Hi there!I am José Ayala, a mathematician at the Universidad Arturo Prat located in a lovely city in northern Chile called Iquique. I am interested in differential geometry, low dimensional topology and dynamics. I completed my Ph.D. at The University of Melbourne under the outstanding guidance of Hyam Rubinstein. In my dissertation, I answered questions raised in 1961 about the connected components of the spaces of bounded curvature paths. Basically, I gave necessary and sufficient conditions for any two bounded curvature paths to lie in the same connected component. I classified the homotopy classes of planar bounded curvature paths. Currently, I am working in a couple of problems in 3manifolds. Especially I am interested in the classification of Heegaard splittings of surface bundles. I am also applying normal surface theory to compute the Heegaard genus of triangulated knot complement for knots with a small crossing number. 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.

