Singularities in Geometric Flows
Closing Date: 31 December 2020Apply Now
The candidate will analyze singularity formation in geometric evolution equations through the application of (local) a priori estimates and/or the analysis of ancient solutions.
Geometric evolution equations deform geometric objects (metrics, connections, maps or interfaces, for instance) according to some natural geometric law, often a gradient flow, which can be interpreted as a nonlinear parabolic partial differential equation. These equations tend to deform initial data towards some canonical equilibrium state, and therefore have a huge potential for organizing the geometric objects concerned.
For instance, geometric evolution equations have been crucial in the resolution of the Poincare conjecture in topology, the 1/4-pinched differentiable sphere conjecture in geometry, and the Riemannian Penrose conjecture in general relativity. They also arise naturally in various physical situations involving surface tension, in the dynamics of interfaces in materials science, in image processing, and even in the modeling of bushfire fronts.
Their analysis is complicated, however, by the onset of ‘singularities’, which generally form before an equilibrium state can be reached. Understanding and overcoming singularity formation is therefore key to unlocking their potential and is for this reason the most prominent theme of research in the area. This project concerns the deformation of submanifolds by their curvature.
There are two main approaches to the analysis of singuarities.
- The first is the method of a priori 'pinching' estimates. These estimates control the curvature and its derivatives, and hold up to the onset of singularities. 'Integrating' then estimates in regions of high curvature then yields a geometric description of such regions.
- The second approach is the method of scaling. In this case, one magnifies the solution in a neighbourhood of the singularity. Under certain conditions, this magnification process yields a limit solution. Moreover, such limits are of a very special form: they are 'ancient solutions'. Understanding ancient solutions therefore allows us to understand singularity formation.
This project involves the application of one or both of these approaches to study singularity formation in a range of settings.
PhD Scholarship details
Funding: $28,092 per annum (2020 rate) indexed annually. For a PhD candidate, the living allowance scholarship is for 3.5 years and the tuition fee scholarship is for four years. For an MPhil candidate, the living allowance and tuition fee scholarships are two years.
Supervisor: Mathew Langford
Available to: Domestic students
- The applicant will need to meet the minimum eligibility criteria for admission.
- A strong background in partial differential equations and differential geometry.
Interested applicants should send an email expressing their interest along with scanned copies of their academic transcripts, CV, a brief statement of their research interests and a proposal that specifically links them to the research project.
Please send the email expressing interest to Mathew.Langford@newcastle.edu.au by 5pm on 31 December 2020.
Applications Close 31 December 2020 Apply Now
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