I study dynamical systems theory. As its name suggests, the theory explores systems where something moves over time. To describe a dynamical system, we need two ingredients: a phase space containing some points to tell what moves and some self-maps of that space to describe how our points move. A map applied to a point in the space carries it to a new position given by the map's value at that point. Our main goal is to understand patterns carved by points travelling through the phase space.

Usually, the phase space is endowed with a structure (for example, topological, smooth, or probabilistic), and the self-maps responsible for the evolution are compatible with it (are continuous, differentiable, or measure-preserving). The choice of the structure divides the dynamical systems theory into subareas, each having its own methods.

My interests pivot around topological dynamics and ergodic theory. These are essential branches of dynamical systems with broad connections to smooth and low dimensional dynamics, symbolic dynamics, thermodynamic formalism, fractal geometry, deterministic chaos, and other areas of mathematics, like functional analysis, measure theory, (differential) geometry, combinatorics, number theory, descriptive set theory, etc.

My research efforts usually revolve around the following strongly interconnected problems:

• Classification: We divide dynamical systems into categories (equivalence classes) based on their characteristics. We sometimes obtain anti-classification results, which rigorously prove that classification is impossible in a particular sense.
• Genericity: We determine which dynamical systems are typical inside a specific collection of systems and what behaviour to expect from these typical instances.
• Flexibility: We construct examples to show that certain restrictions on dynamical properties in a given family of systems are best possible; within these constraints, all possibilities are realised by some systems from the family.
• Complexity: We develop ways to measure the complexity of a system and its evolution. The best-known example measure of complexity is entropy.

In addition, I occasionally apply dynamical systems theory to other areas of mathematics. To accomplish this, I set static objects in motion to gain insight that would be unavailable if they stayed put.

This description would have to be much longer to be complete. Any mathematical object that moves (changes, evolves) or can get moving is a potential subject of my next inquiry.

I'm working in the area of dynamical systems and ergodic theory with connections to other research areas like number theory, probability theory or mathematical physics, particularly aperiodic order.
One of my particular interests are dynamical systems where extreme events play a crucial role. Differently to independent, identically distributed (i.i.d.) random variables, can extreme events in the dynamical systems context occur in clusters and make it behave differently as in the i.i.d. context. Closely related to this is my research in infinite ergodic theory. Here, often an orbit stays a long time in an infinite orbit and it takes many steps until the orbit returns to a finite measure set. Often, nice toy models for the above described phenomena can be found in the area of (generalized) continued fractions.
Another research interest are diffraction measures of aperiodic systems like the Thue Morse measure and their scaling properties. Here, I also use methods originating from statistical physics like the thermodynamic formalism which can also be used to obtain information about the Hausdorff dimension of Birkhoff averages.

A complete list of my publications and preprints and some further informations can be found on my personal homepage: https://tanja-schindler.github.io/

Besides my mathematical interests while being in Poland I'm also interested in learning the Polish language and given that there are a number of nature areas around I would also like to get to know more about the lokal plants.