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1. The stable exotic Cuntz algebras are higher-rank graph algebras

   https://doi.org/10.1090/bproc/180  

   Boersema, Jeffrey ; Browne, Sarah ; Gillaspy, Elizabeth ( March 2024 , Proceedings of the American Mathematical Society, Series B)

   For each odd integer$n \geq 3$, we construct a rank-3 graph$\Lambda _n$with involution$\gamma _n$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution$(\Lambda ,\gamma )$can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution$(\Lambda , \gamma )$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$is stably isomorphic to the suspension$S \mathbb {R}$. In the Appendix, we show that the$i$-fold suspension$S^i \mathbb {R}$is stably isomorphic to a graph algebra iff$-2 \leq i \leq 1$.

    

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2. A proof of the Kahn–Kalai conjecture

   https://doi.org/10.1090/jams/1028  

   Park, Jinyoung ; Pham, Huy ( August 2023 , Journal of the American Mathematical Society)

   Proving the “expectation-threshold” conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property$\mathcal {F}$on a finite set$X$,\[$p_c(\mathcal {F})=O(q(\mathcal {F})\log \ell (\mathcal {F})),$\]where$p_c(\mathcal {F})$and$q(\mathcal {F})$are the threshold and “expectation threshold” of$\mathcal {F}$, and$\ell (\mathcal {F})$is the maximum of$2$and the maximum size of a minimal member of$\mathcal {F}$.

    

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3. 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$\epsilon > 0$there exists$N > 0$such that, if$|G| \ge N$and$S, T$are normal subsets of$G$with at least$\epsilon |G|$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$\mathrm {PSL}_n(q)$where$q$is fixed and$n\to \infty$. However, in the case$S=T$and$G$alternating this holds with an explicit bound on$N$in terms of$\epsilon$.

   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$g \in G$,$P_{w_1(G),w_2(G)}(g)$denotes the probability that$g_1g_2 = g$where$g_i \in w_i(G)$are chosen uniformly and independently, then, as$|G| \to \infty$, the distribution$P_{w_1(G),w_2(G)}$tends to the uniform distribution on$G$with respect to the$L^{\infty }$norm.

    

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4. Braid group action and quasi-split affine 𝚤quantum groups I

   https://doi.org/10.1090/ert/657  

   Lu, Ming ; Wang, Weiqiang ; Zhang, Weinan ( October 2023 , Representation Theory of the American Mathematical Society)

   This is the first of our papers on quasi-split affine quantum symmetric pairs$\big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big )$, focusing on the real rank one case, i.e.,$\mathfrak g = \mathfrak {sl}_3$equipped with a diagram involution. We construct explicitly a relative braid group action of type$A_2^{(2)}$on the affine$\imath$quantum group$\widetilde {{\mathbf U}}^\imath$. Real and imaginary root vectors for$\widetilde {{\mathbf U}}^\imath$are constructed, and a Drinfeld type presentation of$\widetilde {{\mathbf U}}^\imath$is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine$\imath$quantum groups in the sequels.

    

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5. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin ; Owen, Art ( March 2023 , Mathematics of Computation)

   Let$f$be analytic on$[0,1]$with$|f^{(k)}(1/2)|\leqslant A\alpha ^kk!$for some constants$A$and$\alpha >2$and all$k\geqslant 1$. We show that the median estimate of$\mu =\int _0^1f(x)\,\mathrm {d} x$under random linear scrambling with$n=2^m$points converges at the rate$O(n^{-c\log (n)})$for any$c> 3\log (2)/\pi ^2\approx 0.21$. We also get a super-polynomial convergence rate for the sample median of$2k-1$random linearly scrambled estimates, when$k/m$is bounded away from zero. When$f$has a$p$’th derivative that satisfies a$\lambda$-Hölder condition then the median of means has error$O( n^{-(p+\lambda )+\epsilon })$for any$\epsilon >0$, if$k\to \infty$as$m\to \infty$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number.

    

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