Objective Alpha shapes were invented in the early 80s of last century, and their implementation in three dimensions in the early 90s was at the forefront of the exact arithmetic paradigm that enabled fast and correct geometric software. In the late 90s, alpha shapes motivated the development of the wrap algorithm for surface reconstruction, and of persistent homology, which was the starting point of rapidly expanding interest in topological algorithms aimed at data analysis questions. We now see alpha shapes, wrap complexes, and persistent homology as three aspects of a larger theory, which we propose to fully develop. This viewpoint was a long time coming and finds its clear expression within a generalizedversion of discrete Morse theory. This unified framework offers new opportunities, including(I) the adaptive reconstruction of shapes driven by the cavity structure;(II) the stochastic analysis of all aspects of the theory;(III) the computation of persistence of dense data, both in scale and in depth;(IV) the study of long-range order in periodic and near-periodic point configurations.These capabilities will significantly deepen as well as widen the theory and enable new applications in the sciences. To gain focus, we concentrate on low-dimensional applications in structural molecular biology and particle systems. Fields of science natural sciencescomputer and information sciencesdata sciencenatural sciencesearth and related environmental sciencesphysical geographycartographygeographic information systemsnatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesbiological sciencesmolecular biology Keywords computational geometry computational topology algorithms alpha shapes persistent homology Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2017-ADG - ERC Advanced Grant Call for proposal ERC-2017-ADG See other projects for this call Funding Scheme ERC-ADG - Advanced Grant Host institution INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA Net EU contribution € 1 678 432,00 Address Am Campus 1 3400 Klosterneuburg Austria See on map Region Ostösterreich Niederösterreich Wiener Umland/Nordteil Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 678 432,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA Austria Net EU contribution € 1 678 432,00 Address Am Campus 1 3400 Klosterneuburg See on map Region Ostösterreich Niederösterreich Wiener Umland/Nordteil Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 678 432,00
