José J. Garcia

Welcome! I am a PhD candidate in the Department of Mathematics and Statistics at the University of New Mexico. My research lies at the intersection of linear algebra, quantum mechanics, condensed matter physics, and network science, where I develop mathematical and computational methods for understanding complex systems. I am particularly interested in how mathematical structure can reveal hidden patterns in physical and data-driven systems.

A significant portion of my research focuses on composite operator approaches to quantum systems and the computation of topological invariants for physical systems. This work has applications to topological materials and non-Hermitian physics, where many theoretical challenges remain open. I am also interested in approximate joint eigenvectors of nearly commuting matrices, graph sparsification, and computational approaches to large-scale scientific problems.

Alongside my research, I am passionate about teaching and mentoring students. Over the past six years, I have taught undergraduate courses including Calculus, Linear Algebra, and Ordinary Differential Equations. One of my favorite parts of teaching is helping students realize that mathematics is more than a collection of formulas and procedures; it is a language for describing patterns, solving problems, and communicating ideas. I strive to create learning environments where students feel comfortable asking questions, making mistakes, and developing confidence in their ability to think mathematically.

I am also interested in mentoring undergraduate research and reading courses. Whether a student is encountering advanced mathematics for the first time or considering graduate school, I enjoy helping them develop the skills and confidence needed to tackle challenging problems independently. Some of the most rewarding moments in teaching occur when students begin to see connections between mathematics, computation, and the world around them.

I have also had the opportunity to collaborate with researchers at Sandia National Laboratories on projects involving quantum systems, topological materials, and scientific computing.

Beyond research and teaching, I am actively involved in the mathematical community. I currently serve as Treasurer of the University of New Mexico Association for Women in Mathematics Student Chapter and as an officer of both the UNM SIAM Student Chapter and the Mathematics and Statistics Graduate Student Organization. Through these organizations, I enjoy supporting outreach, professional development, student engagement, and building a welcoming mathematical community.

When I am not doing mathematics, I enjoy spending time with family, camping, traveling, corresponding with pen pals, watching anime, and especially reading manga.