Research

Research

On this page you can find a short description of my research interests, planned visits and conference attendances, as well as my full list of publications. For a list of (selected) past scientific activities, please see my CV which can be found here.

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Research Interests

My main interest is the dimension theory of sets and measures arising from stochastic processes and symbolic dynamical systems. I study their interplay with fractal geometry, number theory, and geometric measure theory. Currently, I am studying continuum tree spaces and their relationship to analytical properties and embedding problems of random metric spaces such as the Brownian continuum random tree and Brownian map. I investigate small scale behaviour of sets and measures in connection to the notions of Assouad dimensions.

Funding Information

From September 2022 to August 2024 my research is funded by the Horizons Europe Marie Skłodowska-Curie Individual Fellowship “Dimension and Dynamics”. See the European Commission page for more details of the scheme.

From June 2020 to July 2022 my research was funded by the Senior Lise Meitner Fellowship M-2813 from the FWF (Austrian Research Fund).

Publications

Submitted

  1. On continuum real trees of circle maps and their graphs.
    Maik Gröger and Sascha Troscheit.
    Preprint, available at arXiv:2401.08479, (2024).

  2. Interpolating with generalized Assouad dimensions.
    Amlan Banaji, Alex Rutar, and Sascha Troscheit.
    Preprint, available at arXiv:2308.12975, (2023).

  3. Stability relations for Hilbert space operators and a problem of Kaplansky.
    Laurent W. Marcoux, Heydar Radjavi, Sascha Troscheit, and Yuanhang Zhang.
    Preprint, available arXiv:2306.11202, (2023).

Published

  1. On the Minkowski content of self-similar random homogeneous iterated function systems.
    Sascha Troscheit.
    Annales Fennici Mathematici, 48(1), (2023), 345–359.
    doi:10.54330/afm.129638 and arXiv:2103.04664.

  2. Box-counting dimension in one-dimensional random geometry of multiplicative cascades.
    Kenneth J. Falconer and Sascha Troscheit.
    Communications in Mathematical Physics, 399, (2023), 57–83.
    doi:10.1007/s00220-022-04558-9 and arXiv:2203.15315.

  3. Dynamically defined subsets of generic self-affine sets.
    Balázs Bárány and Sascha Troscheit.
    Nonlinearity, 35, (2022), 4986–5013.
    doi:10.1088/1361-6544/ac8219 and arXiv:2109.06727.

  4. Analogues of Khintchine’s theorem for random attractors.
    Simon Baker and Sascha Troscheit.
    Transactions of the American Mathematical Society, 375(2), (2022), 1411–1441.
    doi:10.1090/tran/8537 and arXiv:2010.07197.

  5. Regularity versus smoothness of measures.
    Jonathan M. Fraser and Sascha Troscheit.
    Pacific Journal of Mathematics, 311(2), (2021), 257–275.
    doi:10.2140/pjm.2021.311.257 and arXiv:1912.07292.

  6. On quasisymmetric embeddings of the Brownian map and continuum trees.
    Sascha Troscheit.
    Probability Theory and Related Fields, 179(3), (2021), 1023–1046.
    doi:10.1007/s00440-020-01024-2 and arXiv:1912.07291.

  7. Exact Hausdorff and packing measures for random self-similar code-trees with necks.
    Sascha Troscheit.
    Studia Mathematica, 257(3), (2021), 253–285.
    doi:10.4064/sm190327-26-8 and arXiv:1710.07167.

  8. Lower Assouad dimension of measures and regularity.
    Kathryn E. Hare and Sascha Troscheit.
    Mathematical Proceedings of the Cambridge Philosophical Society, 170(2), (2021), 379–416.
    doi:10.1017/S0305004119000458 and arXiv:1812.05573.

  9. The Assouad spectrum of random self-affine carpets.
    Jonathan M. Fraser and Sascha Troscheit.
    Ergodic Theory and Dynamical Systems, 41(10), (2021), 2927–2945.
    doi:10.1017/etds.2020.93
    Preprint in Institut Mittag-Leffler preprints, 14801(40), (2018).
    IML Preprint and arXiv:1805.04643.

