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1. ℝ-motivic v1-periodic homotopy

   https://doi.org/10.2140/pjm.2024.330.43  

   Belmont, Eva; Isaksen, Daniel C; Kong, Hana Jia (July 2024, Pacific Journal of Mathematics)

   We compute the $$v_1$$-periodic $$\mathbb{R}$$-motivic stable homotopy groups. The main tool is the effective slice spectral sequence. Along the way, we also analyze $$\mathbb{C}$$-motivic and $$\eta$$-periodic $$v_1$$-periodic homotopy from the same perspective. 

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2. The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere

   https://doi.org/10.1112/plms.70006  

   Kong, Hana Jia; Quigley, J D (November 2024, Proceedings of the London Mathematical Society)

   Abstract We compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields. 

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3. Classical Stable Homotopy Groups of Spheres via $$\mathbb {F}_2$$-Synthetic Methods

   https://doi.org/10.1007/s42543-025-00098-y  

   Burklund, Robert; Isaksen, Daniel_C; Xu, Zhouli (April 2025, Peking Mathematical Journal)

   Abstract We study the$$\mathbb {F}_2$$ $F_2$-synthetic Adams spectral sequence. We obtain new computational information about$$\mathbb {C}$$ $C$-motivic and classical stable homotopy groups. 

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4. Stable homotopy groups of spheres

   https://doi.org/10.1073/pnas.2012335117  

   Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (September 2020, Proceedings of the National Academy of Sciences)

   We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence. 

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5. The homotopy Leray spectral sequence

   https://doi.org/10.1090/conm/745/15021  

   Asok, Aravind; Deglise, Frederic; Nagel, Johannes (April 2020, Contemporary Mathematics: Motivic Homotopy Theory and Refined Enumerative Geometry)

   In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its E2-page. Our description of the E2-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost’s cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work. 

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