Professor Nalini Joshi - MIMS Distinguished Visitor 12-2005/01-2006
Short CV
Nalini Joshi is professor of mathematics at the School of Mathematics and Statistics of the University of Sydney. She received her education at the University of Sydney, graduating in 1982 and Princeton University obtaining her PhD in 1986.
She has held academic positions at the:
- Australian National University
- University of New South Wales
- University of Adelaide
- University of Sydney
- Research Institute for Mathematical Sciences at Kyoto University
She has held visiting positions at: University of Colorado, Boulder, Princeton University, Rutgers University, University of Exeter, Isaac Newton Institute of Mathematical Sciences, Cambridge University.
Research interests
What distinguishes order from chaos? How can we identify systems that are integrable and only have ordered solutions? There is strong evidence that the answer depends on the complex analytic properties of the systems of interest. The Painlevé equations (six classical nonlinear ordinary differential equations) are prototypical examples of this evidence. Much of my research concentrates on these deep and beautiful equations.
These equations were named after Paul Painlevé, a French mathematician and aviation enthusiast who was also a prime minister (twice) early in this century. The central idea that links complex analyticity and integrability was initiated by Sofia Kovalevskaya, one of my mathematical heroes.
Asymptotics is closely related to the techniques used for finding singularity structure. Pierre Boutroux initiated a deep and extensive asymptotic study of the Painlevé equations. George Gabriel Stokes was an amazing mathematical physicist who initiated many profound asymptotic studies, some of which are only now being extended to nonlinear equations.
Research while at MIMS
Marta Mazzocco and I plan to investigate the asymptotic behaviour at infinity of the generic solutions of the second Painlevé hierarchy. The second Painlevé hierarchy is an infinite sequence of nonlinear ordinary differential equations obtained by recurrence relations applied to the celebrated second Painlevé equation. This hierarchy is a symmetry reduction of the mKdV hierarchy and for this reason, it is believed that its elements all posses the Painlevé Property, i.e. all the movable singularities of all solutions are poles. We conjecture that, in fact, all solutions of every equation in the hierarchy are meromorphic in the complex x-plane. Since ∞ is a non-Fuchsian singularity for each equation in the hierarchy, one expects that the generic solution possesses asymptotic behaviours that are given by (meromorphic combinations of) hyperelliptic functions. It is our aim to prove this fact.
Contact details while at MIMS
Further information
- Professor Joshi's webpage