Matching is a mysterious phenomenon that has recently been observed for several parametrised families of interval maps in deterministic and random settings. It is the property that for each critical point the (random) orbits of the left and right limit merge after some finite number of steps, and that the (expected value of the) derivatives of both orbits are also equal at that time; this assures the stability of this phenomenon under small perturbations of the parameter. Since most of the dynamical behaviour of systems are encoded in the possible trajectories of the critical points, knowledge on when and how matching occurs can help in finding explicit expression for the natural invariant measure. Once this measure is found, one is able to obtain essential information regarding the system, such as the frequency the orbits enter a specific region, the entropy, the Lyapunov exponents, mixing rates etc., and to make comparisons as the parameter varies. There are many theorems that assert the existence of such invariant measures, but there are few results that give a recipe for an explicit formula, which is essential in describing the exact asymptotic behaviour. We propose a new methodology to construct such measures and to uncover their properties and behaviour under parametrised perturbations. Our aim is to give a systematic way of analysing these systems and to provide a new approach, with the help of matching, to relate non-isomorphic systems that exhibit similar matching phenomena and to extract information from one system to the other.
In this project, you will be investigating questions such as:
- investigate the arithmetic, metric and matching properties of various deterministic and random algorithms generating r-continued fractions;
- investigate the symbolic structure of matching sequences. Explain why we see connections between different quantities regarding different dynamical systems? The sequences obtained seem to have similar behaviour with regard to the phenomenon of matching. The structure of such sequences will be studied using the tools of the field Combinatorics on Words.
As our PhD candidate, you have an MSc degree in mathematics with emphasis on dynamical systems, in particular ergodic theory and symbolic dynamics. Candidates in the final stage of obtaining this degree will also be considered. Having a good knowledge of Mathematica and/or Sage is considered an advantage.
- a position for four years;
- a full-time gross salary, ranging from €2,395 in the first year to €3,061 in the fourth year, in scale P (PhD-Promovendus);
- benefits including 8% holiday bonus and 8.3% end-of-year bonus;
- a pension scheme, partially paid parental leave, and flexible employment conditions based on the Collective Labour Agreement Dutch Universities (cao).
In addition to the employment conditions laid down in the cao for Dutch Universities, Utrecht University has a number of its own arrangements. For example, there are agreements on professional development, leave arrangements, and sports. We also give you the opportunity to expand your terms of employment via the Employment Conditions Selection Model. This is how we like to encourage you to continue to grow.
More information about working at the Faculty of Science can be found here.
About the organization
The Mathematical Institute of the Department of Mathematics organises and teaches the Bachelor’s curricula in Mathematics as well as the (English-taught) Master’s programme in Mathematics and some service teaching in mathematics. The institute currently comprises about 35 faculty members, 11 Postdocs, and 29 PhD students, and teaches 500 students. The institute is internationally recognised for its research in both pure and applied mathematics. It houses the Utrecht Geometry Centre and maintains a long-standing tradition of transdisciplinary collaboration with other scientific fields including, among others, atmosphere/ocean sciences, computational life sciences, mathematical biology, medical and seismic imaging, and theoretical physics. The Mathematical Institute has close ties with the Departments of Biology, Physics, Information and Computer Sciences, the Faculty of Geosciences, UMCU (Utrecht University Medical Center), and institutes such as IMAU (Institute for Marine and Atmospheric research), RIVM (The Dutch National Institute for Public Health and the Environment) and CWI (the National Research Institute for Mathematics and Computer science). We participate in university focus areas in Complex Systems and Applied Data Science.
At the Faculty of Science, there are 6 departments to make a fundamental connection with: Biology, Chemistry, Information and Computing Sciences, Mathematics, Pharmaceutical Sciences, and Physics. Each of these is made up of distinct institutes that work together to focus on answering some of humanity’s most pressing problems. More fundamental still are the individual research groups – the building blocks of our ambitious scientific projects.
Utrecht University is a friendly and ambitious university at the heart of an ancient city. We love to welcome new scientists to our city – a thriving cultural hub that is consistently rated as one of the world’s happiest cities. We are renowned for our innovative interdisciplinary research and our emphasis on inspirational research and excellent education. We are equally well-known for our familiar atmosphere and the can-do attitude of our people. This fundamental connection attracts Researchers, Professors and PhD candidates from all over the globe, making both the university and the Faculty of Science a vibrant international and wonderfully diverse community
If you have any questions, please contact Karma Dajani, via firstname.lastname@example.org.
Do you have a question about the application procedure? Please send an email to email@example.com.
There is no deadline. Applications are assessed on a rolling basis until the position is filled. We would like to fill the position by December 1, 2021, at the latest.
Everyone deserves to feel at home at our university. We welcome employees with a wide variety of backgrounds and perspectives. If you have the expertise and the experience to excel in this role, then simply respond via the “Apply now” button! Please submit the following material:
- a cover letter including an explanation why you consider pursuing a PhD in Ergodic Theory;
- your curriculum vitae;
- a list of all courses taken for your Bachelor’s and Master’s degrees together with grades and explanations of the grades;
- if available, (drafts of) your Master’s or Bachelor’s thesis;
- the names, contact information, and e-mail addresses of at least two academic referees, one of which is preferably your Master’s thesis supervisor.
* Please also arrange for the referees to send their reference letters to firstname.lastname@example.org.
Please note: Due to the current situation regarding the Coronavirus (COVID-19) the process of selection and interviews is subject to change. Initial interviews will most likely be conducted online