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1. 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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2. A note on surfaces in ℂℙ² and ℂℙ²#ℂℙ²

   https://doi.org/10.1090/bproc/218  

   Marengon, Marco; Miller, Allison; Ray, Arunima; Stipsicz, András (June 2024, Proceedings of the American Mathematical Society, Series B)

   In this brief note, we investigate the ${\mathbb{C}\mathbb{P}}^2$-genus of knots, i.e., the least genus of a smooth, compact, orientable surface in ${\mathbb{C}\mathbb{P}}^2\setminus <\#comment/>\stackrel{˚<\#comment/>}{B^4}$bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the ${\mathbb{C}\mathbb{P}}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in ${\mathbb{C}\mathbb{P}}^2\mathrm{\#<\#comment/>}{\mathbb{C}\mathbb{P}}^2$. 

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3. Bohr recurrence and density of non-lacunary semigroups of ℕ

   https://doi.org/10.1090/proc/17006  

   Frantzikinakis, Nikos; Host, Bernard; Kra, Bryna (January 2025, Proceedings of the American Mathematical Society)

   A subset $R$of integers is a set of Bohr recurrence if every rotation on ${\mathbb{T}}^d$returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!2^m{3}^n:<\#comment/>k,m,n\in <\#comment/>\mathbb{N}\}$is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$is a real polynomial with at least one non-constant irrational coefficient, then the set $\left\{P\right(2^m{3}^n):<\#comment/>m,n\in <\#comment/>\mathbb{N}\}$is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl. 

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4. 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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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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