@inproceedings{6628,
abstract = {Fejes Tóth [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(d−1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space.},
author = {Vegter, Gert and Wintraecken, Mathijs},
booktitle = {The 31st Canadian Conference in Computational Geometry},
location = {Edmonton, Canada},
pages = {275--279},
title = {{The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds}},
year = {2019},
}
@article{6634,
abstract = {In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.},
author = {Akopyan, Arseniy and Hubard, Alfredo and Karasev, Roman},
journal = {Topological Methods in Nonlinear Analysis},
number = {2},
pages = {457--490},
publisher = {Akademicka Platforma Czasopism},
title = {{Lower and upper bounds for the waists of different spaces}},
doi = {10.12775/TMNA.2019.008},
volume = {53},
year = {2019},
}
@inproceedings{6648,
abstract = {Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory
needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.},
author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert},
booktitle = {35th International Symposium on Computational Geometry},
isbn = {9783959771047},
location = {Portland, OR, United States},
pages = {31:1--31:14},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Topological data analysis in information space}},
doi = {10.4230/LIPICS.SOCG.2019.31},
volume = {129},
year = {2019},
}
@article{6671,
abstract = {In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points.},
author = {Boissonnat, Jean-Daniel and Lieutier, André and Wintraecken, Mathijs},
issn = {2367-1734},
journal = {Journal of Applied and Computational Topology},
number = {1-2},
pages = {29–58},
publisher = {Springer Nature},
title = {{The reach, metric distortion, geodesic convexity and the variation of tangent spaces}},
doi = {10.1007/s41468-019-00029-8},
volume = {3},
year = {2019},
}
@article{6756,
abstract = {We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 𝕊2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3σ and 4σ at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck’s predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.},
author = {Pranav, Pratyush and Adler, Robert J. and Buchert, Thomas and Edelsbrunner, Herbert and Jones, Bernard J.T. and Schwartzman, Armin and Wagner, Hubert and Van De Weygaert, Rien},
issn = {14320746},
journal = {Astronomy and Astrophysics},
publisher = {EDP Sciences},
title = {{Unexpected topology of the temperature fluctuations in the cosmic microwave background}},
doi = {10.1051/0004-6361/201834916},
volume = {627},
year = {2019},
}
@article{6793,
abstract = {The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.},
author = {Akopyan, Arseniy and Izmestiev, Ivan},
issn = {14692120},
journal = {Bulletin of the London Mathematical Society},
number = {5},
pages = {765--775},
publisher = {London Mathematical Society},
title = {{The Regge symmetry, confocal conics, and the Schläfli formula}},
doi = {10.1112/blms.12276},
volume = {51},
year = {2019},
}
@article{6828,
abstract = {In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type to the category of finite-dimensional modules of the graded affine Hecke algebra of type . Using results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category as a full subcategory, our results generalize results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between finite-dimensional representations of and representations of the symmetric group .},
author = {Brown, Adam},
issn = {0021-8693},
journal = {Journal of Algebra},
pages = {261--289},
publisher = {Elsevier},
title = {{Arakawa-Suzuki functors for Whittaker modules}},
doi = {10.1016/j.jalgebra.2019.07.027},
volume = {538},
year = {2019},
}
@inproceedings{6989,
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 hole(s) to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability. },
author = {Aichholzer, Oswin and Akitaya, Hugo A and Cheung, Kenneth C and Demaine, Erik D and Demaine, Martin L and Fekete, Sandor P and Kleist, Linda and Kostitsyna, Irina and Löffler, Maarten and Masárová, Zuzana and Mundilova, Klara and Schmidt, Christiane},
booktitle = {Proceedings of the 31st Canadian Conference on Computational Geometry},
location = {Edmonton, Canada},
pages = {164--170},
publisher = {Canadian Conference on Computational Geometry},
title = {{Folding polyominoes with holes into a cube}},
year = {2019},
}
@article{87,
abstract = {Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in Rn+1, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the n-sphere is isometric to the standard n-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the n-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
journal = {Annals of Applied Probability},
number = {5},
pages = {3215 -- 3238},
publisher = {Institute of Mathematical Statistics},
title = {{Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics}},
doi = {10.1214/18-AAP1389},
volume = {28},
year = {2018},
}
@inproceedings{188,
abstract = {Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex.},
author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert},
location = {Budapest, Hungary},
pages = {35:1 -- 35:13},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Smallest enclosing spheres and Chernoff points in Bregman geometry}},
doi = {10.4230/LIPIcs.SoCG.2018.35},
volume = {99},
year = {2018},
}