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1. Total power operations in spectral sequences

   https://doi.org/10.1090/tran/9073  

   Balderrama, William (May 2024, Transactions of the American Mathematical Society)

   We describe how power operations descend through homotopy limit spectral sequences. We apply this to describe how norms appear in the $C_2$-equivariant Adams spectral sequence, to compute norms on ${\mathrm{\pi <\#comment/>}}_0$of the equivariant $KU$-local sphere, and to compute power operations for the $K\left(1\right)$-local sphere. An appendix contains material on equivariant Bousfield localizations which may be of independent interest. 

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2. The Hilbert scheme of infinite affine space and algebraic K-theory

   https://doi.org/10.4171/jems/1340  

   Hoyois, Marc; Jelisiejew, Joachim; Nardin, Denis; Totaro, Burt; Yakerson, Maria (April 2025, Journal of the European Mathematical Society)

   We study the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})from an\mathbf{A}^{1}-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})is\mathbf{A}^{1}-equivalent to the Grassmannian of(d-1)-planes in\mathbf{A}^{\infty}. We then describe the\mathbf{A}^{1}-homotopy type of\mathrm{Hilb}_{d}(\mathbf{A}^{n})in a certain range, fornlarge compared tod. For example, we compute the integral cohomology of\mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C})in a range. We also deduce that the forgetful map\mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect}from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an\mathbf{A}^{1}-equivalence after group completion. This implies that the moduli stack\mathcal{FF}\mathrm{lat}, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum\mathrm{kgl}representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the\mathrm{kgl}-homology of smooth proper schemes over a perfect field. 

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3. Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere

   https://doi.org/10.1090/cams/17  

   Carmeli, Shachar; Yuan, Allen (March 2023, Communications of the American Mathematical Society)

   We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the $K\left(1\right)$-local sphere ${\mathbb{S}}_{K\left(1\right)}$at the prime $2$, in particular realizing the non- $2$-adic rational element $1+\mathrm{\epsilon <\#comment/>}\in <\#comment/>{\mathrm{\pi <\#comment/>}}_0{\mathbb{S}}_{K\left(1\right)}$as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in ${\mathrm{\pi <\#comment/>}}_0{\mathbb{S}}_{K\left(1\right)}$. 

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4. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

   https://doi.org/10.1090/jams/1045  

   Annala, Toni; Hoyois, Marc; Iwasa, Ryomei (March 2024, Journal of the American Mathematical Society)

   We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-category satisfies ${\mathbb{P}}^1$-homotopy invariance and weighted ${\mathbb{A}}^1$-homotopy invariance, which we use in place of ${\mathbb{A}}^1$-homotopy invariance to obtain analogues of several key results from ${\mathbb{A}}^1$-homotopy theory. These allow us in particular to define a universal oriented motivic ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$. 

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5. Higher arity stability and the functional order property

   https://doi.org/10.1007/s00029-025-01055-4  

   Abd_Aldaim, A; Conant, G; Terry, C (July 2025, Selecta Mathematica)

   Abstract Thek-dimensional functional order property ($$\operatorname {FOP}_k$$ ${FOP}_k$) is a combinatorial property of a$$(k+1)$$ $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf [59, 60], which identified$$\operatorname {NFOP}_2$$ ${NFOP}_2$as a ternary analogue of stability in the context of two finitary combinatorial problems related to hypergraph regularity and arithmetic regularity. In this paper we show$$\operatorname {NFOP}_k$$ ${NFOP}_k$has equally strong implications in model-theoretic classification theory, where its behavior as a$$(k+1)$$ $(k+1)$-ary version of stability is in close analogy to the behavior ofk-dependence as a$$(k+1)$$ $(k+1)$-ary version of$$\operatorname {NIP}$$ $NIP$. Our results include several new characterizations of$$\operatorname {NFOP}_k$$ ${NFOP}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when$$k=2$$ $k=2$. As a corollary of our collapsing theorem, we show$$\operatorname {NFOP}_k$$ ${NFOP}_k$is closed under Boolean combinations, and that$$\operatorname {FOP}_k$$ ${FOP}_k$can always be witnessed by a formula where all but one variable have length 1. When$$k=2$$ $k=2$, we prove a composition lemma analogous to that of Chernikov and Hempel from the setting of 2-dependence. Using this, we provide a new class of algebraic examples of$$\operatorname {NFOP}_2$$ ${NFOP}_2$theories. Specifically, we show that ifTis the theory of an infinite dimensional vector space over a fieldK, equipped with a bilinear form satisfying certain properties, thenTis$$\operatorname {NFOP}_2$$ ${NFOP}_2$if and only ifKis stable. Along the way we provide a corrected and reorganized proof of Granger’s quantifier elimination and completeness results for these theories. 

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