Mathematical Sciences: Non-Linear Problems in Differential Geometry and Mathematical Physics

主要研究人员将研究几何分析中在变分和数学物理中出现的问题。这些问题包括水平集的平均曲率运动，周期胞腔中电弱理论的多涡旋的存在性，以及常高斯曲率曲面的边值问题。此外，他们还将研究与极小曲面有关的各种问题，如周期极小曲面的刚性和对称性损失。数学物理中的许多问题都可以用极小化问题来表示。例如，给定空间中的一条闭合曲线，人们可能想要找到一个平均曲率等于零的曲面，该曲面以给定的曲线为边界。这个问题可以重述为求以给定曲线为边界的最小面积曲面的问题。这样的表面在物理环境中以气泡的形式出现，例如肥皂泡。这类问题称为变分问题。主要研究人员将使用分析和几何的技术来研究这些问题。

The principal investigators will study problems in geometric analysis that arise in the calculus of variations and mathematical physics. These problems include the motion of level sets by mean curvature, existence of multivortices of the electroweak theory in a periodic cell, and boundary value problems for surfaces of constant Gauss curvature. In addition they will study various problems associated with minimal surfaces such as rigidity and symmetry loss for periodic minimal surfaces. Many problems in mathematical physics may be framed in terms of a minimization problem. For example, given a closed curve in space one might want to find a surface with average curvature equal to zero and which has the given curve as its boundary. This problem may be restated as the problem of finding the surface of minimum area with the given curve as boundary. Such a surface occurs in a physical setting as a bubble, for example, a soap bubble. This type of problem is known as a variational problem. The principal investigators will study these problems using techniques from analysis and geometry.