Periodic Reporting for period 1 - DimDyn (Dimension and Dynamics)
Reporting period: 2022-09-01 to 2024-08-31
The main aim of this project was to tackle several fundamental questions in dynamical systems and probability theory by developing groundbreaking new techniques. These techniques are rooted in dimension theory and capitalise on the extensive progress made in the field in recent years. To further this aim, our intention was to build bridges between the often disjoint domains of dynamical systems, probability theory, mathematical physics, and metric geometry. The project also sought to elevate the funded researcher's international profile, enhancing their potential to secure a permanent academic position and enable them to continue to develop these new areas of research.
Research activities
The project centred on several key research objectives, leading to advances in dynamical systems and probability theory:
1) Local Structures of Sets and Measures
I investigated the Assouad spectrum's behaviour and its role in describing local relative "density". In collaboration with Amlan Banaji and Alex Rutar, we explored the general characteristics of the spectrum,
while joint work with Antti Käenmäki established that well-behaved sets contain measures that capture this spectrum’s nuances.
2) Dynamical Sets and Measures
I studied self-affine sets that stem from diagonal and anti-diagonal linear parts in collaboration with Demi Allen, Antti Käenmäki, Daniel Prokaj, and Karoly Simon. This lead to the closing of a last gap in our understanding of planar self-affine sets. Additionally, I explored a dynamical covering problem for self-similar sets with Balazs Barany and Henna Koivusalo.
3) Embedability of sets with minimal structure
In joint work with Efstathios-Konstantinos Chrontsios-Garitsis, we realised that the Assouad spectrum gives sufficient information on its own to extend the classical Assouad embedding theorem.
4) Stochastically Self-Similar Sets
In joint work with Jayadev Athreya, we proved that Galton-Watson trees (with minimal assumptions) are almost surely quasi-isometric. This implies that these random trees are generically "similar" in a very precise sense, which has implications for the analysis of stochastically self-similar sets. Ongoing work with Ville Suomala seeks to extend these findings to show that Mandelbrot percolation is almost surely quasisymmetric.
Collaborative work with Simon Baker, Henna Koivusalo, and Xintian Zhang also advanced harmonic analysis on stochastically self-similar sets, examining Fourier decay properties of stochastically self-similar measures.
5) Interdisciplinary Connections.
Partnered with Laurent Marcoux, Heydar Radjavi, and Yuanhang Zhang we investigated stability of operators. After translating their problems into the language of measure theory, I provided examples of measures that proved to be important examples in their investigations.
In separate work with Maik Gröger we examined a construction of continuum trees arising in probability theory. We applied this construction to real valued functions and were able to progress our understanding of such classical objects by bridging dimension theory and probability theory.
Of the research activities highlighted above, the majority has been completed and published (or submitted to be published). These research activities not only achieved their primary objectives but also inspired new projects, many of which are now ongoing and described in the next section.
Research Visits and Networking
In-person collaborations were integral to the project's advancement. Notable research exchanges included:
- A November-December 2022 visit to the University of Washington, University of North Texas (Denton), and University of British Columbia, where discussions with Jayadev Athreya, Pablo Shmerkin, Alexia Yavicoli, Pieter Allart, and Mariusz Urbanski contributed significantly to the project.
- Collaborative work with Demi Allen during visits in October and December 2022.
- A February 2023 seminar and research stay at the University of Jyväskylä with Tuomas Orponen.
- A month-long visit to the Banach Centre in Warsaw (May 2023) as part of a thematic semester on Dynamical Systems.
- Further visits to the University of Bremen, University of Bristol, and multiple research institutions across Europe, where I collaborated with a diverse array of experts, including Maik Gröger, Simon Baker, and Henna Koivusalo.
- Extended stays in Budapest (February/March 2024) and a 7 week visit to the Fields Institute (May/June 2024) as part of their thematic programme on Randomness and Geometry.
Conference activities
As part of this project the University of Oulu hosted two events. The first event was a smaller workshop on Shrinking Targets in January 2024 with three speakers: Simon Baker, Markus Myllyoja, and Xintian Zhang that was broadcast as part of the One World Fractals network.
We also held a larger conference in June 2024 on the main topics of this action. It was designed to support emerging talent and build bridges between Geometric Measure Theory, Dynamical Systems, and Probability Theory by showcasing the exceptional research of (mostly) early and mid-career mathematicians.
