On 24 January 2026, at VNUHCM–University of Science (HCMUS), researcher Phan Ngọc Yến (specialising in Algebra and Number Theory) successfully defended her doctoral dissertation at the institutional level. The research, titled “Distance Functions, Least Squares Problems, and Related Issues”, was conducted under the academic supervision of Associate Professor Đinh Trung Hòa and Dr Nguyễn Anh Thi.

The dissertation focuses on theoretical problems within algebra, operator theory, and matrix equations, employing an approach aligned with mathematical models in quantum information theory. This area of inquiry is highly specialised, requiring the rigorous integration of modern mathematical tools and a robust theoretical foundation.

The primary substance of the thesis involves the construction and analysis of several novel distance functions based on matrix means, while simultaneously examining the least squares problem corresponding to these functions. Building upon this framework, the researcher investigated non-linear matrix equations related to weighted geometric means, specifically focusing on proving the existence of solutions and analysing the relationships between systems of equations.

The new distance functions proposed in the dissertation satisfy essential properties of quantum information theory, including data processing inequalities and the midpoint property. Furthermore, the thesis establishes and examines a class of novel distance functions generated by monotonic inequalities. This work extends results concerning matrix means and trace functions within operator theory, providing a theoretical basis for developing mathematical models and broadening research approaches in interdisciplinary studies.