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

等周不等式是具有内在几何内容的尖锐不等式的例子。经典的二维尖锐不等式将平面内封闭区域的面积与其周长进行比较。由于这种清晰度，可以立即推断出在所有具有固定周长的区域中，圆产生的面积最大。更广泛地说，这种风格的等周不等式可以用来识别类内对象的最大化或最小化，它们在物理问题(例如，液滴形成)中有应用。此外，这些物理应用通常要求证明基本的尖锐不等式的一个“稳定”版本(例如，考虑外部扰动力的影响)。这个广泛的项目的主要目的是了解在调和分析、几何分析和微分方程中出现的一些中心不等式的尖锐和稳定性。其中包括强大的BrasCamp-Lieb不等式，它可以被视为一种推广的等周型不等式，在上述领域和其他领域都有令人着迷的广泛应用。该项目还包括重要微分方程解的尖锐时空不等式，包括薛定谔方程和波动方程。

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.