Singularities in Characteristic Zero and Singularities in Positive Characteristic

Karl Schwede提出的程序旨在研究高维代数几何与正特征交换代数之间的关系。 在过去的30年里，一本字典已经开发出来，通过简化到特征p，将来自这两个不同数学领域的看似无关的概念联系起来。 然而，这本词典似乎是不完整的。 例如，虽然有一个很好的正特性类似物的“日志终端奇点”，没有已知的类似物的“终端奇点”。 Schwede将尝试填补这些空白。 此外，受此字典的启发，施韦德将探讨当地和全球的几何品种的积极特征和特征零。 一个问题，施韦德计划探讨的是如何测试理想（积极的特征模拟乘数理想）的行为下双有理地图。 另一个问题，施韦德计划研究的是是否杜波依斯奇点变形。代数几何是一个中心的重要和非常活跃的数学领域与许多其他领域的密切联系，包括领域作为不同的弦理论和编码理论。 代数几何是对由多项式方程（如y = x^2）的解组成的几何对象（称为代数簇）的研究。 在最基本的水平，施韦德计划研究之间的关系几何这些代数品种，代数性质的方程本身。 在过去，这种相互作用导致了几何和代数理论的新见解。 在代数几何中，过去30年来的主要研究领域之一是代数簇的分类--“最小模型程序”。 为了做到这一点，必须研究奇异簇（奇异簇的一个例子是二次锥的解集z^2 = x^2 + y^2）。 施韦德提出的具体问题，研究将有望导致更深入地了解品种和奇异性，出现在这种分类。

The program proposed by Karl Schwede aims to study relations between higher dimensional algebraic geometry and positive characteristic commutative algebra. Over the past 30 years, a dictionary has been developed linking, by reduction to characteristic p, seemingly unrelated concepts coming from these two distinct areas of mathematics. However, the dictionary as it exists seems incomplete. For example, while there is a good positive characteristic analog of ``log terminal singularities'', there is no known analog of ``terminal singularities''. Schwede will attempt to fill in these gaps. Furthermore, inspired by this dictionary, Schwede will explore both the local and global geometry of varieties in positive characteristic and characteristic zero. One problem that Schwede plans to explore is how the test ideal (a positive characteristic analog of the multiplier ideal) behaves under birational maps. Another problem that Schwede plans to study is whether Du Bois singularities deform.Algebraic geometry is a centrally important and very active field of mathematics with strong ties to many other areas including fields as disparate as string theory and coding theory. Explicitly, algebraic geometry is the study of geometric objects (called algebraic varieties) made up of the solutions to polynomial equations (such as y = x^2). At the most basic level, Schwede plans to study relations between the geometry of these algebraic varieties, with algebraic properties of the equations themselves. In the past, this interplay has led to new insights in both the geometric and algebraic theories. In algebraic geometry, one of the major areas of research in the last 30 years has been the classification of algebraic varieties -- the "minimal model program". In order to accomplish this, one must study singular varieties (an example of a singular variety is the solution set to the quadric cone, z^2 = x^2 + y^2). The particular questions that Schwede proposes to study will hopefully lead to a deeper understanding of the varieties and the singularities that appear in this classification.