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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 and motivic homotopy theory

   Isaksen, Daniel C; Wang, Guozhen; Xu, Zhouli (December 2023, European Mathematical Society Press)

   Beliaev, Dmitry; Smirnov, Stanislav (Ed.)

   We consider the problem of computing the stable homotopy groups of spheres, including applications and history. We describe a new technique that yields streamlined computations through dimension 61 and gives new computations through dimension 90 with very few exceptions. We discuss questions and conjectures for further study, including a new approach to the computation of motivic stable homotopy groups over arbitrary base fields. We provide complete charts for the Adams spectral sequence through dimension 90. 

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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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