Mathematical Sciences: Minimal Surfaces, Complex and Symplectic Geometry

主要研究人员将为Kahler流形发展一个极小曲面理论。将特别强调复切点与这些曲面的黎曼几何和稳定性之间的关系。他还将研究向量丛的特征形式和丛映射的奇异性之间的关系。他将把他过去在调和映射方面的工作推广到无限的格拉斯曼尼亚人。一项新的辛拓扑研究将开始。极小曲面是数学家的肥皂片。通过将普通的弯曲金属丝浸入肥皂和水溶液中，可以形成奇异的极小表面。近年来，在全面了解这些表面方面取得了很大进展。现在，首席研究员和其他几何学家正在使用这些新理论来建立与新兴的卡勒流形几何物理的联系。

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.