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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. 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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3. The two-phase problem for harmonic measure in VMO and the chord-arc condition

   https://doi.org/10.1090/btran/197  

   Tolsa, Xavier; Toro, Tatiana (November 2024, Transactions of the American Mathematical Society, Series B)

   Let ${\mathrm{\Omega <\#comment/>}}^{+}\subset <\#comment/>{\mathbb{R}}^{n+1}$be a bounded $\mathrm{\delta <\#comment/>}$-Reifenberg flat domain, with $\mathrm{\delta <\#comment/>}>0$small enough, possibly with locally infinite surface measure. Assume also that ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}={\mathbb{R}}^{n+1}\setminus <\#comment/>\stackrel{¯<\#comment/>}{{\mathrm{\Omega <\#comment/>}}^{+}}$is an NTA (non-tangentially accessible) domain as well and denote by ${\mathrm{\omega <\#comment/>}}^{+}$and ${\mathrm{\omega <\#comment/>}}^{-<\#comment/>}$the respective harmonic measures of ${\mathrm{\Omega <\#comment/>}}^{+}$and ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}$with poles $p^{\pm <\#comment/>}\in <\#comment/>{\mathrm{\Omega <\#comment/>}}^{\pm <\#comment/>}$. In this paper we show that the condition that $\log <\#comment/>\frac{d{\mathrm{\omega <\#comment/>}}^{-<\#comment/>}}{d{\mathrm{\omega <\#comment/>}}^{+}}\in <\#comment/>\mathrm{VMO}<\#comment/>\left({\mathrm{\omega <\#comment/>}}^{+}\right)$is equivalent to ${\mathrm{\Omega <\#comment/>}}^{+}$being a chord-arc domain with inner unit normal belonging to $\mathrm{VMO}<\#comment/>\left({\mathcal{H}}^n{|}_{\mathrm{\partial <\#comment/>}{\mathrm{\Omega <\#comment/>}}^{+}}\right)$. 

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4. 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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5. Dens, nests and the Loehr-Warrington conjecture

   https://doi.org/10.1090/jams/1057  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G (June 2025, Journal of the American Mathematical Society)

   We prove and extend the longest-standing conjecture in ‘ $q,t$-Catalan combinatorics,’ namely, the combinatorial formula for ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$conjectured by Loehr and Warrington, where $s_{\mathrm{\mu <\#comment/>}}$is a Schur function and $\mathrm{\nabla <\#comment/>}$is an eigenoperator on Macdonald polynomials. Our approach is to establish a stronger identity of infinite series of $G{L}_l$characters involvingSchur Catalanimals; these were recently shown by the authors to represent Schur functions $s_{\mathrm{\mu <\#comment/>}}[-<\#comment/>M{X}^{m,n}]$in subalgebras $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)\subset <\#comment/>\mathcal{E}$isomorphic to the algebra of symmetric functions $\mathrm{\Lambda <\#comment/>}$over $\mathbb{Q}(q,t)$, where $\mathcal{E}$is the elliptic Hall algebra of Burban and Schiffmann. We establish a combinatorial formula for Schur Catalanimals as weighted sums of LLT polynomials, with terms indexed by configurations of nested lattice paths callednests, having endpoints and bounding constraints controlled by data called aden. The special case for $\mathrm{\Lambda <\#comment/>}\left(X^{m,1}\right)$proves the Loehr-Warrington conjecture, giving ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. In general, for $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)$our formula implies a new $(m,n)$version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduce to our previous shuffle theorem for paths under any line. Both this and the $(m,n)$Loehr-Warrington formula generalize the $(km,kn)$shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our formula here unifies these two generalizations. 

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