Singularities in Characteristic Zero and Singularities in Positive Characteristic

由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，人们开发了一本字典，通过还原特征p，将来自这两个不同数学领域的看似不相关的概念联系起来。然而，现存的词典似乎不完整。例如，虽然有一个很好的“日志终端奇点”的正特性模拟，但没有已知的“终端奇点”的模拟。瑞士将试图填补这些空白。此外，受该词典的启发，Schwede将探索正特征和特征零变异的局部和全局几何。Schwede计划探索的一个问题是测试理想（乘数理想的正特征模拟）在两族映射下的行为。Schwede计划研究的另一个问题是杜波依斯奇点是否会变形。代数几何是数学中一个非常重要和活跃的中心领域，与许多其他领域有很强的联系，包括弦理论和编码理论等不同的领域。明确地说，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（称为代数变量）的研究。在最基本的层面上，Schwede计划研究这些代数变量的几何关系，以及方程本身的代数性质。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，近30年来的主要研究领域之一是代数变体的分类——“最小模型程序”。为了做到这一点，必须研究奇异变分（奇异变分的一个例子是二次锥的解集，z^2 = x^2 + y^2）。Schwede提出要研究的具体问题，有望使我们对这种分类中出现的多样性和独特性有更深入的了解。

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.