Motivic homotopy theory

在这个项目中，PI打算使用动机同伦理论为代数几何问题的研究创造新的工具。PI计划将经典的阻碍理论转移到动力设置中，其具体目标是了解在代数封闭场上寻找代数纤维束截面的障碍。PI计划研究代数共体，这是复共体拓扑理论的代数版本，并进一步研究它与Donaldson-Thomas理论的联系。此外，PI计划进一步研究delign - goncharov动机基群。最后，PI计划对动机波斯特尼科夫塔进行进一步的研究，目的是为了更好地理解这个塔的各种有趣的代数变异的广义上同调理论，以及光滑投影变异的动机。同伦理论是拓扑学的一个分支，研究高维曲线、曲面和形状的基本性质。另一方面，代数几何试图理解方程解的性质，即使一个人实际上不能明确地解出方程。在代数和拓扑学这两个看似无关的领域之间建立类比，往往是解决这两个领域难题的有效方法。Morel和Voevodsky将整个拓扑分支——稳定同伦理论——转移到代数集合中，使稳定同伦理论的思想适用于代数和数论问题。PI计划从同伦理论中获取一些特定的结构，使它们适应这个新的环境，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.