Mathematics · Global Analysis · Geometric PDE · Variational Methods
Carlos Daniel Velázquez Mendoza
I am a Ph.D. student in the Graduate Program in Mathematical Sciences at Universidad Nacional Autónoma de México (UNAM). My academic interests include teaching and research in global analysis, partial differential equations on Riemannian manifolds and vector bundles, variational calculus on manifolds and vector bundles, geometric analysis, and topology, particularly realcompact spaces.
My current work focuses on developing new intrinsic results in global analysis, connecting tools from differential geometry, functional analysis, Sobolev theory, PDE, and variational methods. I am especially interested in geometric formulations of PDE problems on vector bundles.
I teach at the Faculty of Sciences of UNAM, where I have taught several undergraduate courses and have also assisted in graduate courses related to nonlinear analysis on Riemannian manifolds and variational methods on Riemannian manifolds. I also teach undergraduate courses at Universidad Anáhuac.
Carlos Daniel Velázquez Mendoza
M.Sc. in Mathematics
Ph.D. student, UNAM
Academic profile
Research
My research lies in global analysis, geometric analysis, Sobolev spaces on manifolds and vector bundles, geometric PDE, differential operators, and variational methods.
Undergraduate teaching
I have taught undergraduate courses at the Faculty of Sciences of UNAM and at Universidad Anáhuac. I am currently teaching at both institutions.
Graduate-level support
I have assisted in graduate courses on nonlinear analysis on Riemannian manifolds and variational methods on Riemannian manifolds.
Mathematical exposition
I develop notes, guides, and manuscripts in calculus, global analysis, differential equations, differential geometry, and functional analysis.
Featured work
Geometric Integration by Parts and Sobolev Spaces on Vector Bundles
With María de los Ángeles Sandoval Romero.
arXiv:2602.01016, 2026.
Análisis global
Una introducción al análisis no lineal y sus métodos variacionales en variedades riemannianas.
Book manuscript in progress with María de los Ángeles Sandoval Romero and Romulo Diaz Carlos.
Current project
Ground state solutions for quasilinear equations on vector bundles
Work in progress with María de los Ángeles Sandoval Romero and Romulo Diaz Carlos on quasilinear elliptic equations on vector bundles from the perspective of geometric analysis and variational methods. The project studies intrinsic formulations through the formal adjoint of the covariant derivative and divergence-form problems on Riemannian manifolds with boundary.