matrices can be applied to linear preserver problems. Linear
preservers are linear maps on linear spaces of matrices that leave certain properties or relations invariant.
There are various other research fields which are connected to the geometry of matrices, e.g., Laguerre geometry,
special relativity, ring geometry, and polar spaces.
We are going to study the following questions:
1. Adjacency preserving mappings of geometry of alternate matrices and application to linear preserver problems.
2. Distance k preserving mappings.
3. Adjacency preserving mappings between two different spaces.
4. Quasi-commutativity preserver problems.