Mathematical Sciences: Minimal Surfaces, Complex and Symplectic Geometry

首席研究员将开发Kahler流形的最小曲面理论。特别强调的是复切点与黎曼几何的关系以及这些曲面的稳定性。他还将研究向量束的特征形式与束映射的奇异性之间的关系。他将把他过去关于谐波图的研究扩展到无限格拉斯曼人。一项新的辛拓扑研究将开始。最小曲面是数学家们的肥皂膜。将普通的弯曲电线浸入肥皂和水溶液中可以形成奇异的最小表面。近年来，在全面了解这些表面方面取得了巨大进展。现在，首席研究员和其他几何学家正在利用这些新理论，与新兴的卡勒流形几何物理学建立联系。

The principal investigator will develop a theory of minimal surfaces for Kahler manifolds. Special emphasis will be paid to the relationship between the complex tangent points and the Riemannian geometry and stability properties of these surfaces. He will also investigate the relationship between the characteristic forms of vector bundles and the singularities of bundle maps. And he will extend his past work on harmonic maps to infinite Grassmannians. A new study of symplectic topology will begin. Minimal surfaces are the mathematicians' soap films. Exotic minimal surfaces can be formed by dipping ordinary curved wires into a soap and water solution. Great progress toward a complete understanding of these surfaces has been made in recent years. Now, the principal investigator and other geometers are using these new theories to build connections with the emerging geometrical physics of Kahler manifolds.