# Research Interests

My aim is to identify and characterize novel spatial structures in physics and mathematics. Prominent examples are hyperuniformity, quasicrystals, fractals, or rigidity properties. Moreover, I want to learn what geometry can tells us about the physical properties. Finally, I optimize the complex structures for innovative material design.

Thus, my research combines surprising physical phenomena with rigorous mathematics and statistics. Therefore, I work together with mathematicians and theoretical as well as experimental physicists.

### Examples

Together with Paul Steinhardt and Salvatore Torquato from Princeton University, I showed that it is possible to construct foam-based heterostructures with complete photonic band gaps. The image above shows an eigenmode of the electric field in such a “phoamtonic” heterostructure. Recently, we discovered a suprising universal behavior in the gap sensitivity of crystal and disordered networks.

Disordered hyperuniformity is defined by an anomalous suppression of large-scale density fluctuations in a system which can be locally amorphous and isotropic like a liquid but at the same time macroscopically uniform like a crystal. Two recent examples of my work on hyperuniformity are how to cloak the underlying long-range order of randomly perturbed lattices and a characterization of local number fluctuations.

In amorphous tessellations with certain extreme geometrical properties, my coauthors and I discovered a universal hidden order. Applying an iterative local optimization of the so-called Quantizer energy, we show that a broad range of disordered point patterns converge to the apparently same effectively hyperuniform state.

Together with Günter Last from Karlsruhe and D. Yogeshwaran from Bangalore, I defined a stable matching of point processes. The resulting point pattern combines local properties of one of its ancestors with global properties of the other. For example, if we match the ideal gas with a lattice, results in a hyperuniform point process. We also proved that the resulting process is “number rigid”, that is, the number of points in an observation window are determined by the configuration of points outside.

In contrast to previous belief, I found an example of a “strongly rigid point process” which is hyperfluctuating (that is, the diametric opposite of hyperuniform). Günter Last and I recently published a rigorous proof.

My work on Minkowski functionals, as powerful tools for the quantification of complex shapes, ranges from fundamental insights in stochastic geometry to applications in physics and biology across length scales. I used them, for example, to detect faint signals in noisy gray scale images together with Klaus Mecke from Erlangen or to predict the adhesion forces of bacteria on nanorough surfaces together with Karin Jacobs from Saarland University.