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

主要研究人员将研究几何问题， 变分法和数学中出现的分析 物理学这些问题包括水平集的平均运动 的电弱理论的多点存在性 一个周期的细胞，和边界值问题的表面， 常高斯曲率此外，他们还将研究各种 与最小表面相关的问题，例如刚性和 周期极小曲面的对称性损失 数学物理中的许多问题可以用 一个最小化问题。例如，给定中的闭合曲线， 人们可能想找到一个平均曲率等于 零，并以给定的曲线作为其边界。这个问题 可以重新表述为寻找最小曲面的问题 以给定曲线为边界的区域。这样的表面发生在 物理设置为气泡，例如肥皂泡。这种类型 问题被称为变分问题。 校长 研究人员将使用以下技术来研究这些问题： 分析和几何。

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.