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1. Soliton resolution for energy-critical wave maps in the equivariant case

   https://doi.org/10.1090/jams/1012  

   Jendrej, Jacek; Lawrie, Andrew (December 2024, Journal of the American Mathematical Society)

   We consider the equivariant wave maps equation ${\mathbb{R}}^{1+2}\to <\#comment/>{\mathbb{S}}^2$, in all equivariance classes $k\in <\#comment/>\mathbb{N}$. We prove that every finite energy solution resolves, continuously in time, into a superposition of asymptotically decoupling harmonic maps and free radiation. 

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2. Type-I permanence

   https://doi.org/10.1090/ert/648  

   Chirvasitu, Alexandru (July 2023, Representation Theory of the American Mathematical Society)

   We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}⊴<\#comment/>\mathbb{E}$of locally compact groups and a twisted action $(\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>})$thereof on a (post)liminal $C^{\ast <\#comment/>}$-algebra $A$the twisted crossed product $A{⋊<\#comment/>}_{\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>}}\mathbb{E}$is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}⊴<\#comment/>\mathbb{E}$is type-I as soon as $\mathbb{E}$is. This happens for instance if $\mathbb{N}$is discrete and $\mathbb{E}$is Lie, or if $\mathbb{N}$is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group $\mathbb{G}$type-I-preserving if all semidirect products $\mathbb{N}⋊<\#comment/>\mathbb{G}$are type-I as soon as $\mathbb{N}$is, andlinearlytype-I-preserving if the same conclusion holds for semidirect products $V⋊<\#comment/>\mathbb{G}$arising from finite-dimensional $\mathbb{G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. 

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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. 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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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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