Biological "units" (cells, bacteria, animals, units, …) are out of equilibrium: there is an energy uptake from the environment that is translated into activity, examples of which are motion or the generation of stresses. Active matter is a conceptual term for systems that are comprised of such active units. As the energy source as well as the specific machinery that generates the activity are often not explicitly modelled, such active matter systems are to an extent outside the paradigms of standard statistical physics, for example usual conservation laws for energy or momentum may not hold.

The general type of questions is very similar: Given a specific set of dynamical rules (equations of motion), what behaviour would one observe in a large ensemble of these systems? Consequently, techniques from standard passive statistical physics can be adapted towards active matter. Any analytical description has to deal with the abundance of information in these systems, actually tracking the evolution of every microscopic degree of freedom is generally neither feasible nor constructive. Identifying the relevant information (often in form of slow modes) is, therefore, an essential task. This step is often guided by numerical insight into the model via simulation. Having identified a physically interesting set of variables, reduction of the model then constitutes a technical challenge: We explicitly want to neglect information about the system and describe it solely in terms of the interesting variables. Working along these lines, might for example result in the effective description of a deterministic system being stochastic, with a noise term originating in the reduced description.

Addressing both aspects, is by no-means straightforward and the way to a efficient and insightful treatment problem-specific, which results in broad variety of approaches.

Condensed matter is considered soft, when at least some of the relevant energy scales are comparable with the one set by room temperature or, more generally, when thermal excitations are sufficient to deform or otherwise structurally alter it. Conceptually, this means that entropy is very important and on at least equal relevance as the elastic energies in the system. An instructive example of this is the entropic elasticity of polymeric materials, which is not generated in a simple mechanic way, by extending molecular bonds, but entropically by constraining  the thermally fluctuating polymers. One way, an abundance of very soft modes is often generated is by means of disorder. Generally speaking, disorder generates frustration and frustration leads to a high number of states with similar energetic properties.

A prominent representation of the interesting dynamics that can emerge on meso- to macroscopic scales is pattern formation. The evolution of a system that would statically allow for spatially uniform states, dynamically shows a richer, scale-dependent behaviour. A striking historical example for the counter-intuitiveness that can be involved is the Turing mechanism in reaction-diffusion systems which has for example been used to describe the patterns in animals. Here, the formation of ordered patterns is driven by diffusion, which would be expected to contribute to greater uniformity in the system.

Kinetic roughening, the evolution of structured interfaces due to dynamical instabilities, occurs in a lot of growth or otherwise driven processes. The description of such rough interfaces requires its own (fractal) language, that conceptually resembles the tools used to describe critical systems near a phase-transition. For example, encoding the complex geometry into roughness exponents.

These pattern problems can be fruitful in every of the branches of contemporary physics. We numerically predicted the existence of a novel building block, double-finger structure now called doublon, in phase transitions with diffusion transport by means of front-tracking algorithm. Analytically, describing the selection rules poses an interesting challenge as the highest derivative (the curvature) enters with an asymptotically small prefactor (the surface tension), thus requiring a singular pertubation theory that is conceptually similar to the WKB approach to quantum mechanics. Finally, we were in collaboration with experimentalists from the University of Paris also able to validate the predictions  experimentally.