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1. Canonical Stratifications Along Bisheaves

   https://doi.org/10.1007/978-3-030-43408-3_15  

   Nanda, V; Patel, A (June 2020, Topological Data Analysis)

   null (Ed.)

   A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible. 

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2. Fair canonical correlation analysis

   Zhou, Zhuoping; Ataee-Tarzanagh, Davoud; Hou, Bojian; Tong, Boning; Xu, Jia; Feng, Yanbo; Long, Qi; Shen, Li (December 2023, NeurIPS’23: 37th Conference on Neural Information Processing Systems)
3. Canonical description of cosmological backreaction

   https://doi.org/10.1088/1475-7516/2021/03/083  

   Bojowald, Martin; Ding, Ding (March 2021, Journal of Cosmology and Astroparticle Physics)

   Abstract Canonical methods of quasiclassical dynamics make it possible to go beyond a strict background approximation for cosmological perturbations by including independent fields such as correlation degrees of freedom. New models are introduced and analyzed here for cosmological dynamics in the presence of quantum correlations between background and perturbations, as well as cross-correlations between different modes of a quantum field. Evolution equations for moments of a perturbation state reveal conditions required for inhomogeneity to build up out of an initial vacuum. A crucial role is played by quantum non-locality, formulated by canonical methods as an equivalent local theory with non-classical degrees of freedom given by moments of a quantum state. 

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4. Azumaya Algebras and Canonical Components

   https://doi.org/10.1093/imrn/rnaa209  

   Chinburg, Ted; Reid, Alan W; Stover, Matthew (September 2020, International Mathematics Research Notices)

   Abstract Let $$M$$ be a compact 3-manifold and $$\Gamma =\pi _1(M)$$. Work by Thurston and Culler–Shalen established the $${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$$ character variety $$X(\Gamma )$$ as fundamental tool in the study of the geometry and topology of $$M$$. This is particularly the case when $$M$$ is the exterior of a hyperbolic knot $$K$$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $$X(\Gamma )$$, as well as distinguished points on the canonical component, when $$\Gamma $$ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields. 

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5. Canonical resolutions over Koszul algebras

   https://doi.org/10.1007/978-3-030-91986-3_11  

   Faber, Eleonore; Juhnke-Kubitzke, Martina; Lindo, Haydee; Miller, Claudia; R. G., Rebecca; Seceleanu, Alexandra (January 2021, Association for Women in Mathematics series)

   Miller, Claudia; Striuli, Janet; Witt, Emily (Ed.)

   We generalize Buchsbaum and Eisenbud’s resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras. 

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