A: Arithmetik und Geometrie von Calabi-Yau Räumen; B: Lagrangesche Zyklen und Lagrangesche Singularitäten; C: Geometrie von Hilbertschemata; D: Irreduzible holomorph sympletkrischen Mannigfaltigkeiten; E: Flächensingularitäten mit rationalen Homologiesphä

Calabi-Yau空间是代数几何和复分析中的重要对象。镜像对称现象在物理学和几何学之间提供了一种意想不到的联系。一个特别有趣的子类是由刚性卡-丘空间提供的，它具有算术意义。拉格朗日奇点在数学的几个部分中起着重要的作用。他们的系统研究是由Arnold和Givental发起的。消失圈的拉格朗日性质是Fano簇的范畴镜像对称的基础。这些关系的许多实例仍有待详细研究。希尔伯特格式提供了辛奇点的自然分解。它们可以作为研究轨道上同调与归结上同调之间关系的一般定理的典型例子。不可约全纯流形很难构造。只有两个系列和两个例外的例子是已知的变形。研究辛奇点作为构造新例子的工具是一个重大挑战。拓扑上最简单的一类曲面奇点是那些其链环是有理同调球面的曲面奇点。泛交换覆盖是完全交的猜想是该领域的一个突出的公开问题。

Calabi-Yau spaces are objects of key interest in algebraic geometry and complex analysis. The phenomenon of mirror symmetry gives an unexpected relation between physics and geometry. A particular interesting subclass of Calabi-Yaus is that provided by rigid Calabi-Yau spaces, which are of arithmetical significance. Lagrangian singularities play a significant role in several parts of mathematics. Their systematic study was initiated by Arnold and Givental. The Lagrangian nature of the vanishing cycles is at the basis of categorical mirror symmetry for Fano varieties. Many instances of these relations are still to be studied in detail. Hilbert schemes provide natural resolutions of symplectic singularities. They can be studied as archetypical examples for general conjectures on the relation between orbifold cohomology and the cohomology of resolutions. Irreducible holomorphic manifolds are hard to construct. Only two series and two exceptional examples are known up to deformation. It is major challenge to study symplectic singularities as a tool to construct new examples. The topologically simplest class of surface singularities are those whose links are rational homology spheres. The surprising conjecture that the universal abelian cover should be a complete intersection is one of the outstanding open problems in the field.