  10. Quasi-doubling of self-similar measures with overlaps.
    Kathryn E. Hare, Kevin G. Hare, and Sascha Troscheit.
    Journal of Fractal Geometry, 7(3), (2020), 233–270.
    doi:10.4171/JFG/91 and arXiv:1807.09198.

  11. Self-conformal sets with positive Hausdorff measure.
    Jasmina Angelevska, Antti Käenmäki, and Sascha Troscheit.
    Bulletin of the London Mathematical Society, 52(1), (2020), 200–223.
    doi:10.1112/blms.12320, arXiv:1803.09113.
    Calculations for Example 7.6: Calculations.cdf.

  12. Assouad spectrum thresholds for some random constructions.
    Sascha Troscheit.
    Canadian Mathematical Bulletin, 63(2), (2020), 434–453.
    doi:10.4153/S0008439519000547 and arXiv:1906.02555.

  13. The Mass Transference Principle: Ten years on.
    Demi Allen and Sascha Troscheit.
    Horizons of Fractal Geometry and Complex Dimensions, AMS Contemporary Mathematics Series, 731, (2019), 1–33.
    doi:10.1090/conm/731 and arXiv:1704:06628.

  14. The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra.
    Jonathan M. Fraser, Kathryn E. Hare, Kevin G. Hare, Sascha Troscheit, and Han Yu.
    Annales Academiæ Scientiarum Fennicæ, 44(1), (2019), 379–387.
    doi:10.5186/aasfm.2019.4419 and arXiv:1804.09607.

  15. Random ubiquitous transformation semigroups.
    Julius Jonušas and Sascha Troscheit.
    Semigroup Forum, 99(3), (2019), 655–678.
    doi:10.1007/s00233-018-09992-7 and arXiv:1705.05709.

  16. The quasi-Assouad dimension of stochastically self-similar sets.
    Sascha Troscheit.
    Proceedings A of the Royal Society of Edinburgh, 150(1), (2020), 261–275.
    doi:10.1017/prm.2018.112 and arXiv:1709.02519.

  17. Local dimensions of random homogeneous self-similar measures: strong separation and finite type.
    Kathryn E. Hare, Kevin G. Hare, and Sascha Troscheit.
    Mathematische Nachrichten, 291(16), (2018), 2397–2426.
    doi:10.1002/mana.201700466 and arXiv:1711.09718.

  18. On the inverse problem for Channell collisionless plasma equilibria.
    Oliver Allanson, Sascha Troscheit, and Thomas Neukirch.
    IMA Journal of Applied Mathematics, hxy026, (2018), 25pp.
    doi:10.1093/imamat/hxy026.

  19. The Assouad dimension of randomly generated fractals.
    Jonathan M. Fraser, Jun Jie Miao, and Sascha Troscheit.
    Ergodic Theory and Dynamical Systems, 38(3), (2018), 982–1011.
    doi:10.1017/etds.2016.64 and arXiv:1410.6949.

  20. The box dimension of random box-like self-affine sets.
    Sascha Troscheit.
    Indiana University Mathematics Journal, 67(2), (2018), 495–535.
    doi:10.1512/iumj.2018.67.7295 and arXiv:1512.07022.

  21. On the dimensions of attractors of random self-similar graph directed iterated function systems.
    Sascha Troscheit.
    Journal of Fractal Geometry, 4(3), (2017), 257–303.
    doi:10.4171/JFG/51 and arXiv:1511.03461.

  22. From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials.
    Oliver Allanson, Thomas Neukirch, Sascha Troscheit, and Fiona Wilson.
    Journal of Plasma Physics, 82, (2016).
    doi:10.1017/S0022377816000519.

  23. An exact collisionless equilibrium for the Force-Free Harris Sheet with low plasma beta.
    Oliver Allanson, Thomas Neukirch, Fiona Wilson, and Sascha Troscheit.
    Physics of Plasmas, 22, 102116 (2015).
    doi:10.1063/1.4934611.

  24. Hölder differentiability of self-conformal devil’s staircases.
    Sascha Troscheit.
    Mathematical Proceedings of the Cambridge Philosophical Society, 156(2), (2014), 295–311.
    doi:10.1017/S0305004113000698 and arXiv:1301.1286.

