Extremisers and near-extremisers for central inequalities in harmonic analysis, geometric analysis and PDE

Isoperimetric inequalities are examples of sharp inequalities with intrinsic geometric content. The classical sharp inequality in two dimensions compares the area of an enclosed region in the plane with the length of its perimeter. Thanks to the sharpness, it is possible to deduce immediately that amongst all regions with a fixed perimeter length, the circle produces the maximum area. More generally, isoperimetric inequalities of this flavour can be used to identify maximising or minimising objects within a class, and they have applications in physical problems (for example, liquid droplet formation). Furthermore, these physical applications often demand that a "stable" version of the underlying sharp inequality is proved (to allow for the effect of external perturbative forces, for instance).

The main thrust of this broad project is to understand the sharpness and stability of some central inequalities which occur in harmonic analysis, geometric analysis and differential equations. These include the powerful Brascamp-Lieb inequality, which can be viewed as a generalised isoperimetric-type inequality and has fascinating and wide-ranging applications in the above fields and beyond. The project also encompasses sharp space-time inequalities for solutions of important differential equations, including the Schrodinger and wave equations.

In the short term, the primary beneficiaries will be the proposer and the post-doctoral researcher. The model is that the 12 month period of the post-doctoral researcher will coincide with the first 12 months of the project. During this time, leading experts on aspects of the proposal will be invited to spend extended visits at the School of Mathematics in Birmingham. This strategy for the structure of the project will maximise the impact of the project in several ways. It is anticipated that the scientific impetus provide by these visits during the early phase of the project will have an important role to play in the quality of the overall research output from the project. The visits will also provide a platform for the post-doctoral researcher, and the other members of the Analysis Group at Birmingham, to interact with leading mathematicians in their field. It is also intended that current proposal's links to aspects of arithmetic combinatorics will contribute to the development of strong ties between the Analysis Group and the highly successful Combinatorics Group at Birmingham.

At an intermediate time-scale, the beneficiaries are expected to include mathematicians working in the allied fields of euclidean harmonic analysis, geometric analysis and dispersive partial differential equations. A major feature of the current proposal is its breadth and timely response to recent breakthroughs in these prominent fields. These fields are particularly strongly represented in the UK and therefore the current proposal has an unusually inclusive position on the UK mathematical landscape.

In the longer term, impact beyond academic beneficiaries is likely to arise from the fundamental scientific nature of the current proposal and the resulting applications to understanding models of various important physical phenomena. At the centre of the current proposal is the powerful Brascamp-Lieb inequality; this inequality and closely related geometric inequalities enjoy a remarkable number applications. To give a flavour, these inequalities have direct connections to inequalities involving entropy and, for example, there are consequences of such inequalities to aspects of information theory. Also, Brascamp-Lieb inequalities have direct links to probability theory and the results here may be used to determine whether certain statistical tests are unbiased. Nonlinear generalisations of the Brascamp-Lieb inequality established by the proposer have found applications to the equations governing the propagation of Langmuir waves in a plasma. Further applications of such inequalities can be made to obstacle scattering in diffraction tomography. Also, closely related are isoperimetric-type inequalities which, for example, can be used to explain the shape of liquid droplets and the formation of crystals.

Included in the remainder of the proposal are questions concerning fundamental equations from quantum and classical physics, the Schrodinger equations and wave equations. For these differential equations, Strichartz estimates provide a means for understanding their solutions and the current proposal intends to generate novel viewpoints and tools for treating such estimates.