More Like this

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$-2 \leq i \leq 1$.

    

   more » « less
2. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

   https://doi.org/10.1090/tran/8976  

   Livshyts, Galyna ( June 2023 , Transactions of the American Mathematical Society)

   We show that for any even log-concave probability measure$\mu$on$\mathbb {R}^n$, any pair of symmetric convex sets$K$and$L$, and any$\lambda \in [0,1]$,$\begin{equation*} \mu ((1-\lambda ) K+\lambda L)^{c_n}\geq (1-\lambda ) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*}$where$c_n\geq n^{-4-o(1)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures.

    

   more » « less
3. Residual intersections and linear powers

   https://doi.org/10.1090/btran/127  

   Eisenbud, David ; Huneke, Craig ; Ulrich, Bernd ( October 2023 , Transactions of the American Mathematical Society, Series B)

   If$I$is an ideal in a Gorenstein ring$S$, and$S/I$is Cohen-Macaulay, then the same is true for any linked ideal$I’$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal$L_{n}$of minors of a generic$2 \times n$matrix when$n>3$.

   In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of$I$. For example, suppose that$K$is the residual intersection of$L_{n}$by$2n-4$general quadratic forms in$L_{n}$. In this situation we analyze$S/K$and show that$I^{n-3}(S/K)$is a self-dual maximal Cohen-Macaulay$S/K$-module with linear free resolution over$S$.

   The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.

    

   more » « less
4. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan ; Smirnov, Ilya ( February 2023 , Proceedings of the American Mathematical Society)

   Let$(R,\mathfrak {m})$be a Noetherian local ring of dimension$d\geq 2$. We prove that if$e(\widehat {R}_{red})>1$, then the classical Lech’s inequality can be improved uniformly for all$\mathfrak {m}$-primary ideals, that is, there exists$\varepsilon >0$such that$e(I)\leq d!(e(R)-\varepsilon )\ell (R/I)$for all$\mathfrak {m}$-primary ideals$I\subseteq R$. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of$I$.

    

   more » « less
5. 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.

    

   more » « less