Computational motivic homotopy theory

我在这个项目中的主要目标是在动力同伦理论中进行计算，即代数变异的同伦理论。Voevodsky现在著名的对Milnor猜想的证明主要是基于动机同伦理论中的技术。具体来说，我对亚当斯谱序列的动机版本感兴趣，基本场为R或C。只关注R或C的一个原因是，在这些场上进行彻底的计算是可能的，但计算仍然表现出非经典现象。动机亚当斯谱序列有一个主要的缺点，即它不收敛。即使谱序列不收敛，仍然可以得出许多关于动力稳定同伦群的结论。我相信这些和其他进一步的计算将对进一步研究动机同伦理论具有重要的指导意义。动机计算也有可能揭示一些关于经典稳定同伦群的新东西。同伦理论是研究具有某种变形的几何物体的一种技术。同主题方法通常允许具体的计算，否则是无法实现的。自20世纪中期以来，稳定同伦群的计算一直是拓扑学研究的一个重要课题。尽管取得了巨大的进展，但仍有许多未知之处。20世纪90年代，Fabien Morel和Vladimir Voevodsky发展了动机同伦理论。与经典同伦理论类似，动力同伦理论允许我们研究代数对象的某些变形。我的目标是在动力同伦理论中进行一些基本的计算。目的是更好地理解新的代数理论和经典的几何理论。

My main goal in this project is to carry out computations in motivic homotopy theory, i.e., the homotopy theory of algebraic varieties.Voevodsky's now famous proof of the Milnor Conjecture is based primarily on techniques in motivic homotopy theory. Specifically, I am interested in the motivic version of the Adams spectral sequence, with base field R or C. One reason to focus only on R or C is that thorough computations are possible over these fields, but the computations still exhibit non-classical phenomena. The motivic Adams spectral sequence has a major deficiency in that it is not known to converge. Even if the spectral sequence does not converge, it is nonetheless possible to draw many conclusions about motivic stable homotopy groups. I believe that these and other further computations will be an important guide to further study of motivic homotopy theory. There is also a chance that motivic computations will reveal something new about classical stable homotopy groups.Homotopy theory is a technique for studying geometric objects up to certain kinds of deformations. Homotopical approaches often allow for concrete calculations that are otherwise inaccessible. Computations of stable homotopy groups have been a major topic of research in topology since the middle of the 20th century. Although tremendous progress has been made, much remains unknown. In the 1990's, Fabien Morel and Vladimir Voevodsky developed motivic homotopy theory. In analogy to classical homotopy theory, motivic homotopy theory allows us to study algebraic objects up to certain kinds of deformations. My goal is to carry out some fundamental computations in motivic homotopy theory. The goal is both a better understanding of the new algebraic theory as well as the classical geometric theory.