@inproceedings{13048,
  abstract     = {In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(K′)) ≲ dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.},
  author       = {Lieutier, André and Wintraecken, Mathijs},
  booktitle    = {Proceedings of the 55th Annual ACM Symposium on Theory of Computing},
  isbn         = {9781450399135},
  location     = {Orlando, FL, United States},
  pages        = {1768--1776},
  publisher    = {Association for Computing Machinery},
  title        = {{Hausdorff and Gromov-Hausdorff stable subsets of the medial axis}},
  doi          = {10.1145/3564246.3585113},
  year         = {2023},
}

@article{10754,
  abstract     = {Targeting dysregulated Ca2+ signaling in cancer cells is an emerging chemotherapy approach. We previously reported that store-operated Ca2+ entry (SOCE) blockers, such as RP4010, are promising antitumor drugs for esophageal cancer. As a tyrosine kinase inhibitor (TKI), afatinib received FDA approval to be used in targeted therapy for patients with EGFR mutation-positive cancers. While preclinical studies and clinical trials have shown that afatinib has benefits for esophageal cancer patients, it is not known whether a combination of afatinib and RP4010 could achieve better anticancer effects. Since TKI can alter intracellular Ca2+ dynamics through EGFR/phospholipase C-γ pathway, in this study, we evaluated the inhibitory effect of afatinib and RP4010 on intracellular Ca2+ oscillations in KYSE-150, a human esophageal squamous cell carcinoma cell line, using both experimental and mathematical simulations. Our mathematical simulation of Ca2+ oscillations could fit well with experimental data responding to afatinib or RP4010, both separately or in combination. Guided by simulation, we were able to identify a proper ratio of afatinib and RP4010 for combined treatment, and such a combination presented synergistic anticancer-effect evidence by experimental measurement of intracellular Ca2+ and cell proliferation. This intracellular Ca2+ dynamic-based mathematical simulation approach could be useful for a rapid and cost-effective evaluation of combined targeting therapy drugs.},
  author       = {Chang, Yan and Funk, Marah and Roy, Souvik and Stephenson, Elizabeth R and Choi, Sangyong and Kojouharov, Hristo V. and Chen, Benito and Pan, Zui},
  issn         = {14220067},
  journal      = {International Journal of Molecular Sciences},
  number       = {3},
  publisher    = {MDPI},
  title        = {{Developing a mathematical model of intracellular Calcium dynamics for evaluating combined anticancer effects of afatinib and RP4010 in esophageal cancer}},
  doi          = {10.3390/ijms23031763},
  volume       = {23},
  year         = {2022},
}

@article{10773,
  abstract     = {The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.},
  author       = {Biswas, Ranita and Cultrera Di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  pages        = {811--842},
  publisher    = {Springer Nature},
  title        = {{Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics}},
  doi          = {10.1007/s00454-022-00371-2},
  volume       = {67},
  year         = {2022},
}

@inproceedings{10828,
  abstract     = {Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown.},
  author       = {Heiss, Teresa and Tymochko, Sarah and Story, Brittany and Garin, Adélie and Bui, Hoa and Bleile, Bea and Robins, Vanessa},
  booktitle    = {2021 IEEE International Conference on Big Data},
  isbn         = {9781665439022},
  location     = {Orlando, FL, United States; Virtuell},
  pages        = {3824--3834},
  publisher    = {IEEE},
  title        = {{The impact of changes in resolution on the persistent homology of images}},
  doi          = {10.1109/BigData52589.2021.9671483},
  year         = {2022},
}

@inproceedings{11428,
  abstract     = {The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.},
  author       = {Chambers, Erin and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs},
  booktitle    = {38th International Symposium on Computational Geometry},
  editor       = {Goaoc, Xavier and Kerber, Michael},
  isbn         = {978-3-95977-227-3},
  issn         = {1868-8969},
  location     = {Berlin, Germany},
  pages        = {66:1--66:9},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{A cautionary tale: Burning the medial axis is unstable}},
  doi          = {10.4230/LIPIcs.SoCG.2022.66},
  volume       = {224},
  year         = {2022},
}