Future visits and conference attendances

There's no better present than a future.
— (Death in) Hogfather, Terry Pratchett

2024

  • Uppsala University, Uppsala, Sweden: 24 – 26 Apr 2024.
  • Canada Visit:
    • Randomness and Geometry Programme at the Fields Institute, Toronto ON, Canada: 6 May – 16 Jun 2024. link
    • University of Waterloo, Waterloo ON, Canada: 8 – 10 May 2024.
    • On the Interface of Geometric Measure Theory and Harmonic Analysis, BIRS (Banff International Research Station, Banff AB, Canada: 9 – 14 Jun 2024. link
  • Fractals and Hyperbolic Dynamical Systems Programme at the Rényi Institute, Budapest, Hungary: 26 August – 6 September 2024. link
  • Fractal Geometry and Stochastics 7, Chemnitz, Germany: 23 Sep – 27 Sep 2024. link

Past visits and conference attendances

The story so far:
In the beginning the Universe was created.
This has made a lot of people very angry and been widely regarded as a bad move.
— The Restaurant at the End of the Universe, Douglas Adams

Here I am recording conferences and past visits, including some links, for “posterity”. These listings are incomplete and only date back to 2020.

2024

  • Annual General Meeting of the Finnish Mathematical Society, University of Helsinki, Helsinki, Finland: 25 Mar 2024.
  • Budapest University of Technology and Economics, Budapest, Hungary: 7 Feb – 1 Mar 2024.

2023

  • CIRM Luminy, France: 20 – 25 Nov 2023.
  • Scuola Normale Superiore di Pisa, Pisa, Italy: 27 Nov – 1 Dec 2023.
  • Universiteit Leiden, Netherlands: 2 Nov 2023.
  • Probability and Geometry in, on and of non-Euclidian spaces, CIRM Luminy, France: 2 – 6 Oct 2023. link
  • Quasiworld Workshop, University of Helsinki, Helsinki, Finland: 14 – 18 August 2023. link
  • University of Bristol: 10 – 14 July 2023.
  • Fractal Geometry Workshop, ICMS, Edinburgh, Scotland: 3 – 7 July 2023. link
  • University of Bremen: 26 – 30 June 2023.
  • IMPAN, Warsaw, Poland. Simons Semester: 01 – 27 May 2023. link
    • Thermodynamic Formalism: Non-additive Aspects and Related Topics, Bedlewo, Poland: 14 – 19 May 2023. link
  • University of Jyväskylä, Jyväskylä, Finland: 29 January – 02 February 2023.

2022

  • University of Exeter, UK: 12 – 21 December 2022.
  • North America Visit:
    • 7 – 13 November 2022: University of Washington, Seattle WA, USA;
    • 14 – 24 November 2022: University of British Columbia, Vancouver, Canada;
    • 25 November – 2 December 2022: University of North Texas, Denton TX, USA.
  • Fractals and Related Fields 4, Isle de Proquerolles: 5 – 9 Sep 2022. link
  • LMS Scheme 2 Lecture Tour of the UK:
    • 11 – 15 July 2022: University of Bristol;
    • 18 – 22 July 2022: University of Birmingham;
    • 25 – 29 July 2022: University of Glasgow.
  • Geometry of Deterministic and Random Fractals, Budapest: 27 June – 1 July 2022. link
  • Lund University, Lund, Sweden: 8 – 10 June 2022.
  • Workshop on overlapping and affine IFSs, University of Bristol, Bristol, UK: 2 May – 13 May 2022. link
  • Probability, Analysis, and Dynamics, University of Bristol, Bristol, UK: 6 April – 11 April 2022. link
  • The Analysis and Geometry of Random Spaces workshop, Berkeley CA, USA: 27 March – 4 April 2022. link

2021

  • University of Oulu, Oulu, Finland: 25 October – 14 November 2021.
  • University of St Andrews, Scotland: 11 – 25 September 2021.
  • Geometric Measure Theory workshop, Cortona, Italy: 29 August – 3 September 2021. link
  • Jagiellonian University, Krakow, Poland: 20 – 27 August 2021.
  • First Dynamical Systems Summer Meeting, Bedlewo, Poland: 16 – 20 August 2021. link

2020

  • University of Bielefeld, Bielefeld, Germany: 17 – 20 February 2020.
  • BME Budapest, Budapest, Hungary: 31 January – 7 February 2020.