Motivic homotopy theory

在这个项目中，PI打算使用motivic同伦理论为代数几何问题的研究创造新的工具。PI计划将经典的障碍理论转移到motivic环境中，具体目标是理解在代数闭域上找到代数纤维束截面的障碍。PI计划 研究代数配边，复配边的拓扑理论的代数版本，并进一步研究其与唐纳森-托马斯理论的联系。 此外，PI计划进一步研究Deligne-Goncharov motivic基本群。最后，PI计划进一步研究动机Postnikov塔，目的是为了更好地理解这座塔，以便更好地理解代数簇上各种有趣的广义上同调理论，以及光滑投射簇的动机。同伦理论是拓扑学的一个分支，它处理曲线，曲面和高维形状的基本性质。另一方面，代数几何试图理解方程解的性质，即使人们实际上不能明确地解出方程。在代数和拓扑这两个看似不相关的领域之间建立类比，往往是解决这两个领域难题的一种卓有成效的方法。 莫雷尔和Voevodsky转移了整个分支的拓扑，所谓的稳定同伦理论，代数设置，使思想从稳定同伦理论适用于问题的代数和数论。PI计划从同伦理论中提取一些特定的结构，使它们适应这种新的设置，并使用这些结构来解决代数几何中的问题。

In this project, the PI intends to use motivic homotopy theory to create new tools for the study of problems in algebraic geometry. The PI plans to transfer classical obstruction theory to the motivic setting, with the specific goal of understanding the obstructions to finding sections to algebraic fiber bundles over an algebraically closed field. The PI plans to study algebraic cobordism, an algebraic versions of the topological theory of complex cobordism, and to further examine its connection with Donaldson-Thomas theory. Additionally, the PI plans a further study of the Deligne-Goncharov motivic fundamental group. Finally, the PI plans a further study of the motivic Postnikov tower, with the goal of gaining a better understanding of this tower for a variety of interesting generalized cohomology theories on algebraic varieties, as well as for the motives of smooth projective varieties.Homotopy theory is a branch of topology, which deals with fundamental properties of curves, surfaces and shapes of higher dimension. Algebraic geometry, on the other hand, tries to understand the properties of solutions of equations, even when one cannot actually solve the equation explicitly. Creating analogies between the seemingly unrelated fields of algebra and topology has often been a fruitful approach to solving difficult problems in both fields. Morel and Voevodsky have transferred an entire branch of topology, called stable homotopy theory, to the algebraic setting, making ideas from stable homotopy theory applicable to problems in algebra and number theory. The PI plans to take a number of specific constructions from homotopy theory, adapt them to this new setting, and use these constructions to solve problems in algebraic geometry.