Geometry of Lipschitz mappings

Non-differentiable functions and deformations have become essential in a wide range of applications of today's Mathematical Analysis, from understanding the motion of fluids to controlling robots and modelling stock exchange prices. An important class of functions that model distortions and deformations are Lipschitz maps. One of the key problems in studying the local behaviour of Lipschitz maps is whether they look like affine or smooth ones close to some points, or even close to many points. The goal of the proposed research is to contribute to the solution of such key open problems. The research will employ new geometric constructions and methods; for instance, it is proposed that one can form a set consisting of points distributed so sparsely that they almost avoid any curve, yet any Lipschitz map will look like a smooth map (be differentiable) at a typical point of this 'thin' set. This will in particular lead to a solution of a long-standing open question in geometric measure theory.

The proposed research will have impact in several areas over different time periods. One of the most immediate benefits of the programme will be the advancement in the area of mathematics known as geometric nonlinear functional analysis and in research fields where the theory of Lipschitz functions plays a significant role - PDE, calculus of variations, optimal control. Thus both the results and the methodology of the proposed research will be of broad appeal to pure and applied mathematicians. Alongside the impact on knowledge, great benefits will be brought about by training the research assistant. As part of the training, the RA will write research papers, regularly give seminars and attend conferences. Another avenue of impact, also in the near future - in the next 5 years - would be to generate interest in the United Kingdom and beyond in the proposed research field. It is expected that the results of this programme will attract the attention of researchers working in different fields such as ergodic theory, partial differential equations and combinatorics. It is anticipated that as this research area takes off in the United Kingdom, more job openings and research positions in the field will appear. The proposer intends to use regular meetings of the Analysis Working Group in the University of Birmingham for presenting most recent achievements and to use the ongoing Analysis Seminar series in Birmingham which the proposer will be organising this academic year. She would also draw on the experience of experts at Birmingham for generating contacts in different fields. It is proposed to use the grant's travel budget to disseminate results of the project via conferences in US, Europe and the UK and departmental seminars. A final and very important longer term goal is to raise the profile of the United Kingdom itself in the research fields of geometric nonlinear functional analysis and nonlinear Banach space theory. It is anticipated that this will be achieved to some extent in the next 5-10 years; by generating results that interest world class experts, by the new questions that arise from them and the new avenues of exploration they open, and by proposing international collaborations. For the latter, the proposer already has contacts in the USA, France, Israel, the Czech Republic and Italy, who would be interested in inviting her as a visiting researcher. She would also present her work on the international arena in conferences and extended workshops. For 10 years from now and beyond the proposer expects to see new research groups in the UK arise in this area, who will actively contribute to research at national and international level.