---
OA_place: publisher
_id: '21021'
abstract:
- lang: eng
  text: This thesis examines how geometry and topology intersect in the representation,
    transformation, and analysis of complex shapes. It considers how continuous manifolds
    relate to their discrete analogues, how topological structures evolve in persistence
    vineyards, and how tools from topological data analysis can illuminate problems
    in mathematical physics. Central to this exploration is the question of how structure,
    both geometric and topological, persists or changes under approximation, sampling,
    or deformation. The work develops new approaches to skeletal and grid-based representations
    of surfaces, reveals the full expressive capacity of persistence vineyards, and
    applies topological methods to the longstanding problem of equilibria in electrostatic
    fields. These threads braid together into a broader understanding of how topology
    and geometry inform one another across theory, computation, and application.
acknowledged_ssus:
- _id: M-Shop
- _id: ScienComp
acknowledgement: "The research presented in this thesis was funded by the DFG Collaborative
  Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’.\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Christopher D
  full_name: Fillmore, Christopher D
  id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
  last_name: Fillmore
citation:
  ama: Fillmore CD. Braiding geometry and topology to study shapes and data. 2026.
    doi:<a href="https://doi.org/10.15479/AT-ISTA-21021">10.15479/AT-ISTA-21021</a>
  apa: Fillmore, C. D. (2026). <i>Braiding geometry and topology to study shapes and
    data</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT-ISTA-21021">https://doi.org/10.15479/AT-ISTA-21021</a>
  chicago: Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes
    and Data.” Institute of Science and Technology Austria, 2026. <a href="https://doi.org/10.15479/AT-ISTA-21021">https://doi.org/10.15479/AT-ISTA-21021</a>.
  ieee: C. D. Fillmore, “Braiding geometry and topology to study shapes and data,”
    Institute of Science and Technology Austria, 2026.
  ista: Fillmore CD. 2026. Braiding geometry and topology to study shapes and data.
    Institute of Science and Technology Austria.
  mla: Fillmore, Christopher D. <i>Braiding Geometry and Topology to Study Shapes
    and Data</i>. Institute of Science and Technology Austria, 2026, doi:<a href="https://doi.org/10.15479/AT-ISTA-21021">10.15479/AT-ISTA-21021</a>.
  short: C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute
    of Science and Technology Austria, 2026.
corr_author: '1'
date_created: 2026-01-20T21:38:40Z
date_published: 2026-01-21T00:00:00Z
date_updated: 2026-07-22T06:33:54Z
day: '21'
ddc:
- '514'
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
- _id: UlWa
doi: 10.15479/AT-ISTA-21021
file:
- access_level: open_access
  checksum: 4c0889130095c31d4e5088c5b8dfd607
  content_type: application/pdf
  creator: cfillmor
  date_created: 2026-01-26T19:44:46Z
  date_updated: 2026-01-30T11:40:09Z
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  date_created: 2026-01-26T19:46:20Z
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  file_name: Thesis.zip
  file_size: 166080788
  relation: source_file
file_date_updated: 2026-01-30T11:40:09Z
has_accepted_license: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: '122'
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '20260'
    relation: part_of_dissertation
    status: public
  - id: '21051'
    relation: part_of_dissertation
    status: public
  - id: '21050'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
title: Braiding geometry and topology to study shapes and data
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2026'
...