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1. Noncommutative resolutions using syzygies: NONCOMMUTATIVE RESOLUTIONS USING SYZYGIES

   https://doi.org/10.1112/blms.12210  

   Dao, Hailong; Iyama, Osamu; Iyengar, Srikanth B.; Takahashi, Ryo; Wemyss, Michael; Yoshino, Yuji (October 2018, Bulletin of the London Mathematical Society)
2. Stratified Noncommutative Geometry

   https://doi.org/10.1090/memo/1485  

   Ayala, David; Mazel-Gee, Aaron; Rozenblyum, Nick (May 2024, Memoirs of the American Mathematical Society)

   We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable $\mathrm{\infty <\#comment/>}$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as ${\mathbb{E}}_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory. 

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3. Operator noncommutative functions

   https://doi.org/10.4153/S0008439522000339  

   Augat, Meric; McCarthy, John E. (June 2023, Canadian Mathematical Bulletin)

   Abstract We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g.,$$B({\mathcal H})$$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the$$k{\mathrm {th}}$$directional derivative of any NC function at a scalar point is ak-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball. 

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4. Noncommutative quasi-resolutions

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   Qin, X.-S.; Wang, Y.-H.; Zhang, J.J. (October 2019, Journal of Algebra)
5. Noncommutative rational Clark measures

   https://doi.org/10.4153/S0008414X22000384  

   Jury, Michael T.; Martin, Robert T.W.; Shamovich, Eli (July 2022, Canadian Journal of Mathematics)

   Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $$\mathbb {C} ^d$$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions. 

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