The Minimal Model Program in Positive and Mixed Characteristics

代数几何是数学中一个活跃而有影响的领域。它的目标是理解代数变体，代数变体是由代数方程描述的几何形状。这样的对象不仅在许多数学领域是必不可少的，而且在其他学科如工程、计算机科学或生物学中也自然而然地出现。主要研究者的研究围绕着对所有代数变种进行分类的最终目标展开。PI计划利用其他基本数学领域的最新突破，即数论和交换代数，以显著扩展这种分类的范围。主要研究人员打算应用与F-正则性、+-正则性和混合特征完善相关的新技术，以便更好地理解具有正特征和混合特征的方案的几何和交换代数。该项目的第二个方面是通过关注第三维度和第四维度，在积极和混合特征方面进一步推进最小模型方案。PI计划应用这些结果来洞察奇点和射影变种从正特征到特征零的可升性，以及有限域上变种上有理点的存在。这一奖项反映了NSF的法定使命，并通过使用基金会的智力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.