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 and discrete geometry, viewed through the lens of the calculus of variations. I study spaces of realised 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 a variational problem may be read from the realised geometry of the object and its active constraints. Contacts, self contacts, apertures, thickness, and boundary data determine the admissible motions and the relevant first and second variations. This leads to intrinsic tools such as rolling spaces, contact strata, homogeneity defects, length and perimeter criteria, and obstruction profiles.

Through this intrinsic approach I study rigidity, deformation, and gordian phenomena. The central questions are 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 language of realised geometry, active constraints, and first and second variation.