Career Development
A major part of this grant was the continual development of the funded researcher. The activities of the grant lead to various short-listings for positions at major European research institutions
such as Aalto University, CNRS/Paris Créteil, and TU Chemnitz. Further, the researcher was invited to speak at various high-profile events such as the Fractal Geometry and Stochastics 7 conference, as well as a mini course lecturer at the aforementioned Fields Institute event. The researcher has also started supervising and mentoring two PhD students during the action and now holds a (tenured) Associate Professorship at Uppsala University.
The project centred on several key research objectives, leading to advances in dynamical systems and probability theory:
1) Local Structures of Sets and Measures
I investigated the Assouad spectrum's behaviour and its role in describing local relative "density". In collaboration with Amlan Banaji and Alex Rutar, we explored the general characteristics of the spectrum,
while joint work with Antti Käenmäki established that well-behaved sets contain measures that capture this spectrum’s nuances.
2) Dynamical Sets and Measures
I studied self-affine sets that stem from diagonal and anti-diagonal linear parts in collaboration with Demi Allen, Antti Käenmäki, Daniel Prokaj, and Karoly Simon. This lead to the closing of a last gap in our understanding of planar self-affine sets. Additionally, I explored a dynamical covering problem for self-similar sets with Balazs Barany and Henna Koivusalo.
3) Embedability of sets with minimal structure
In joint work with Efstathios-Konstantinos Chrontsios-Garitsis, we realised that the Assouad spectrum gives sufficient information on its own to extend the classical Assouad embedding theorem.
4) Stochastically Self-Similar Sets
In joint work with Jayadev Athreya, we proved that Galton-Watson trees (with minimal assumptions) are almost surely quasi-isometric. This implies that these random trees are generically "similar" in a very precise sense, which has implications for the analysis of stochastically self-similar sets. Ongoing work with Ville Suomala seeks to extend these findings to show that Mandelbrot percolation is almost surely quasisymmetric.
Collaborative work with Simon Baker, Henna Koivusalo, and Xintian Zhang also advanced harmonic analysis on stochastically self-similar sets, examining Fourier decay properties of stochastically self-similar measures.
5) Interdisciplinary Connections.
Partnered with Laurent Marcoux, Heydar Radjavi, and Yuanhang Zhang we investigated stability of operators. After translating their problems into the language of measure theory, I provided examples of measures that proved to be important examples in their investigations.
In separate work with Maik Gröger we examined a construction of continuum trees arising in probability theory. We applied this construction to real valued functions and were able to progress our understanding of such classical objects by bridging dimension theory and probability theory.
Of the research activities highlighted above, the majority has been completed and published (or submitted to be published). These research activities not only achieved their primary objectives but also inspired new projects, many of which are now ongoing and described in the next section.
Research Visits and Networking
In-person collaborations were integral to the project's advancement. Notable research exchanges included:
- A November-December 2022 visit to the University of Washington, University of North Texas (Denton), and University of British Columbia, where discussions with Jayadev Athreya, Pablo Shmerkin, Alexia Yavicoli, Pieter Allart, and Mariusz Urbanski contributed significantly to the project.
- Collaborative work with Demi Allen during visits in October and December 2022.
- A February 2023 seminar and research stay at the University of Jyväskylä with Tuomas Orponen.
- A month-long visit to the Banach Centre in Warsaw (May 2023) as part of a thematic semester on Dynamical Systems.
- Further visits to the University of Bremen, University of Bristol, and multiple research institutions across Europe, where I collaborated with a diverse array of experts, including Maik Gröger, Simon Baker, and Henna Koivusalo.
- Extended stays in Budapest (February/March 2024) and a 7 week visit to the Fields Institute (May/June 2024) as part of their thematic programme on Randomness and Geometry.
Conference activities
As part of this project the University of Oulu hosted two events. The first event was a smaller workshop on Shrinking Targets in January 2024 with three speakers: Simon Baker, Markus Myllyoja, and Xintian Zhang that was broadcast as part of the One World Fractals network.
We also held a larger conference in June 2024 on the main topics of this action. It was designed to support emerging talent and build bridges between Geometric Measure Theory, Dynamical Systems, and Probability Theory by showcasing the exceptional research of (mostly) early and mid-career mathematicians.
Career Development
A major part of this grant was the continual development of the funded researcher. The activities of the grant lead to various short-listings for positions at major European research institutions
such as Aalto University, CNRS/Paris Créteil, and TU Chemnitz. Further, the researcher was invited to speak at various high-profile events such as the Fractal Geometry and Stochastics 7 conference, as well as a mini course lecturer at the aforementioned Fields Institute event. The researcher has also started supervising and mentoring two PhD students during the action and now holds a (tenured) Associate Professorship at Uppsala University.
