@inbook{74,
abstract = {We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class
of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.
We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian
measures.},
author = {Akopyan, Arseniy and Karasev, Roman},
booktitle = {Geometric Aspects of Functional Analysis},
editor = {Klartag, Bo'az and Milman, Emanuel},
isbn = {9783030360191},
issn = {16179692},
pages = {1--27},
publisher = {Springer Nature},
title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}},
doi = {10.1007/978-3-030-36020-7_1},
volume = {2256},
year = {2020},
}
@phdthesis{7460,
abstract = {Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.
For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries.},
author = {Ölsböck, Katharina},
issn = {2663-337X},
keywords = {shape reconstruction, hole manipulation, ordered complexes, Alpha complex, Wrap complex, computational topology, Bregman geometry},
pages = {155},
publisher = {IST Austria},
title = {{The hole system of triangulated shapes}},
doi = {10.15479/AT:ISTA:7460},
year = {2020},
}
@article{7554,
abstract = {Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
issn = {10957219},
journal = {Theory of Probability and its Applications},
number = {4},
pages = {595--614},
publisher = {SIAM},
title = {{Weighted Poisson–Delaunay mosaics}},
doi = {10.1137/S0040585X97T989726},
volume = {64},
year = {2020},
}
@article{7567,
abstract = {Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/d−−√) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d2). We conjecture that both bounds are optimal for triangulations in Euclidean space.},
author = {Choudhary, Aruni and Kachanovich, Siargey and Wintraecken, Mathijs},
issn = {1661-8289},
journal = {Mathematics in Computer Science},
publisher = {Springer Nature},
title = {{Coxeter triangulations have good quality}},
doi = {10.1007/s11786-020-00461-5},
year = {2020},
}
@unpublished{7568,
abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e.manifolds defined as the zero set of some multivariate multivalued functionf:Rd→Rd−n.A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear(PL) approximation based on a triangulationTof the ambient spaceRd. In this paper, we giveconditions under which the PL-approximation of an isomanifold is topologically equivalent to theisomanifold. The conditions can always be met by taking a sufficiently fine triangulationT.},
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
booktitle = {EUROCG 2020},
pages = {8},
title = {{The topological correctness of the PL-approximation of isomanifolds}},
year = {2020},
}
@article{7666,
abstract = {Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.},
author = {Edelsbrunner, Herbert and Ölsböck, Katharina},
issn = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{Tri-partitions and bases of an ordered complex}},
doi = {10.1007/s00454-020-00188-x},
year = {2020},
}
@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 = {21996768},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{When different norms lead to same billiard trajectories?}},
doi = {10.1007/s40879-020-00405-0},
year = {2020},
}
@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 = {0179-5376},
journal = {Discrete & Computational Geometry},
publisher = {Springer Nature},
title = {{Sheaf-theoretic stratification learning from geometric and topological perspectives}},
doi = {10.1007/s00454-020-00206-y},
year = {2020},
}
@phdthesis{7944,
abstract = {This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.},
author = {Masárová, Zuzana},
isbn = {978-3-99078-005-3},
issn = {2663-337X},
keywords = {reconfiguration, reconfiguration graph, triangulations, flip, constrained triangulations, shellability, piecewise-linear balls, token swapping, trees, coloured weighted token swapping},
pages = {160},
publisher = {IST Austria},
title = {{Reconfiguration problems}},
doi = {10.15479/AT:ISTA:7944},
year = {2020},
}
@inproceedings{7952,
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: ℝ^d → ℝ^(d-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 ℝ^d. 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},
booktitle = {36th International Symposium on Computational Geometry},
isbn = {978-3-95977-143-6},
issn = {1868-8969},
location = {Zürich, Switzerland},
pages = {20:1--20:18},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The topological correctness of PL-approximations of isomanifolds}},
doi = {10.4230/LIPIcs.SoCG.2020.20},
volume = {164},
year = {2020},
}