Positivity of Cycles

代数几何是研究由多项式方程定义的形状。 代数几何领域是现代数学的核心，描述在物理学，计算机科学和生物学的某些子领域中发挥重要作用的空间。 形状最重要的特征是它的曲率--它局部偏离平面的程度。 当前研究的一个活跃主题是曲率如何影响形状的其他属性。 例如，它似乎会影响定义多项式方程的整数解的数量。 本研究计划旨在深化曲率及其在代数几何中的意义的研究。该计划的第一个目标是从数值的角度研究子簇的几何。 更确切地说，该项目解决了如何在伪有效锥的位置的一个数字类涉及到这个类的倍数的代表的渐近几何。 这是约数的一个很好的故事，最近的工作表明所有的子变种都有一个类似的美丽理论。 其次，该项目将研究Manin猜想预测有界高度的合理点的增长率。 使用最近开发的几何技术（称为最小模型程序），研究人员将分析马宁猜想的几何基础。 这项工作预计将提供积极的证据的猜想，并确定新的例子，躺在边界上的已知技术。

Algebraic geometry is the study of shapes defined by polynomial equations. The field of algebraic geometry is central to modern mathematics, describing spaces that play important roles in physics, computer science, and certain subfields of biology. The most important feature of a shape is its curvature -- how much it deviates locally from a flat plane. An active topic of current research is how the curvature affects the other properties of a shape. For example, it seems to influence the number of integer solutions to the defining polynomial equations. This research project aims to deepen the study of curvature and its implications in algebraic geometry.The first goal of the project is to study the geometry of subvarieties from a numerical perspective. More precisely, the project addresses how the position of a numerical class in the pseudo-effective cone relates to the asymptotic geometry of representatives of multiples of this class. This is a well-established story for divisors, and recent work suggests a similar beautiful theory for all subvarieties. Second, the project will study Manin's conjecture predicting the growth rate of rational points of a bounded height. Using recently developed geometric techniques (known as the minimal model program), the investigator will analyze the geometric underpinnings of Manin's conjecture. This work is anticipated to provide positive evidence for the conjecture and to identify new examples that lie on the boundary of known techniques.