Mathematical Sciences: Applications in the Geometric Calculus of Variations

9626560皮茨该项目属于最小曲面理论领域。研究者计划利用变化的几何演算来考虑最小曲面的去广格化和构造。此外，研究者提出证明任何紧致黎曼流形包含无限多个不同的极小超曲面，再次使用变分方法。最小表面最早出现在比利时物理学家约瑟夫·高原在1850年左右做的一个肥皂膜实验中，高原观察到一个肥皂膜表面由一个封闭的电线轮廓包围，在所有由相同的电线轮廓跨越的表面中，它的表面积最小。最小曲面被证明有其他显著的数学性质（任何一点的平均曲率必须等于零）；他们的研究涉及现代数学的几个分支。此外，最小表面可用于模拟凝聚态物质，例如，复合聚合物系统。

9626560 Pitts The project lies in the area of minimal surface theory. The investigator plans to utilize the geometric calculus of variations to consider desingularization and construction of minimal surfaces. In addition, the investigator proposes to show that any compact Riemannian manifold contains infinitely many distinct minimal hypersurfaces, again using the variational approach. Minimal surfaces first arose in a soap film experiment done by the Belgian physicist Joseph Plateau around 1850, where Plateau observed that a soap film surface bounded by a closed wire contour has the least surface area amongst all surfaces spanned by the same wire contour. Minimal surfaces turn out to have other remarkable mathematical properties (the average curvature at any point must equal zero); their study involves several branches of modern mathematics. Additionally, minimal surfaces can be used to model condensed matter, e.g., compound polymer systems.