Stable Homotopy Groups of Spheres and Motivic Homotopy Theory

Explore the fundamental problem of computing stable homotopy groups of spheres in this 46-minute lecture by Guozhen Wang and Zhouli Xu. Delve into classical methods, understand their limitations through Mahowald's Uncertainty Principles, and discover a new technique using motivic homotopy theory. Learn how this innovative approach streamlines computations through previously known ranges and extends new calculations up to dimension 90. Gain insights into the connections between stable homotopy groups and other areas of topology, including cobordism theory and the classification of smooth structures on spheres. Examine topics such as the generalized Poincaré conjecture, stable range computations, stemwise computations, and the Classical Adams Spectral Sequence. Investigate motivic stable homotopy groups of spheres, motivic generalized homology theory, and the strategy of stem-wise computations. Conclude with a discussion on general questions, the Chow heart SH(K), Postnikov-Whitehead Tower, and future conjectures in this field of study.