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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. Shadow line distributions

   https://doi.org/10.1090/mcom/4110  

   Balakrishnan, Jennifer; Çiperiani, Mirela; Mazur, Barry; Rubin, Karl (July 2025, Mathematics of Computation)

   Let $E$be an elliptic curve over $\mathbb{Q}$with Mordell–Weil rank $2$and $p$be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free ${\mathbb{Z}}_p$-submodule of rank $1$, in $E\left(K\right)\otimes <\#comment/>{\mathbb{Z}}_p$given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$; we call it theshadow line. When the twist of $E$by $K$has analytic rank $1$, the shadow line is conjectured to lie in $E\left(\mathbb{Q}\right)\otimes <\#comment/>{\mathbb{Z}}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E\left(\mathbb{Q}\right)\otimes <\#comment/>{\mathbb{Z}}_p$as $K$varies, framing conjectures based on the computations we have made. 

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3. 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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4. 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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5. Random walk on group extensions

   https://doi.org/10.1090/tran/9309  

   Golsefidy, S Alireza; Srinivas, Srivatsa (January 2025, Transactions of the American Mathematical Society)

   We study random walks on various group extensions. Under certain bounded generation and bounded scaled conditions, we estimate the spectral gap of a random walk on a quasi-random-by-nilpotent group in terms of the spectral gap of its projection to the quasi-random part. We also estimate the spectral gap of a random-walk on a product of two quasi-random groups in terms of the spectral gap of its projections to the given factors. Based on these results, we estimate the spectral gap of a random walk on the ${\mathbb{F}}_q$-points of a perfect algebraic group $\mathbb{G}$in terms of the spectral gap of its projections to the almost simple factors of the semisimple quotient of $\mathbb{G}$. These results extend a work of Lindenstrauss and Varjú and an earlier work of the authors. Moreover, using a result of Breuillard and Gamburd, we show that there is an infinite set $\mathcal{P}$of primes of density one such that, if $k$is a positive integer and $\mathbb{G}=\mathbb{U}⋊<\#comment/>({\mathrm{SL}}_2{)}_{\mathbb{Q}}^m$is a perfect group and $\mathbb{U}$is a unipotent group, then the family of all the Cayley graphs of $\mathbb{G}\left(\mathbb{Z}/\underset{i=1}{\overset{k}{\prod <\#comment/>}}{p}_i\mathbb{Z}\right)$, $p_i\in <\#comment/>\mathcal{P}$, is a family of expanders. 

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