This is an exciting opportunity for an enthusiastic and motivated student to pursue a cotutelle PhD in Pure Mathematics at Deakin University, Melbourne Burwood Campus and IIT-Madras, Chennai India, working with Dr. Lashi Bandara (Deakin) and Dr. Aprameyan Parthasarathy (IIT-Madras).
The successful candidate will spend a total of 3 years at Deakin University and 1 year at IIT-Madras, receiving a stipend of AUD$35,500 per annum (tax exempt). On the mandatory year at IIT-Madras, they will be provided travel insurance through Deakin University.
Melbourne has an active mathematical culture including nearby at Monash and Melbourne Universities. Chennai is similar with institutes including the Chennai Mathematical Institute and the Institute for Mathematical Sciences at Chennai.
The resolution of the Kato Square Root problem by Auscher-Hofmann-Lacey-McIntosh-Tchamitchian in 2002, which took in excess of 30 years, was a major achievement in mathematics. First proposed by Kato in the 1960s, the problem was to identify the domain of the square root of elliptic differential operators with complex and measurable coefficients in divergence form. Such equations arise naturally in partial differential equations and have applications to problems in engineering and physics.
Novel methods linking operator theory and real-variable harmonic analysis were developed in its resolution. These methods have become more widely applicable in signal processing through wavelet theory. A generalisation of this problem was studied and resolved in the setting of Lie groups in 2012 by Bandara-ter Elst-McIntosh. This concerned a subelliptic version of the problem, arising from algebraic vectorfields of a dimension less than that of the Lie group. At the same time, geometric generalisations were also obtained but in the elliptic setting.
This leaves a major research gap in the geometric subelliptic setting. The aim of the proposed project would be to study this problem for subelliptic geometric operators in Riemannian symmetric spaces, of which the Lie groups results are a special case. Its objectives are to identify sufficiently general geometric conditions under which the subelliptic Kato square root problem admits a resolution. This would likely begin with considering the special case of curved Lie groups before moving on to the more general setting of Riemannian symmetric spaces.
REQUIRED CRITERIA
DESIRABLE CRITERIA
Link to the official project page:
Please follow directions there (the "Apply Now" button is on the top right).
Information regarding the cotutelle program can be found on:
If you like more information about this opportunity or project, please email Lashi Bandara or Aprameyan Parthasarathy