Singularities in Characteristic Zero and Singularities in Positive Characteristic

该程序由Karl Schwede提出，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，通过将来自这两个截然不同的数学领域的看似不相关的概念归结为特征p，一本词典已经开发出来。然而，现有的词典似乎并不完整。例如，虽然有一个很好的正特性模拟的“对数终端奇点”，但没有已知的模拟“终端奇点”。施韦德将试图填补这些空白。此外，受这本词典的启发，Schwede将在正特征和特征零两个方面探索变种的局部和全局几何。施韦德计划探索的一个问题是，测试理想(乘数理想的正特征模拟)在出生映射下的表现如何。Schwede计划研究的另一个问题是Du Bois奇点是否变形。代数几何是一个非常重要和非常活跃的数学领域，与许多其他领域有很强的联系，包括弦理论和编码理论等截然不同的领域。显然，代数几何是研究由多项式方程(如y=x^2)的解组成的几何对象(称为代数簇)。在最基本的层面上，施韦德计划研究这些代数变体的几何与方程本身的代数性质之间的关系。在过去，这种相互作用在几何和代数理论中都带来了新的见解。在代数几何中，过去30年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，我们必须研究奇异簇(奇异簇的一个例子是二次锥解集，z^2=x^2 y^2)。施韦德建议研究的特定问题有望使人们更深入地理解这种分类中出现的变种和奇点。

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.