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1. Integral metaplectic modular categories

   https://doi.org/10.1142/S0218216520500327  

   Deaton, Adam; Gustafson, Paul; Mavrakis, Leslie; Rowell, Eric C.; Poltoratski, Sasha; Timmerman, Sydney; Warren, Benjamin; Zhang, Qing (April 2020, Journal of Knot Theory and Its Ramifications)

   A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories [Formula: see text] with the fusion rules of [Formula: see text] we determine the finite group [Formula: see text] for which [Formula: see text] is braided equivalent to [Formula: see text]. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point. 

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2. Mapping class group representations from non-semisimple TQFTs

   https://doi.org/10.1142/S0219199721500917  

   De_Renzi, Marco; Gainutdinov, Azat M; Geer, Nathan; Patureau-Mirand, Bertrand; Runkel, Ingo (February 2023, Communications in Contemporary Mathematics)

   In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category [Formula: see text]. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when [Formula: see text] is the category of finite-dimensional representations of the small quantum group of [Formula: see text], the action of all Dehn twists for surfaces without marked points has infinite order. 

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3. Jointly low-rank and bisparse recovery: Questions and partial answers

   https://doi.org/10.1142/S0219530519410094  

   Foucart, Simon; Gribonval, Rémi; Jacques, Laurent; Rauhut, Holger (January 2020, Analysis and Applications)

   We investigate the problem of recovering jointly [Formula: see text]-rank and [Formula: see text]-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that [Formula: see text] measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when [Formula: see text] for some exponent [Formula: see text]. We show that this is feasible for [Formula: see text], and that the proposed analysis cannot cover the case [Formula: see text]. The precise value of the optimal exponent [Formula: see text] is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements. 

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4. Rigid local systems with monodromy group the Conway group Co2

   https://doi.org/10.1142/S1793042120500189  

   Katz, Nicholas M.; Rojas-León, Antonio; Tiep, Pham Huu (March 2020, International Journal of Number Theory)

   null (Ed.)

   We first develop some basic facts about hypergeometric sheaves on the multiplicative group [Formula: see text] in characteristic [Formula: see text]. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic [Formula: see text]. One of these, of rank [Formula: see text] in characteristic [Formula: see text], turns out to have the Conway group [Formula: see text], in its irreducible orthogonal representation of degree [Formula: see text], as its arithmetic and geometric monodromy groups. 

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5. Classifying module categories for generalized Temperley–Lieb–Jones ∗-2-categories

   https://doi.org/10.1142/S0129167X20500275  

   Ferrer, Giovanni; Hernández Palomares, Roberto (April 2020, International Journal of Mathematics)

   Generalized Temperley–Lieb–Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong ∗-pseudofunctors into the ∗-2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to ∗-equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami’s classification of fiber functors on TLJ categories and DeCommer and Yamashita’s classification of unitary modules for [Formula: see text]. 

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