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1. Probabilistic convergence and stability of random mapper graphs

   https://doi.org/10.1007/s41468-020-00063-x  

   Brown, Adam ; Bobrowski, Omer ; Munch, Elizabeth ; Wang, Bei ( March 2021 , Journal of Applied and Computational Topology)

   null (Ed.)

   Abstract We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $${\mathbb {X}}$$ X equipped with a continuous function $$f: {\mathbb {X}}\rightarrow \mathbb {R}$$ f : X → R . We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $$\mathbb {R}$$ R . We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $$({\mathbb {X}}, f)$$ ( X , f ) when it is applied to points randomly sampled from a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of $$({\mathbb {X}}, f)$$ ( X , f ) , a constructible $$\mathbb {R}$$ R -space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $$({\mathbb {X}},f)$$ ( X , f ) to the mapper of a super-level set of a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Finally, building on the approach of Bobrowski et al. (Bernoulli 23 (1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data. 

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2. Intersection complexes and unramified 𝐿-factors

   https://doi.org/10.1090/jams/990  

   Sakellaridis, Yiannis ; Wang, Jonathan ( July 2022 , Journal of the American Mathematical Society)

   Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G ˇ \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G ˇ \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties. 

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3. Development and stochastic validation of a parameterized model of maize stalk flexure and buckling

   https://doi.org/10.1093/insilicoplants/diad010  

   Ottesen, Michael ; Carter, Joseph ; Hall, Ryan ; Liu, Nan-Wei ; Cook, Douglas D. ; Zhu, ed., Xin-Guang ( August 2023 , in silico Plants)

   Maize stalk lodging is the structural failure of the stalk prior to harvest and is a major problem for maize (corn) producers and plant breeders. To address this problem, it is critical to understand precisely how geometric and material parameters of the maize stalk influence stalk strength. Computational models could be a powerful tool in such investigations, but current methods of creating computational models are costly, time-consuming and, most importantly, do not provide parameterized control of the maize stalk parameters. The purpose of this study was to develop and validate a parameterized 3D model of the maize stalk. The parameterized model provides independent control over all aspects of the maize stalk geometry and material properties. The model accurately captures the shape of actual maize stalks and is predictive of maize stalk stiffness and strength. The model was validated using stochastic sampling of material properties to account for uncertainty in the values and influence of mechanical tissue properties. Results indicated that buckling is influenced by material properties to a greater extent that flexural stiffness. Finally, we demonstrate that this model can be used to create an unlimited number of synthetic stalks from within the parameter space. This model will enable the future implementation of parameter sweep studies, sensitivity analysis and optimization studies, and can be used to create computational models of maize stalks with any desired combination of geometric and material properties.

    

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4. Geometry of canonical genus 4 curves

   https://doi.org/10.1112/plms.12577  

   Rezaee, Fatemeh ( January 2024 , Proceedings of the London Mathematical Society)

   We apply the machinery of Bridgeland stability conditions on derived categories of coherent sheaves to describe the geometry of classical moduli spaces associated with canonical genus 4 space curves via an effective control over its wall‐crossing. This article provides the first description of a moduli space of Pandharipande–Thomas stable pairs that is used as an intermediate step toward the description of the associated Hilbert scheme, which in turn is the first example where the components of a classical moduli space were completely determined via wall‐crossing. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme. There are several long standing open problems regarding classical sheaf theoretic moduli spaces, and the present work will shed light on further studies of such moduli spaces such as Hilbert schemes of curves and moduli of stable pairs that are very hard to tackle without the wall‐crossing techniques.

    

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5. Random Variables with Measurability Constraints with Application to Opportunistic Scheduling

   Neely, Michael J. ( July 2022 , Arxiv:2207.02345v1)

   This paper proves a representation theorem regarding sequences of random elements that take values in a Borel space and are measurable with respect to the sigma algebra generated by an arbitrary union of sigma algebras. This, together with a related representation theorem of Kallenberg, is used to characterize the set of multidimensional decision vectors in a discrete time stochastic control problem with measurability and causality constraints, including opportunistic scheduling problems for timevarying communication networks. A network capacity theorem for these systems is refined, without requiring an implicit and arbitrarily complex extension of the state space, by introducing two measurability assumptions and using a theory of constructible sets. An example that makes use of well known pathologies in descriptive set theory is given to show a nonmeasurable scheduling scheme can outperform all measurable scheduling schemes. 

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