Analytical and geometrical problems in calculus of variations and partial differential equations

首席研究员的研究将集中在不同的数学领域。首先，他计划研究最优运输理论。这个问题包括寻找最便宜的方式将质量分布从一个地方转移到另一个地方，最近在气象学、生物学和种群动力学中有许多应用，或者例如研究天线设计问题。研究者已经研究这个课题好几年了。他还打算研究不可压缩流体的欧拉方程，并研究来自量子物理学的半经典极限问题。最后，他希望将他最近的一些结果应用于经典等周不等式的一些“改进”版本，以研究在外部电位作用下晶体形状的稳定性。上述研究在许多不同的领域都有应用：最优运输问题显然适用于经济学，但它也被证明是非常跨学科的，与其他数学领域，如几何、概率和偏微分方程，以及物理学和生物学都有联系。欧拉方程和半经典极限是物理学和量子物理学中的经典问题，对它们的数学研究可以加深对物理现象本身的理解。最后，对外部电位作用下晶体的刚性的研究将有助于解释目前在实验中观察到但尚未完全理解的许多现象。

The research of the principal investigator will be focused on different mathematical areas. First of all, he plans to work on optimal transport theory. This problem, which consists in finding the cheapest way to transport a distribution of mass from one place to another, has recently found many applications in meteorology, biology and populations dynamics, or for instance to study the antenna design problem. The investigator has already worked on this subject for several years. He also intends to study Euler equations for incompressible fluids and to work on problems of semiclassical limit coming from quantum physics. Finally, he wishes to apply some of his recent results on some "improved" version of the classical isoperimetric inequalities to study the stability of the shapes of crystals under the action of an external potential.The research described above has applications in many different areas: the optimal transport problem has obvious application to economics, but it has also shown to be very interdisciplinary, with links with other areas of mathematics like geometry, probability and partial differential equations, and also with physics and biology. Euler equations and semiclassical limits are classical problem in physics and quantum physics, and their mathematical study may increase deeper understanding of the physical phenomena themselves. Finally, the study of the rigidity of crystals under exterior potential should help to explain many phenomena which are currently observed in experiments but not yet completely understood.