Computational differential topology

Denis Blackmore, Yuriy Mileyko

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Some of the more differential aspects of the nascent field of computational topology are introduced and treated in considerable depth. Relevant categories based upon stratified geometric objects are proposed, and fundamental problems are identified and discussed in the context of both differential topology and computer science. New results on the triangulation of objects in the computational differential categories are proven, and evaluated from the perspective of effective computability (algorithmic solvability). In addition, the elements of innovative, effectively computable approaches for analyzing and obtaining computer generated representations of geometric objects based upon singularity/stratification theory and obstruction theory are formulated. New methods for characterizing complicated intersection sets are proven using differential analysis and homology theory. Also included are brief descriptions of several implementation aspects of some of the approaches described, as well as applications of the results in such areas as virtual sculpting, virtual surgery, modeling of heterogeneous biomaterials, and high speed visualizations.

Original languageEnglish (US)
Pages (from-to)35-92
Number of pages58
JournalApplied General Topology
Volume8
Issue number1
DOIs
StatePublished - 2007

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Keywords

  • Decidable
  • Effectively computable
  • Embeddings
  • Homology
  • Isotopy
  • Obstructions
  • Shape
  • Sl-projective varieties
  • Stratification
  • Varieties

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