Research

Self-similar sets.
Connected questions.

I study how self-similar sets connect, how their topology changes across parameter space, and how this geometry relates to roots of integer polynomials.

The doctoral thesis ↗Download bibliography

Publications & preprints

2013–2026

2026

Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

Preprint

Bernat EspigulearXiv:2606.23278

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigule2026gaskets,
  author = {Espigule, Bernat},
  title = {Explicit interpolations among the {Sierpi\'nski}, {Rauzy}, and {Apollonian} gaskets},
  year = {2026},
  eprint = {2606.23278},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.23278},
  url = {https://arxiv.org/abs/2606.23278}
}

2026

Infinitely many holes in connectedness loci for collinear affine iterated function systems

Preprint

Bernat EspigulearXiv:2606.01467

arXiv ↗PDF ↗

Cite

@misc{espigule2026holes,
  author = {Espigule, Bernat},
  title = {Infinitely many holes in connectedness loci for collinear affine iterated function systems},
  year = {2026},
  eprint = {2606.01467},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.01467},
  url = {https://arxiv.org/abs/2606.01467}
}

2026

Finite capture and the closure of roots of restricted polynomials

Preprint

Bernat Espigule, David JuherarXiv:2603.07397

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigulejuher2026capture,
  author = {Espigule, Bernat and Juher, David},
  title = {Finite capture and the closure of roots of restricted polynomials},
  year = {2026},
  eprint = {2603.07397},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2603.07397},
  url = {https://arxiv.org/abs/2603.07397}
}

2024

Collinear Fractals and Bandt's Conjecture

Journal article

Bernat Espigule, David Juher, Joan SaldañaFractal and Fractional 8(12), 725

DOI ↗arXiv ↗PDF ↗

Cite

@article{espigulejuhersaldana2024bandt,
  author = {Espigule, Bernat and Juher, David and Salda{\~n}a, Joan},
  title = {Collinear Fractals and {Bandt}'s Conjecture},
  year = {2024},
  journal = {Fractal and Fractional},
  volume = {8},
  number = {12},
  pages = {725},
  doi = {10.3390/fractalfract8120725},
  url = {https://doi.org/10.3390/fractalfract8120725}
}

2019

Families of connected self-similar sets generated by complex trees

Preprint

Bernat EspigulearXiv:1902.11282

arXiv ↗PDF ↗

Cite

@misc{espigule2019families,
  author = {Espigule, Bernat},
  title = {Families of connected self-similar sets generated by complex trees},
  year = {2019},
  eprint = {1902.11282},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.1902.11282},
  url = {https://arxiv.org/abs/1902.11282}
}

2019

A Family of Fern-like Ternary Complex Trees

Proceedings article

Bernat EspiguléBridges 2019, pp. 147–154

Publication ↗PDF ↗

Cite

@inproceedings{espigule2019ferns,
  author = {Espigul{\'e}, Bernat},
  title = {A Family of Fern-like Ternary Complex Trees},
  year = {2019},
  booktitle = {Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture},
  pages = {147--154},
  publisher = {Tessellations Publishing},
  address = {Phoenix, Arizona},
  editor = {Goldstine, Susan and McKenna, Douglas and Fenyvesi, Krist{\'o}f},
  url = {https://archive.bridgesmathart.org/2019/bridges2019-147.html}
}

2013

Generalized self-contacting symmetric fractal trees

Journal article

Bernat Espigulé PonsSymmetry: Culture and Science 24(1–4), 320–338

Publication ↗PDF ↗

Cite

@article{espigule2013symmetry,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Generalized self-contacting symmetric fractal trees},
  year = {2013},
  journal = {Symmetry: Culture and Science},
  volume = {24},
  number = {1--4},
  pages = {320--338},
  url = {https://journal-scs.symmetry.hu/content-pages/volume-24-numbers-1-4-pages-001-512-2013/}
}

2013

Unfolding Symmetric Fractal Trees

Proceedings article

Bernat Espigulé PonsBridges 2013, pp. 295–302

Publication ↗PDF ↗

Cite

@inproceedings{espigule2013unfolding,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Unfolding Symmetric Fractal Trees},
  year = {2013},
  booktitle = {Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture},
  pages = {295--302},
  publisher = {Tessellations Publishing},
  address = {Phoenix, Arizona},
  editor = {Hart, George W. and Sarhangi, Reza},
  url = {https://archive.bridgesmathart.org/2013/bridges2013-295.html}
}

Citations use the titles and author credits of the linked public records. LaTeX citations containing \url require the url or hyperref package.

Theses

2026

Collinear Fractals and Connectedness Loci: Topology, Finite Capture, and Restricted Polynomial Roots

Doctoral thesis · Universitat de Girona
Corrected circulation edition, 15 September 2026 · defence pending

Thesis page ↗

Cite

@unpublished{espigule2026thesis,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Collinear Fractals and Connectedness Loci: Topology, Finite Capture, and Restricted Polynomial Roots},
  year = {2026},
  url = {https://complextrees.com/research/thesis/},
  note = {Doctoral thesis, Universitat de Girona. Corrected circulation edition, 15 September 2026. Thesis deposited 9 September 2026; defence pending}
}

2018

Complex trees and their families of connected self-similar sets

Master’s thesis · Universitat de Barcelona
Supervisors: Núria Fagella and Xavier Jarque

Institutional repository ↗

Cite

@mastersthesis{espigule2018master,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Complex trees and their families of connected self-similar sets},
  school = {Universitat de Barcelona},
  year = {2018},
  url = {https://hdl.handle.net/2445/129385}
}

Talks & materials

All recordings & slides ↗

Browse the talks archive for research recordings, teaching materials and historical presentations. The citable audiovisual record below remains part of this bibliography.

2026

Finite capture and the closure of roots of restricted polynomials

Recorded talk

Bernat EspiguleInstitut Henri Poincaré · recorded 27 March 2026

DOI ↗

Cite

@misc{espigule2026ihptalk,
  author = {Espigule, Bernat},
  title = {Finite capture and the closure of roots of restricted polynomials},
  year = {2026},
  howpublished = {Institut Henri Poincar{\'e}, audiovisual resource},
  note = {Recorded 27 March 2026},
  doi = {10.57987/IHP.2026.T1.WS3.016},
  url = {https://doi.org/10.57987/IHP.2026.T1.WS3.016}
}

Jun 2026

One World Numeration SeminarResearch seminar presentation

Request slides

2025

ICERM · Illustrating Mathematics Reunion/ExpansionOnline research presentation

Math Cartography ↗

Jun 2024

Collinear fractals · GSD-UABResearch talk, 10 June 2024

Slides PDF ↗

2015

Experiences in 3D Printing FractalsEuropean Wolfram Technology Conference

Watch recording ↗