I am José Ayala Hoffmann, a mathematician at the Universidad de Tarapacá in Iquique, on the Pacific coast of northern Chile.

My research lies at the intersection of geometric topology, discrete geometry, and variational analysis. I study spaces of configurations whose topology and rigidity are shaped by contact, thickness, curvature, length, or equal size constraints. The main examples in my work include hard disk clusters, circle packings, immersed flat ribbons, and physical knots and links.

A guiding principle is that the variational problem should be read intrinsically from the realised geometry of the object itself. Contacts, self contacts, apertures, thickness constraints, and boundary data determine the admissible motions and the relevant first and second variation equations. This leads to intrinsic tools such as rolling spaces, contact strata, homogeneity defects, length and perimeter certificates, and obstruction profiles.

Through this intrinsic approach I study rigidity, deformation, and gordian phenomena: when constrained configurations can move, when they are locally or globally minimal, and when geometric obstructions prevent isotopy, realisation, or length preserving deformation. The aim is to connect classical rigidity theory and geometric topology through a common variational language.