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1. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn; Wake, Preston (October 2022, Proceedings of the American Mathematical Society, Series B)

   We show that for primes $N,p\ge <\#comment/>5$with $N\equiv <\#comment/>-<\#comment/>1modp$, the class number of $\mathbb{Q}\left(N^{1/p}\right)$is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N\equiv <\#comment/>-<\#comment/>1modp$, there is always a cusp form of weight $2$and level ${\mathrm{\Gamma <\#comment/>}}_0\left(N^2\right)$whose $\mathrm{\ell <\#comment/>}$th Fourier coefficient is congruent to $\mathrm{\ell <\#comment/>}+1$modulo a prime above $p$, for all primes $\mathrm{\ell <\#comment/>}$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree- $p$extension of $\mathbb{Q}\left(N^{1/p}\right)$. 

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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 single-point Reshetnyak’s theorem

   https://doi.org/10.1090/tran/9415  

   Kangasniemi, Ilmari; Onninen, Jani (May 2025, Transactions of the American Mathematical Society)

   We prove a single-value version of Reshetnyak’s theorem. Namely, if a non-constant map $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$ from a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfies the estimate $\left|Df\right(x){|}^n\le K{J}_f(x)+\mathrm{\Sigma }(x\left)\right|f\left(x\right)-y_0{|}^n$ at almost every $x\in \mathrm{\Omega }$ for some $K\ge 1$, $y_0\in {\mathbb{R}}^n$ and $\mathrm{\Sigma }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\epsilon }\left(\mathrm{\Omega }\right)$, then $f^{-1}\left\{y_0\right\}$ is discrete, the local index $i(x,f)$ is positive in $f^{-1}\left\{y_0\right\}$, and every neighborhood of a point of $f^{-1}\left\{y_0\right\}$ is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$ at every $y_0\in {\mathbb{R}}^n$ is equivalent to assuming that the map $f$ is $K$-quasiregular, even if the choice of $\mathrm{\Sigma }$ is different for each $y_0$. Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$-quasiregularity. As a corollary of our single-value Reshetnyak’s theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calderón problem. 

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5. Infinite sumsets in sets with positive density

   https://doi.org/10.1090/jams/1030  

   Kra, Bryna; Moreira, Joel; Richter, Florian; Robertson, Donald (August 2023, Journal of the American Mathematical Society)

   Motivated by questions asked by Erdős, we prove that any set $A\subset <\#comment/>\mathbb{N}$with positive upper density contains, for any $k\in <\#comment/>\mathbb{N}$, a sumset $B_1+\cdots <\#comment/>+B_k$, where $B_1$, …, $B_k\subset <\#comment/>\mathbb{N}$are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of $k=2$. 

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