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1. Locally Complete Intersection Maps and the Proxy Small Property

   https://doi.org/10.1093/imrn/rnab041  

   Briggs, Benjamin; Iyengar, Srikanth B; Letz, Janina C; Pollitz, Josh (April 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract It is proved that a map $${\varphi }\colon R\to S$$ of commutative Noetherian rings that is essentially of finite type and flat is locally complete intersection if and only if $$S$$ is proxy small as a bimodule. This means that the thick subcategory generated by $$S$$ as a module over the enveloping algebra $$S\otimes _RS$$ contains a perfect complex supported fully on the diagonal ideal. This is in the spirit of the classical result that $${\varphi }$$ is smooth if and only if $$S$$ is small as a bimodule; that is to say, it is itself equivalent to a perfect complex. The geometric analogue, dealing with maps between schemes, is also established. Applications include simpler proofs of factorization theorems for locally complete intersection maps. 

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2. Generalised knotoids

   https://doi.org/10.1017/S0305004124000148  

   ADAMS, COLIN; ROMRELL, ZACHARY; BONAT, ALEXANDRA; CHANDE, MAYA; CHEN, JOYE; JIANG, MAXWELL; SANTIAGO, DANIEL; SHAPIRO, BENJAMIN; WOODRUFF, DORA (July 2024, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices. 

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3. Variational quasi-harmonic maps for computing diffeomorphisms

   https://doi.org/10.1145/3592105  

   Wang, Yu; Guo, Minghao; Solomon, Justin (July 2023, ACM Transactions on Graphics)

   Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation ofvariational quasi-harmonic mapsto obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps. 

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4. Untilting Line Bundles on Perfectoid Spaces

   https://doi.org/10.1093/imrn/rnab314  

   Dorfsman-Hopkins, Gabriel (November 2021, International Mathematics Research Notices)

   Abstract Let $$X$$ be a perfectoid space with tilt $$X^\flat $$. We build a natural map $$\theta :\Pic X^\flat \to \lim \Pic X$$ where the (inverse) limit is taken over the $$p$$-power map and show that $$\theta $$ is an isomorphism if $$R = \Gamma (X,\sO _X)$$ is a perfectoid ring. As a consequence, we obtain a characterization of when the Picard groups of $$X$$ and $$X^\flat $$ agree in terms of the $$p$$-divisibility of $$\Pic X$$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence. 

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5. Guaranteeing completely positive quantum evolution

   https://doi.org/10.1088/1751-8121/ac2e28  

   Dilley, Daniel; Gonzales, Alvin; Byrd, Mark (November 2021, Journal of Physics A: Mathematical and Theoretical)

   Abstract In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming an initial NCP map to a CP map through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems. Previous work have looked at structural physical approximation (SPA), which is a CP approximation of an NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to a set of qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used. 

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