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 a video describing some techniques used to characterise the length minimising bounded curvature paths in homotopy classes. Recently, we extended Dubins´ Explorer features for the flat torus and Klein bottle.