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1. On the maximal spectral type of nilsystems

   https://doi.org/10.1090/bproc/229  

   Ackelsberg, Ethan; Richter, Florian; Shalom, Or (September 2024, Proceedings of the American Mathematical Society, Series B)

   Let $(G/\mathrm{\Gamma <\#comment/>},R_a)$be an ergodic $k$-step nilsystem for $k\ge <\#comment/>2$. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that $L^2\left(G/\mathrm{\Gamma <\#comment/>}\right)$decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math.124(2014), pp. 261–295] for $2$-step nilsystems and a result by Stepin [Uspehi Mat. Nauk24(1969), pp. 241–242] for nilsystems $G/\mathrm{\Gamma <\#comment/>}$with connected, simply connected $G$. 

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2. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$have the property that for each $\mathrm{ϵ<\#comment/>}>0$there exists $N>0$such that, if $|G|\ge <\#comment/>N$and $S,T$are normal subsets of $G$with at least $\mathrm{ϵ<\#comment/>}|G|$elements each, then every non-trivial element of $G$is the product of an element of $S$and an element of $T$. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$where $q$is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$. However, in the case $S=T$and $G$alternating this holds with an explicit bound on $N$in terms of $\mathrm{ϵ<\#comment/>}$. Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$are non-trivial words, $G$is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$, $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$denotes the probability that $g_1{g}_2=g$where $g_i\in <\#comment/>w_i\left(G\right)$are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$, the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$tends to the uniform distribution on $G$with respect to the $L^{\mathrm{\infty <\#comment/>}}$norm. 

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3. Colength one deformation rings

   https://doi.org/10.1090/tran/9191  

   Le, Daniel; Le_Hung, Bao; Morra, Stefano; Park, Chol; Qian, Zicheng (May 2024, Transactions of the American Mathematical Society)

   Let $K/{\mathbb{Q}}_p$be a finite unramified extension, $\stackrel{¯<\#comment/>}{\mathrm{\rho <\#comment/>}}:\mathrm{G}\mathrm{a}\mathrm{l}\left({\stackrel{¯<\#comment/>}{\mathbb{Q}}}_p/K\right)\to <\#comment/>{\mathrm{G}\mathrm{L}}_n\left({\stackrel{¯<\#comment/>}{\mathbb{F}}}_p\right)$a continuous representation, and $\mathrm{\tau <\#comment/>}$a tame inertial type of dimension $n$. We explicitly determine, under mild regularity conditions on $\mathrm{\tau <\#comment/>}$, the potentially crystalline deformation ring $R_{\stackrel{¯<\#comment/>}{\mathrm{\rho <\#comment/>}}}^{\mathrm{\eta <\#comment/>},\mathrm{\tau <\#comment/>}}$in parallel Hodge–Tate weights $\mathrm{\eta <\#comment/>}=(n-<\#comment/>1,\cdots <\#comment/>,1,0)$and inertial type $\mathrm{\tau <\#comment/>}$when theshapeof $\stackrel{¯<\#comment/>}{\mathrm{\rho <\#comment/>}}$with respect to $\mathrm{\tau <\#comment/>}$has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150]. 

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4. On flat manifold bundles and the connectivity of Haefliger’s classifying spaces

   https://doi.org/10.1090/proc/16941  

   Nariman, Sam (November 2024, Proceedings of the American Mathematical Society)

   We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston’s conjecture predicts that every $M$-bundle over a manifold $B$where $\dim <\#comment/>\left(B\right)\le <\#comment/>\dim <\#comment/>\left(M\right)$is cobordant to a flat $M$-bundle. In particular, we study the bordism class of flat $M$-bundles over low dimensional manifolds, comparing a finite dimensional Lie group $G$with ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0\left(G\right)$. 

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5. 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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