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1. 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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2. Stable homotopy groups of spheres: from dimension 0 to 90

   https://doi.org/10.1007/s10240-023-00139-1  

   Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (June 2023, Publications mathématiques de l'IHÉS)

   Abstract Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic Adams spectral sequence for $$C\tau $$ C τ and the algebraic Novikov spectral sequence for $$BP_{*}$$ B P ∗ , we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties. 

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3. 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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4. Generalized $$\mathbb Z$$-homotopy fixed points of $$C_n$$ spectra with applications to norms of $$MU_{\mathbb R}$$

   Hill, Michael A; Zeng, Mingcong (January 2020, New York journal of mathematics)

   null (Ed.)

   We introduce a computationally tractable way to describe the $$\mathbb Z$$-homotopy fixed points of a $$C_{n}$$-spectrum $$E$$, producing a genuine $$C_{n}$$ spectrum $$E^{hn\mathbb Z}$$ whose fixed and homotopy fixed points agree and are the $$\mathbb Z$$-homotopy fixed points of $$E$$. These form the bottom piece of a contravariant functor from the divisor poset of $$n$$ to genuine $$C_{n}$$-spectra, and when $$E$$ is an $$N_{\infty}$$-ring spectrum, this functor lifts to a functor of $$N_{\infty}$$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $$E^{hn\mathbb Z}$$, giving the homotopy groups of the $$\mathbb Z$$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $$\mathbb Z$$-homotopy fixed point case, giving us a family of new tools to simplify slice computations. 

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5. 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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