The Minimal Model Program in Positive and Mixed Characteristics

代数几何是一个活跃而有影响的数学领域。它的目标是理解代数变量，代数变量是由代数方程描述的几何形状。这些对象不仅在许多数学领域是必不可少的，而且在工程、计算机科学或生物学等其他学科中也很自然地出现。首席研究员的研究围绕着对所有代数变种进行分类的最终目标。PI计划利用最近在其他基础数学领域的突破，即数论和交换代数，来显著扩展这种分类的范围。为了更好地理解正特征和混合特征方案的几何和交换代数，本课题拟应用与f正则性、+正则性和混合特征完美性相关的新技术。该项目的第二个方面是通过关注第三和第四维度，进一步推进最小模型计划的积极和混合特征。PI计划应用这些结果来深入了解奇点和射影变异从正特征到特征零的可提性，以及有限域上变异上有理点的存在性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry is an active and influential field of mathematics. Its goal is to understand algebraic varieties, which are geometric shapes described by algebraic equations. Not only are such objects essential in many fields of mathematics, but they also come up naturally in other disciplines such as engineering, computer science, or biology. The principal investigator's research centers around the ultimate goal of classifying all algebraic varieties. The PI plans to capitalize on recent breakthroughs in other fundamental fields of mathematics, namely number theory and commutative algebra, to significantly extend the scope of this classification.The principal investigator intends to apply new techniques, related to F-regularity, +-regularity, and mixed characteristic perfections, in order to provide a better understanding of the geometry and commutative algebra of schemes in positive and mixed characteristics. A second aspect of the project is to further the Minimal Model Program in positive and mixed characteristics by focusing on dimensions three and four. The PI plans to apply these results to obtain insight into the liftability of singularities and projective varieties from positive characteristic to characteristic zero, as well as the existence of rational points on varieties over finite fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.