@article{8773,
abstract = {Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.},
author = {Brown, Adam and Romanov, Anna},
issn = {1088-6826},
journal = {Proceedings of the American Mathematical Society},
keywords = {Applied Mathematics, General Mathematics},
number = {1},
pages = {37--52},
publisher = {American Mathematical Society},
title = {{Contravariant forms on Whittaker modules}},
doi = {10.1090/proc/15205},
volume = {149},
year = {2021},
}
@phdthesis{9056,
abstract = {In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations.
The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics.
We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss
the implications on persistence for periodic data sets.},
author = {Osang, Georg F},
issn = {2663-337X},
pages = {134},
publisher = {IST Austria},
title = {{Multi-cover persistence and Delaunay mosaics}},
doi = {10.15479/AT:ISTA:9056},
year = {2021},
}
@article{9317,
abstract = {Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.},
author = {Edelsbrunner, Herbert and Osang, Georg F},
issn = {14320444},
journal = {Discrete and Computational Geometry},
publisher = {Springer Nature},
title = {{The multi-cover persistence of Euclidean balls}},
doi = {10.1007/s00454-021-00281-9},
year = {2021},
}
@article{9465,
abstract = {Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.},
author = {Edelsbrunner, Herbert and Nikitenko, Anton and Osang, Georg F},
issn = {14208997},
journal = {Journal of Geometry},
number = {1},
publisher = {Springer Nature},
title = {{A step in the Delaunay mosaic of order k}},
doi = {10.1007/s00022-021-00577-4},
volume = {112},
year = {2021},
}
@inproceedings{9604,
abstract = {Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.},
author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza},
booktitle = {Leibniz International Proceedings in Informatics},
isbn = {9783959771849},
issn = {18688969},
location = {Online},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Counting cells of order-k voronoi tessellations in ℝ^{3} with morse theory}},
doi = {10.4230/LIPIcs.SoCG.2021.16},
volume = {189},
year = {2021},
}