Unraveling Topology
Topology, a fascinating branch of mathematics, explores the properties of figures or objects that remain unchanged under continuous deformations—those that do not involve cutting or sewing the object. Consider the everyday example of shoelaces: tying a knot involves passing one end under the other to create a more secure structure. If the ends were glued together, the knot would stay locked within the lace, and the only way to untie it would be to cut it. This concept of topology is crucial not only for understanding abstract mathematical objects, like the very definition of a “knot,” but it also has significant applications in various fields. These range from theoretical physics (including field theories and Feynman path integrals) to the statistical mechanics of soft matter (like polymers, liquid crystals, and other complex fluids), and even to essential biological molecules such as DNA, RNA, and proteins. Due to the vast and specialized nature of these fields, a common language to integrate their shared traits would be highly beneficial. The review titled “Topology in Soft and Biological Matter,” recently published in Physics Reports, seeks to achieve this integration. Starting from the rigorous language of knot theory and field theories in physics to the latest computational techniques and algorithms for quantifying the topological properties of real molecules, the review presents an extensive overview of the current applications of topological concepts in mathematics, physics, chemistry, and biology. This comprehensive review marks the culmination of a four-year journey under the COST Action CA17139 EUTOPIA, coordinated internationally by Prof. Luca Tubiana of the University of Trento. Funded by the European COST program, this initiative brought together scientists from almost all European Union countries and several partners, including Turkey, Israel, and the USA. Achille Giacometti, from the Ca’ Foscari University of Venice, coordinated the EUTOPIA working group “Polymeric and Fibrous Topological Materials,” overseeing the detailed sections on polymers and viscoelasticity. The European Centre for Living Technology (ECLT), directed by Achille Giacometti, hosted the working group that completed the final draft of the review. This review represents a significant collaborative effort, with 59 scientists from various institutions across Europe and its partners contributing to this ambitious project. Through their intense cooperative efforts, the review not only highlights the state of the art but also aims to establish a unified language for topological research across diverse scientific disciplines. The study was published in the Physics Reports journal. Full link: https://doi.org/10.1016/j.physrep.2024.04.002
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