Mathematical Sciences: Differential Structures

Eugenio Calabi将继续研究具有有限多个规定奇点的紧致黎曼曲面上具有常曲率度规存在的障碍；具有极值仿射不变面积凸曲面的Bernstein问题极小子变异的奇异性与其摩尔斯指数的关系。Hermann Gluck将继续他在齐次空间中最小化循环的研究；Blaschke问题和大球对球的振动；以及曲面上摩尔斯-小向量场空间的拓扑结构。沃尔夫冈·齐勒将对爱因斯坦度量、校准和对称空间的刚性进行研究。卡拉比的研究领域是几何变分问题。他将研究的问题包括极值Kahler度量、仿射高原问题以及测地线和最小子变分理论。Gluck将继续寻找李群和齐次空间在其同调类中体积最小的子变种。他还将研究著名的Blaschke猜想，即每个Blaschke流形都与一个圆球体或射影空间等距。他对莫尔斯-小向量场的研究是在二球上寻找这样的场集合的分量的同伦类型。Ziller在爱因斯坦度规上的工作是寻找关于流形的信息，这些流形承认这样的度规，也就是说，具有恒定的里奇曲率。他对对称空间的研究包括在保持曲率边界的同时对度规进行局部改变。这样的变化导致度量与原始度量是等距的。他还将使用校准的方法来研究哪些次格拉斯曼动物在其同源类中具有最小的体积。

Eugenio Calabi will continue studying obstructions to the existence of a metric with constant curvature on a compact Riemann surface with finitely many prescribed singularities; Bernstein problems associated with convex surfaces having extremal, affinely invariant area; relations between the singularities of minimal subvarieties and their Morse index. Hermann Gluck will continue his work on minimizing cycles in homogeneous spaces; the Blaschke problem and fibrations of spheres by great spheres; and the topology of the space of Morse- Smale vector fields on surfaces. Wolfgang Ziller will carry out research into Einstein metrics, calibrations and the rigidity of symmetric spaces. Calabi's research is in the area of geometric variational problems. The problems he will study concern extremal Kahler metrics, the affine Plateau problem and a variational theory of geodesics and minimal subvarieties. Gluck will be continuing the search for the subvarieties of Lie groups and homogeneous spaces which are volume minimizing in their homology classes. He will also work on the famous Blaschke conjecture that every Blaschke manifold is isometric to a round sphere or projective space. His work on Morse-Smale vector fields is concerned with finding the homotopy type of the components of the set of such fields on the two-sphere. Ziller's work on Einstein metrics is concerned with finding information about manifolds which admit such metrics, that is, having constant Ricci curvature. His work on symmetric spaces involves making local changes on the metric whilst preserving the bounds on the curvature. Such changes result in metrics isometric to the original metric. He will also use the method of calibrations to investigate which subgrassmannians have minimal volume in their homology class.