My research interests are in Optimization, Linear Algebra, and Matrix Analysis. To provide a more detailed overview of my work, I have focused on the following key themes:
Operator means theory: This area of study includes various means or averages, with a particular focus on the geometric mean (Karcher mean) and the Wasserstein mean. These means are of particular significance as they are geodesic on the Riemannian manifold associated with the Riemannian distance and the Bures-Wasserstein distance, respectively. Their multi-variable version on positive definite Hermitian matrices and the extension to positive operators.
Linear complementarity problems: There are two main aspects of the linear complementarity problem: the first one is the existence and multiplicity of solutions and the other is in designing algorithms to find a solution. Both of these depend on the properties of the matrix, through which the LCP is defined and this usually involves understanding certain matrix classes that satisfy certain positivity criteria. In fact, some of the matrix classes are defined in terms of properties of the LCP.
Nonnegative matrices and generalizations: My research interests involved the study of certain positivity classes of matrices arising from optimization problems, especially linear complementarity problems. In particular, I have studied the structure and linear preservers of a specific class of matrices called semipositive matrices. Notably, I have solved recent conjecture regarding the linear preservers of semipositive matrices.