I am Jose Ayala, a postdoc at the University of Melbourne, Australia. I am interested in computational geometry and low dimensional topology. I completed my Ph.D. at The University of Melbourne under the outstanding guidance of Hyam Rubinstein. In my dissertation, I answered questions raised by Lester Dubins in 1961 about the connected components of the spaces of bounded curvature paths. Basically, I obtained necessary and sufficient conditions for guaranteeing whether any two given bounded curvature paths are in the same connected component. More generally, I classified the homotopy classes of planar bounded curvature paths.

Currently, I am working in problems in 3-manifolds. I am interested in the classification of Heegaard splittings, and the study of spaces of physical knots.

You can watch some videos describing techniques used to characterise the length minimising bounded curvature paths in homotopy classes and their deformatoins. Recently, we extended Dubins´ Explorer features for the flat torus and Klein bottle.