I am José Ayala Hoffmann, a mathematician at the Universidad de Tarapacá in Iquique, Chile. My research lies at the intersection of geometric topology, variational geometry, and rigidity theory.

My current work develops a unified variational framework centred on disk diagrams, finite packing problems, and perimeter minimisation under geometric constraints, with applications to critical ribbon loops and hopefully the gordian unknot conjecture. The gordian unknot conjecture concerns the existence of geometrically distinct unknots that are topologically trivial yet remain locked under curvature and thickness constraints.

We consider rigidity and obstruction phenomena as intrinsically variational: configurations are organised into contact strata determined by active disk contacts, and admissible deformations are governed by these constraints. By combining explicit geometric constructions, first- and second-order variational analysis, and combinatorial stratification by contact graphs, the framework provides a systematic way to analyse rigidity, flexibility, and minimality in constrained planar and spatial geometric systems.