# Topological Filtering and Topological Active Matter

Vibrations are essential characteristics of matter and its activity. One strategy to alleviate the energetic crisis is to adapt part of the electromagnetic technological demands to vibrational filtering technologies. To this end, we plan to build analogue filters employing vibrational eigenstructure design, that is, leveraging the topological eigenstructures of mechanical and molecular vibrating systems. For example, using spectral decomposition, we analyze the eigenscape of fundamental frequencies of tessellations of molecules at surfaces in order to develop topological indicators for dynamic active matter. These indicators depend on the type of physical property (stiffness, mass, velocity, frequency, rotational angles, etc). Ultimately, we aim to topologically classify dynamical and active matter and to design novel filters that enhance transmission of vibrations across finite materials at finite temperatures. This toolbox is applied across scales to create new ways of transmitting information, energy and matter, help in the design of aids, architectural spaces, garments and reduce the complexity of computer-aided shape designs.

For the past three years our research activities have been focused on: (1) The mathematical description of the vibrations of discrete surfaces based on the spectral analysis of the wave equation (Figure 1). (2) The design, modelling, measurement and analysis of functional (supra)molecular architectures at surfaces (Figure 2, Cojal González *et al.* 2019; Palma 2021). These systems act as a playground for the study of topological properties in materials in the presence of global perturbations, such as temperature. Following our extensive efforts, the time is ripe to advance into the applicability of these research: Can we fabricate ›optical fiber‹-analogues for vibrations, so as to transmit and filter information and energy from A to B? How can topological properties of physical eigenspaces help in the disruptive design of artistic, architectural, computational and cultural environments?

Our current research plan comprises the following packages:

**a. Vibrational and diffusional active matter waveguiding.** We are developing topological spaces of propagating waves of vibrating matter, theoretically and experimentally. In accordance with the principles of topological physics, we have the conjecture that at least two distinct topological (vibrational) spaces may exist in matter, such that each one will transmit waves of interest. Ultimately, we will demonstrate the experimental transmission of diffusive waves of matter, on top of a molecular lattice – a vibrational and diffusional ›optical fiber‹.

**b. Physical eigenspace design.** Here we analyse and design vibrating material across scales, from molecular dynamics up to real-world shapes and models. With physically-motivated modifications to the Laplace operator a more versatile and material-sensitive wave equation will be designed whose fundamental eigenspaces serve as major tools in the filtering and design of shapes. The eigenspace design will drastically reduce the complexity of shape design in computer-aided design, computer animation and architecture.

**c. Filtering in spaces of many body physics.** The Cube of Physics represents our knowledge of physics with eight theories at its corners and six worlds at its faces. While most of the corner theories are well established today, there are puzzling open ends, particularly the so-called Theory of Everything. Moreover, this Cube does not contain many body theories, which pertain to matters of activity, where temperature, statistical physics, and the physics of complex systems play key roles. For this purpose, a four-dimensional Hypercube of Physics shall be explored, with the axes connected to the physical constants G, c-1, h and k_B, and thereby to the corresponding natural units for length, mass, time, and temperature, as introduced originally by Max Planck. The Hypercube shall be employed to classify and explore filtering processes in different pertinent spaces of many body physics.