More Like this

1. Instantons and Bar-Natan homology

   https://doi.org/10.1112/s0010437x2000768x  

   Kronheimer, P. B.; Mrowka, T. S. (March 2021, Compositio Mathematica)

   A spectral sequence is established whose $$E_{2}$$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $$E_{2}$$ page is a characteristic-2 version of $$F_{5}$$ homology in Khovanov's classification. 

   more » « less
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. 

   more » « less
3. A numerical toolkit for multiprojective varieties

   https://doi.org/10.1090/mcom/3566  

   Hauenstein, Jonathan; Leykin, Anton; Rodriguez, Jose; Sottile, Frank (January 2021, Mathematics of Computation)

   null (Ed.)

   A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections and to use this toolkit in an algorithm for numerical irreducible decomposition of multiprojective varieties. The toolkit and decomposition algorithm are illustrated throughout in a series of examples. 

   more » « less
4. Brauer groups and Galois cohomology of commutative ring spectra

   https://doi.org/10.1112/S0010437X21007065  

   Gepner, David; Lawson, Tyler (June 2021, Compositio Mathematica)

   null (Ed.)

   In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $$\infty$$ -categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $$E$$ , we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $$\pi _0(E)$$ , recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $$\mathbb {Z}/8\times \mathbb {Z}/2$$ . Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$ -algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’ $KO$ -algebra with the same coefficient ring as $$\mathrm {End}_{KO}(KU)$$ . This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$ , previously studied by Mathew and Stojanoska. 

   more » « less
5. Families of metrics with positive scalar curvature on spectral sequence cobordisms

   https://doi.org/10.1142/S1793525324500109  

   Gong, Sherry (December 2025, Journal of Topology and Analysis)

   We study families of metrics on the cobordisms that underlie the differential maps in Bloom’s monopole Floer spectral sequence, a spectral sequence for links in [Formula: see text] whose [Formula: see text] page is the Khovanov homology of the link, and which abuts to the monopole Floer homology of the double branched cover of the link. The higher differentials in the spectral sequence count parametrized moduli spaces of solutions to Seiberg–Witten equations, parametrized over a family of metrics with asymptotic behavior corresponding to a configuration of unlinks with 1-handle attachments. For a class of configurations, we construct families of metrics with the prescribed behavior, such that each metric therein has positive scalar curvature. The positive scalar curvature implies that there are no irreducible solutions to the Seiberg–Witten equations and thus, when the spectral sequences are computed with these families of metrics, only reducible solutions must be counted. The class of configurations for which we construct these families of metrics includes all configurations that go into the spectral sequence for [Formula: see text] torus knots, and all configurations that involve exactly two 1-handle attachments. 

   more » « less