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 recently 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 3-manifolds. 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.
In February 2016, I will be visiting Hyam Rubinstein at The University of Melbourne, Australia.
I will give a talk at the VII Workshop Internacional de Matemática to be held in Tacna, Perú from 28th to 30th October 2015.
I was a member of the organizing committee for the 24th Congreso de Matemática Capricornio, COMCA 2015 to be held in Iquique from 5th to 7th August 2015.
In January 2015, I was in Hyperbolic Geometry and Minimal Surfaces at IMPA, Rio de Janeiro, Brasil.
In December 2014, I gave a talk at the Computational Topology and Geometry workshop in Foundations of Computational Mathematics FoCM'14 Montevideo, Uruguay.
From November 1st to 30th 2014 I visited Hyam Rubinstein at The University of Melbourne, Australia.
jayalhoff AT gmail DOT com