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1. 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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2. On symmetric representations of 𝑆𝐿₂(ℤ)

   https://doi.org/10.1090/proc/16205  

   Ng, Siu-Hung; Wang, Yilong; Wilson, Samuel (January 2023, Proceedings of the American Mathematical Society)

   We introduce the notions of symmetric and symmetrizable representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. The linear representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$that are subrepresentations of a symmetric one. 

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3. 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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4. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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5. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

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