This project has had a wide reach in topics and has either opened up or identified multiple areas of new research that have the potential for wide-reaching impacts within mathematics. Of particular note are:
- The opening up of embeddability results using weaker notions of regularity. Due to the ubiquity of embeddings to study sets and measures, such weaker assumptions can be applied in more general
contexts to establish rigidity results via quasi-isometries.
- The structural rigidity of Galton-Watson trees has wide-reaching consequences due to the ubiquity of Galton-Watson processes in probability and dynamics.
- Linking of dimension theory of (graphs of) functions and probability theory sheds surprising new light on classical objects. My project with Maik Gröger has only scratched the surface of these
interactions and further spin-off projects are likely.
To summarise, the projected resulted in, thus far, one publication, three pre-prints submitted for
publication, five manuscripts in preparation, as well as several follow-up projects for which
external funding is sought.
- Published:
Stability relations for Hilbert space operators and a problem of Kaplansky.
Laurent W. Marcoux, Heydar Radjavi, Sascha Troscheit, and Yuanhang Zhang.
To appear in Mathematische Annalen.
- Preprints:
Interpolating with generalized Assouad dimensions.
Amlan Banaji, Alex Rutar, and Sascha Troscheit.
Preprint, available at arXiv:2308.12975 (2023).
On continuum real trees of circle maps and their graphs.
Maik Gröger and Sascha Troscheit.
Preprint, available at arXiv:2401.08479 (2024).
Minkowski weak embedding theorem.
Efstathios Konstantinos Chrontsios Garitsis and Sascha Troscheit.
Preprint, available at arXiv:2408.09063 (2024).
- In preparation:
Dynamical self-similar covering sets
Balazs Barany, Henna Koivusalo, Sascha Troscheit
Fourier decay of stochastically self-similar measures
Simon Baker, Henna Koivusalo, Sascha Troscheit, and Xintian Zhang.
Quasi-isometric rigidity of Galton-Watson trees
Jayadev Athreya and Sascha Troscheit
A variational principle for the Assouad dimension
Antti Käenmäki and Sascha Troscheit
Hausdorff dimension of planar box-like self-affine sets with rotations
Demi Allen, Antti Käenmäki, Daniel Prokaj, Karoly Simon, Sascha Troscheit
- The opening up of embeddability results using weaker notions of regularity. Due to the ubiquity of embeddings to study sets and measures, such weaker assumptions can be applied in more general
contexts to establish rigidity results via quasi-isometries.
- The structural rigidity of Galton-Watson trees has wide-reaching consequences due to the ubiquity of Galton-Watson processes in probability and dynamics.
- Linking of dimension theory of (graphs of) functions and probability theory sheds surprising new light on classical objects. My project with Maik Gröger has only scratched the surface of these
interactions and further spin-off projects are likely.
To summarise, the projected resulted in, thus far, one publication, three pre-prints submitted for
publication, five manuscripts in preparation, as well as several follow-up projects for which
external funding is sought.
- Published:
Stability relations for Hilbert space operators and a problem of Kaplansky.
Laurent W. Marcoux, Heydar Radjavi, Sascha Troscheit, and Yuanhang Zhang.
To appear in Mathematische Annalen.
- Preprints:
Interpolating with generalized Assouad dimensions.
Amlan Banaji, Alex Rutar, and Sascha Troscheit.
Preprint, available at arXiv:2308.12975 (2023).
On continuum real trees of circle maps and their graphs.
Maik Gröger and Sascha Troscheit.
Preprint, available at arXiv:2401.08479 (2024).
Minkowski weak embedding theorem.
Efstathios Konstantinos Chrontsios Garitsis and Sascha Troscheit.
Preprint, available at arXiv:2408.09063 (2024).
- In preparation:
Dynamical self-similar covering sets
Balazs Barany, Henna Koivusalo, Sascha Troscheit
Fourier decay of stochastically self-similar measures
Simon Baker, Henna Koivusalo, Sascha Troscheit, and Xintian Zhang.
Quasi-isometric rigidity of Galton-Watson trees
Jayadev Athreya and Sascha Troscheit
A variational principle for the Assouad dimension
Antti Käenmäki and Sascha Troscheit
Hausdorff dimension of planar box-like self-affine sets with rotations
Demi Allen, Antti Käenmäki, Daniel Prokaj, Karoly Simon, Sascha Troscheit