Persistent homology and duality : doctoral thesis

Kališnik Verovšek, Sara (author), Smrekar, Jaka (mentor), Repovš, Dušan (co-mentor)

An important problem with sensor networks is that they do not provide information about the regions that are not covered by their sensors. If the sensors in a network are static, then the Alexander Duality Theorem from classic algebraic topology is sufficient to determine the coverage of a network. However, in many networks the nodes change position with time. In the case of dynamic sensor networks, we consider the covered and uncovered regions as parametrized spaces with respect to time. Parametrized homology is a variant of zigzag persistent homology that measures how the homology of the levelsets of the space changes as we vary the parameter. We present a few theorems that extend different versions of classical Alexander Duality theorem to the setting of parametrized homology theories. This approach sheds light on the practical problem of 'wandering' loss of coverage within dynamic sensor networks.

• Language: English
• Keywords: Alexander duality, persistent homology, zigzag persistence, levelset zigzag persistence, parametrized homology
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [S. Kališnik]
• Year: 2013
• Number of pages: 90 str.
• PID: 20.500.12556/RUL-95849
• UDC: 515.14(043.3)
• COBISS.SI-ID: 16756057
• Publication date in RUL: 24.10.2017

Secondary language

• Language: Slovenian
• Title: Vztrajna homologija in dualnost

Abstract

Eden izmed večjih problemov pri preučevanju senzorskih omrežij je, da nudijo le informacijo o področju, ki ga senzorji pokrivajo. V statičnih senzorskih omrežjih klasična Aleksandrova dualnost zadošča kot kriterij za pokritost, ampak v mnogo omrežjih se položaj senzorjev spreminja s časom in ta izrek ni dovolj. V primeru dinamičnih senzorskih omrežij sta območji pokritosti in nepokritosti parametrizirana prostora glede na čas. Parametrizirana homologijaje različica cikcak vztrajne homologije, ki meri, kako se homologijanivojnic prostora spreminja, če spreminjamo parameter. V disertaciji predstavimo parametrizirane ekvivalente nekaj različic klasične Aleksandrove dualnosti. Parametrizirana Aleksandrova dualnost nam tudi pomaga pri razumevanju 'problema vsiljivca'.

• Keywords: Aleksandrova dualnost, vztrajna homologija, cikcak vztrajnost, cikcak vztrajnost za nivojnice, parametrizirana homologija