@book{11429,
  abstract     = {This book constitutes the refereed proceedings of the 18th International Symposium on Web and Wireless Geographical Information Systems, W2GIS 2022, held in Konstanz, Germany, in April 2022.
The 7 full papers presented together with 6 short papers in the volume were carefully reviewed and selected from 16 submissions.  The papers cover topics that range from mobile GIS and Location-Based Services to Spatial Information Retrieval and Wireless Sensor Networks.},
  editor       = {Karimipour, Farid and Storandt, Sabine},
  isbn         = {9783031062445},
  issn         = {1611-3349},
  pages        = {153},
  publisher    = {Springer Nature},
  title        = {{Web and Wireless Geographical Information Systems}},
  doi          = {10.1007/978-3-031-06245-2},
  volume       = {13238},
  year         = {2022},
}

@inbook{11440,
  abstract     = {To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation.},
  author       = {Bleile, Bea and Garin, Adélie and Heiss, Teresa and Maggs, Kelly and Robins, Vanessa},
  booktitle    = {Research in Computational Topology 2},
  editor       = {Gasparovic, Ellen and Robins, Vanessa and Turner, Katharine},
  isbn         = {9783030955182},
  pages        = {1--26},
  publisher    = {Springer Nature},
  title        = {{The persistent homology of dual digital image constructions}},
  doi          = {10.1007/978-3-030-95519-9_1},
  volume       = {30},
  year         = {2022},
}

@article{11545,
  abstract     = {We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N . We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings.
We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category
N have the structure of highest weight categories and we establish a BGG reciprocity theorem for N .},
  author       = {Brown, Adam and Romanov, Anna},
  issn         = {0021-8693},
  journal      = {Journal of Algebra},
  keywords     = {Algebra and Number Theory},
  number       = {11},
  pages        = {145--179},
  publisher    = {Elsevier},
  title        = {{Contravariant pairings between standard Whittaker modules and Verma modules}},
  doi          = {10.1016/j.jalgebra.2022.06.017},
  volume       = {609},
  year         = {2022},
}

@article{11658,
  abstract     = {The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.},
  author       = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  journal      = {Leibniz International Proceedings on Mathematics},
  publisher    = {Schloss Dagstuhl - Leibniz Zentrum für Informatik},
  title        = {{Depth in arrangements: Dehn–Sommerville–Euler relations with applications}},
  year         = {2022},
}

@article{11660,
  abstract     = {We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining collections of interrelated sorted lists together with their persistence diagrams. },
  author       = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
  journal      = {LIPIcs},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{A window to the persistence of 1D maps. I: Geometric characterization of critical point pairs}},
  year         = {2022},
}

@article{7791,
  abstract     = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.},
  author       = {Akopyan, Arseniy and Karasev, Roman},
  issn         = {2199-6768},
  journal      = {European Journal of Mathematics},
  number       = {4},
  pages        = {1309 -- 1312},
  publisher    = {Springer Nature},
  title        = {{When different norms lead to same billiard trajectories?}},
  doi          = {10.1007/s40879-020-00405-0},
  volume       = {8},
  year         = {2022},
}

@article{9649,
  abstract     = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently
fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.},
  author       = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
  issn         = {1615-3383},
  journal      = {Foundations of Computational Mathematics },
  pages        = {967--1012},
  publisher    = {Springer Nature},
  title        = {{The topological correctness of PL approximations of isomanifolds}},
  doi          = {10.1007/s10208-021-09520-0},
  volume       = {22},
  year         = {2022},
}

@article{10208,
  abstract     = {It is practical to collect a huge amount of movement data and environmental context information along with the health signals of individuals because there is the emergence of new generations of positioning and tracking technologies and rapid advancements of health sensors. The study of the relations between these datasets and their sequence similarity analysis is of interest to many applications such as health monitoring and recommender systems. However, entering all movement parameters and health signals can lead to the complexity of the problem and an increase in its computational load. In this situation, dimension reduction techniques can be used to avoid consideration of simultaneous dependent parameters in the process of similarity measurement of the trajectories. The present study provides a framework, named CaDRAW, to use spatial–temporal data and movement parameters along with independent context information in the process of measuring the similarity of trajectories. In this regard, the omission of dependent movement characteristic signals is conducted by using an unsupervised feature selection dimension reduction technique. To evaluate the effectiveness of the proposed framework, it was applied to a real contextualized movement and related health signal datasets of individuals. The results indicated the capability of the proposed framework in measuring the similarity and in decreasing the characteristic signals in such a way that the similarity results -before and after reduction of dependent characteristic signals- have small differences. The mean differences between the obtained results before and after reducing the dimension were 0.029 and 0.023 for the round path, respectively.},
  author       = {Goudarzi, Samira and Sharif, Mohammad and Karimipour, Farid},
  issn         = {1868-5145},
  journal      = {Journal of Ambient Intelligence and Humanized Computing},
  keywords     = {general computer science},
  pages        = {2621–2635},
  publisher    = {Springer Nature},
  title        = {{A context-aware dimension reduction framework for trajectory and health signal analyses}},
  doi          = {10.1007/s12652-021-03569-z},
  volume       = {13},
  year         = {2022},
}

