Log Canonical and Rational Singularities

AbstractSandor Kovacs98 18357 科瓦奇教授将研究日志规范奇异与特别考虑到变形，消失定理，与杜波依斯奇点，科勒猜想，和cohomologically无关紧要的奇点连接。 他将试图建立有效性的猜想，非同构的最小模型的代数品种不能变形到对方。 他还将考虑推广的相对德拉姆复杂类似杜波依斯的绝对复杂，试图建立一个相对德拉姆上同调理论的非光滑态射。这是一个代数几何的项目。 在这个现代数学的重要分支中，曲线和曲面等几何对象使用代数构造来建模。 现在代数对象的抽象性质使数学家更容易研究它们。 然而，代数模型的新发现的性质又转化为几何曲线和曲面的新性质。 Kovacs教授对奇点特别感兴趣，奇点是几何物体自我坍缩的地方。 他还希望建立一个完全有效的受过教育的猜测，说小的几何变化，某些重要的对象，代数簇，不能改变代数性质的品种。 随着这些强大的代数技术的发展，其他科学家和数学家将有更多的工具来解决他们的几何问题。代数几何已经成为理论物理学的一个重要工具。

AbstractSandor Kovacs98 18357 Professor Kovacs will study log canonical singularities with special regard to deformations, vanishing theorems, connections with Du Bois singularities, Kollar's conjecture, and cohomologically insignificant singularities. He will try to establish the validity of the conjecture that non-isomorphic minimal models of an algebraic variety cannot be deformed into each other. He will also consider generalizations of the relative de Rham complex similar to Du Bois' absolute complex in an attempt to build a relative de Rham cohomology theory for non smooth morphisms. This is a project in algebraic geometry. In this important branch of modern mathematics, geometric objects like curves and surfaces are modeled using algebraic constructions. The abstract nature of the now algebraic objects makes them easier for mathematicians to study. Nevertheless, newly discovered properties of the algebraic models translate back to new properties of the geometric curves and surfaces. Prof. Kovacs is particularly interested in singularities, places where the geometric objects sort of collapse on themselves. He also hopes to establish the complete validity of an educated guess that says small geometric changes to certain important objects, algebraic varieties, cannot change the algebraic properties of the varieties. As these powerful algebraic techniques are developed, other scientists and mathematicians will have additional tools to use on their geometric problems. Already algebraic geometry has become an important tool in theoretical physics.