Contact Information

School of Mathematics and Statistics
University of Melbourne, Australia

Research Interests

Computational geometry, low dimensional topology.

Education

University of Melbourne, Parkvile, VIC Australia.
Ph.D. in Mathematics, March 2014.
Advisor: Hyam Rubinstein.
Dissertation Topic: Classification of the homotopy classes and minimal length elements in spaces of bounded curvature paths.
Iowa State University, Ames, Iowa USA.
M.Sc. in Mathematics, July 2007.
Advisor: Wolfgang Kliemann.
Thesis Topic: Global behavior of graph dynamics with applications to Markov chains.
Universidad Católica del Norte, Antofagasta, Chile.
B.S. in Mathematics, March 2005.
Advisor: Wolfgang Kliemann.
Thesis Topic: Dynamical systems in the real projective space.

Publications and Preprints

1. J. Díaz and J. Ayala, Census of bounded curvature paths. Accepted in Geometriae Dedicata (supervised research).
2. J. Ayala, D. Kirszenblat, and J.H. Rubinstein, A geometric approach to shortest bounded curvature paths, Communications in Analysis and Geometry, Vol. 26, No 4, 2018.
3. J. Ayala and W. Kliemann, Topological dynamics on finite directed graphs, Topology and its Applications, Vol. 241, pp. 345362, 2018.
4. J. Ayala, On the topology of the spaces of curvature constrained plane curves, Advances in Geometry, Vol 17, No 3, pp. 283292, 2017.
5. J. Ayala and J.H. Rubinstein, The classification of homotopy classes of bounded curvature paths, Israel Journal of Mathematics, Vol. 213, No 1, pp 79–107, 2016.
6. J. Ayala, Length minimising bounded.curvature paths in homotopy classes, Topology and its Applications, Vol. 193, pp. 140151, 2015.
7. J. Ayala, Connectedness and minimal length elements in spaces of bounded curvature paths, Bulletin of the Australian Mathematical Society, Vol. 90, No 1, pp.174176, 2014.
8. J. Ayala and J.H. Rubinstein, Nonuniqueness of the homotopy class of bounded curvature paths (submitted).

Ongoing Research

9. J. Ayala and J.H. Rubinstein, On the classification of Heegaard splittings.
10. J. Ayala, Curvature constrained curves in the punctured disk.
11. J. Ayala, D. Kirszenblat, and J.H. Rubinstein, Isotopies of physical knots.

PeerReviewed Conference Articles

1. J. AyalaHoffmann, M. Brazil, H. Rubinstein, D. Thomas, Extendibility and path components of admissible paths for the Dubins problem, Proceedings of Australian Control Conference, 440444, IEEE  Institute of Electrical and Electronic Engineers, 2011.
2. J. AyalaHoffmann, W. Kliemann, Morse decompositions of semiflows associated with graphs, Proceedings of the 20th International Symposium on Mathematical
Systems and
Networks (MTNS 2012, July 913 2012) Melbourne, Australia.
3. J. AyalaHoffmann, M. Brazil, H. Rubinstein, D. Thomas, A geometric approach to shortest bounded curvature paths in surfaces of constant nonpositive curvature,
Proceedings of the 20th International Symposium on Mathematical Systems and Networks (MTNS 2012, July 913 2012) Melbourne, Australia.

Undergraduate Research

1. J. AyalaHoffmann, P. Corbin, K. Mc Conville, F. Colonius, W. Kliemann, Morse decompositions, attractors and chain recurrence, Proyecciones Journal of Mathematics Vol. 25, No 1, pp. 79109, 2006.
2. V. Ayala, J. AyalaHoffmann, Ivan Tribuzy, Controlability of invariant control systems at uniform time, Kybernetika, Vol. 45, No 3, pp. 405416, 2009.

Languages

Spanish, Portuguese, Italian and English.
