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1. A classification of finite simple amenable Z-stable C*-algebras, II: C*-algebras with rational generalized tracial rank one

   Gong, G.; Lin, Huaxin; Niu, Z. (December 2020, C. R. Math. Rep. Acad. Sci. Canada)

   Elliott, G.A. (Ed.)

   A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possi- ble values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C∗-algebras which satisfy the UCT and have finite rational tracial rank. 

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2. Simple stably projectionless C*-algebras with generalized tracial rank one

   https://doi.org/Volume 14, Issue 1, 2020, pp. 251–347 DOI: 10.4171/JNCG/367  

   Elliott, G. A.; Gong, G; Lin, Huaxin; Niu, Z. (May 2020, Journal of noncommutative geometry)

   We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some struc- tural questions concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras. 

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3. Dynamics of RNA localization to nuclear speckles are connected to splicing efficiency

   https://doi.org/10.1126/sciadv.adp7727  

   Wu, Jinjun; Xiao, Yu; Liu, Yunzheng; Wen, Li; Jin, Chuanyang; Liu, Shun; Paul, Sneha; He, Chuan; Regev, Oded; Fei, Jingyi (October 2024, Science Advances)

   Nuclear speckles are nuclear membraneless organelles in higher eukaryotic cells playing a vital role in gene expression. Using an in situ reverse transcription–based sequencing method, we study nuclear speckle–associated human transcripts. Our data indicate the existence of three gene groups whose transcripts demonstrate different speckle localization properties: stably enriched in nuclear speckles, transiently enriched in speckles at the pre–messenger RNA stage, and not enriched. We find that stably enriched transcripts contain inefficiently excised introns and that disruption of nuclear speckles specifically affects splicing of speckle-enriched transcripts. We further reveal RNA sequence features contributing to transcript speckle localization, indicating a tight interplay between transcript speckle enrichment, genome organization, and splicing efficiency. Collectively, our data highlight a role of nuclear speckles in both co- and posttranscriptional splicing regulation. Last, we show that genes with stably enriched transcripts are over-represented among genes with heat shock–up-regulated intron retention, hinting at a connection between speckle localization and cellular stress response. 

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4. TORSORS AND STABLE EQUIVARIANT BIRATIONAL GEOMETRY

   https://doi.org/10.1017/nmj.2022.29  

   HASSETT, BRENDAN; TSCHINKEL, YURI (October 2022, Nagoya Mathematical Journal)

   Abstract We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups. 

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5. Hilbert’s 17th problem in free skew fields

   https://doi.org/10.1017/fms.2021.54  

   Volčič, Jurij (January 2020, Forum of Mathematics, Sigma)

   Abstract This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $$L\succeq 0$$ if and only if it belongs to the rational quadratic module generated by L . The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils. 

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