@article{10413,
  abstract     = {Motivated by the recent introduction of the intrinsic semilattice entropy, we study generalized quasi-metric semilattices and their categories. We investigate the relationship between these objects and generalized semivaluations, extending Nakamura and Schellekens' approach. Finally, we use this correspondence to compare the intrinsic semilattice entropy and the semigroup entropy induced in particular situations, like sets, torsion abelian groups and vector spaces.},
  author       = {Dikranjan, Dikran and Giordano Bruno, Anna and Künzi, Hans Peter and Zava, Nicolò and Toller, Daniele},
  issn         = {0166-8641},
  journal      = {Topology and its Applications},
  publisher    = {Elsevier},
  title        = {{Generalized quasi-metric semilattices}},
  doi          = {10.1016/j.topol.2021.107916},
  volume       = {309},
  year         = {2022},
}

@article{11938,
  abstract     = {A matching is compatible to two or more labeled point sets of size n with labels {1, . . . , n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled sets of n points in convex position there exists a compatible matching with ⌊√2n + 1 − 1⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ). As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(log n) copies of any set of n points are necessary and sufficient for the existence of labelings of these point sets such that any compatible matching consists only of a single edge.},
  author       = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
  issn         = {1526-1719},
  journal      = {Journal of Graph Algorithms and Applications},
  number       = {2},
  pages        = {225--240},
  publisher    = {Brown University},
  title        = {{On compatible matchings}},
  doi          = {10.7155/jgaa.00591},
  volume       = {26},
  year         = {2022},
}

@article{12307,
  abstract     = {Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this very property makes point set topology perfect for teaching and learning abstract mathematical concepts. It clears our minds of preconceived intuitions and expectations and forces us to think in new and creative ways. In this paper, we present guided investigations into topology through questions and thinking strategies that open up fascinating problems. They are intended for faculty who already teach or are thinking about teaching a class in topology or abstract mathematical reasoning for undergraduates. They can be used to build simple to challenging projects in topology, proofs, honors programs, and research experiences.},
  author       = {Shipman, Barbara A. and Stephenson, Elizabeth R},
  issn         = {1935-4053},
  journal      = {PRIMUS},
  keywords     = {Education, General Mathematics},
  number       = {5},
  pages        = {593--609},
  publisher    = {Taylor & Francis},
  title        = {{Tangible topology through the lens of limits}},
  doi          = {10.1080/10511970.2021.1872750},
  volume       = {32},
  year         = {2022},
}

@article{7905,
  abstract     = {We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.},
  author       = {Brown, Adam and Wang, Bei},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  pages        = {1166--1198},
  publisher    = {Springer Nature},
  title        = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}},
  doi          = {10.1007/s00454-020-00206-y},
  volume       = {65},
  year         = {2021},
}

@article{8248,
  abstract     = {We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.},
  author       = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Lieutier, Andre and Wintraecken, Mathijs},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  pages        = {666--686},
  publisher    = {Springer Nature},
  title        = {{Local conditions for triangulating submanifolds of Euclidean space}},
  doi          = {10.1007/s00454-020-00233-9},
  volume       = {66},
  year         = {2021},
}

@article{8317,
  abstract     = {When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.},
  author       = {Aichholzer, Oswin and Akitaya, Hugo A. and Cheung, Kenneth C. and Demaine, Erik D. and Demaine, Martin L. and Fekete, Sándor P. and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane},
  issn         = {09257721},
  journal      = {Computational Geometry: Theory and Applications},
  publisher    = {Elsevier},
  title        = {{Folding polyominoes with holes into a cube}},
  doi          = {10.1016/j.comgeo.2020.101700},
  volume       = {93},
  year         = {2021},
}

@article{8338,
  abstract     = {Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.},
  author       = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  pages        = {938--976},
  publisher    = {Springer Nature},
  title        = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}},
  doi          = {10.1007/s00454-020-00240-w},
  volume       = {66},
  year         = {2021},
}