Log Canonical and Rational Singularities

Sandor Kovacs98 18357 Kovacs教授将研究对数正则奇点，特别涉及变形、消失定理、与Du Bois奇点的联系、Kollar猜想和上同调微不足道奇点。他将试图证明一个猜想的正确性，即一个代数簇的非同构极小模型不能相互变形。他还将考虑类似于Du Bois绝对复形的相对De Rham复形的推广，试图建立非光滑态射的相对De Rham上同调理论。这是一个代数几何的项目。在现代数学的这个重要分支中，像曲线和曲面这样的几何对象是用代数结构来建模的。现在的代数对象的抽象性质使数学家更容易研究它们。然而，新发现的代数模型的性质可以转化为几何曲线和曲面的新性质。科瓦奇教授对奇点特别感兴趣，奇点是几何物体自行坍塌的地方。他还希望建立一个有经验的猜测的完全有效性，即某些重要对象代数族的微小几何变化不能改变代数族的代数性质。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中的一个重要工具。

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.