Motivic stable homotopy groups

摘要奖：DMS 1202213，主要研究者：丹尼尔C。Isaksen PI将在动机同伦理论中进行计算，即，代数簇的同伦理论 运动同伦理论的主要目的是将同伦理论的有力方法从拓扑学引入代数几何。 PI希望通过使用代数几何的结果来发现有关经典同伦理论的新信息，从而扭转这种关系。 例如，PI已经使用C上的motivic同伦理论在大范围内完全重建了经典的2-完全Adams-Novikov谱序列。 类似地，R上的动机计算揭示了等变稳定同伦理论的新信息。具体来说，PI感兴趣的是亚当斯谱序列的motivic版本，与基域R或C，其中彻底的计算是可能的。同伦理论是一种技术，用于研究几何对象的某些种类的变形。 同伦方法通常允许以其他方式无法实现的具体计算。 自世纪中期以来，稳定同伦群的计算一直是拓扑学研究的一个重要课题。 虽然已经取得了巨大进展，但仍有许多未知数。 在20世纪90年代，Fabien Morel和弗拉基米尔Voevodsky发展了motivic同伦理论。 类似于经典同伦理论，动机同伦理论允许我们研究代数对象的某些种类的变形。 他们的目标是将同伦理论的技巧引入代数领域。 PI的目标是相反的：使用代数结果来获得有关经典同伦理论的新信息。

AbstractAward: DMS 1202213, Principal Investigator: Daniel C. IsaksenThe PI will carry out computations in motivic homotopy theory, i.e., the homotopy theory of algebraic varieties. The main point of motivic homotopy theory is to import the powerful methods of homotopy theory from topology to algebraic geometry. The PI would like to turn this relationship around by using results from algebraic geometry to discover new information about classical homotopy theory. For example, the PI has used motivic homotopy theory over C to completely reconstruct the classical 2-complete Adams-Novikov spectral sequence in a large range. Similarly, motivic computations over R have revealed new information about equivariant stable homotopy theory. Specifically, the PI is interested in the motivic version of the Adams spectral sequence, with base field R or C, where thorough computations are possible.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. Their goal was to import the techniques of homotopy theory into the realm of algebra. The PI's goal is the reverse: to use algebraic results to obtain new information about classical homotopy theory.