This book presents the most important advances in the class of topological materials and discusses the topological characterization, modeling and metrology of materials. Further, it addresses currently emerging characterization techniques such as optical and acoustic, vibrational spectroscopy (Brillouin, infrared, Raman), electronic, magnetic, fluorescence correlation imaging, laser lithography, small angle X-ray and neutron scattering and other techniques, including site-selective nanoprobes. The book analyzes the topological aspects to identify and quantify these effects in terms of topology metrics.

The topological materials are ubiquitous and range from (i) de novo nanoscale allotropes of carbons in various forms such as nanotubes, nanorings, nanohorns, nanowalls, peapods, graphene, etc. to (ii) metallo-organic frameworks, (iii) helical gold nanotubes, (iv) Möbius conjugated polymers, (v) block co-polymers, (vi) supramolecular assemblies, to (vii) a variety of biological and soft-matter systems, e.g. foams and cellular materials, vesicles of different shapes and genera, biomimetic membranes, and filaments, (viii) topological insulators and topological superconductors, (ix) a variety of Dirac materials including Dirac and Weyl semimetals, as well as (x) knots and network structures. Topological databases and algorithms to model such materials have been also established in this book.

In order to understand and properly characterize these important emergent materials, it is necessary to go far beyond the traditional paradigm of microscopic structure-property-function relationships to a paradigm that explicitly incorporates topological aspects from the outset to characterize and/or predict the physical properties and currently untapped functionalities of these advanced materials. Simulation and modeling tools including quantum chemistry, molecular dynamics, 3D visualization and tomography are also indispensable. These concepts have found applications in condensed matter physics, materials science and engineering, physical chemistry and biophysics, and the various topics covered in the book have potential applications in connection with novel synthesis techniques, sensing and catalysis. As such, the book offers a unique resource for graduate students and researchers alike.

Autorentext

Prof. Sanju Gupta is Associate professor and has substantial experience and expertise in functional nanomaterials and characterization with an emphasis on energy, water and biophysics- related research. Additionally, she is working in the area of topological and geometrical aspects of materials and co-authored publications on this topic including an invited feature article in MRS Bulletin and the recent News Feature in MRS Bulletin with the co-editor (Dr. Avadh Saxena) about the underpinning role(s) of topology in the 2016 Physics and Chemistry Nobel Prizes. She has co-organized several conferences and symposia on this topic as well as on advanced nanomaterials and applications. She has won many accolades (e.g. Recipient of NSF Fellowship, NIH Fellowship, DOE Fellowship, JSPS Fellowship and Junior Investigator Award, WISE Award) since her graduate study in addition to being well-placed and known in the materials community for more than two decades including the topology community. As a result, she is co-organizing an Applied Topology conference with colleagues in the Physics and Mathematics departments at WKU in 2018.

Dr. Avadh Saxena is Group Leader of the Condensed Matter and Complex Systems theory group at Los Alamos National Laboratory in New Mexico. He has extensive experience in working on topology related problems both in materials science and condensed matter physics with a large number of publications in this field. He is a Los Alamos National Laboratory Fellow and a Fellow of the American Physical Society. He serves on the advisory boards of several international conferences, has organized numerous workshops and symposia related to topology and functional materials (APS, MRS, SIAM, among others), has co-authored an MRS Bulletin article on this topic and a news Feature on 2016 Nobel Prizes, has co-edited four books (with Springer) and many special issues of research journals. His recent work on topological defects, specifically the magnetic textures called skyrmions, has been featured in Discovery News. He holds Adjunct Professor positions at the University of Barcelona, University of Arizona and Virginia Tech. Recently he was granted the prestigious Affiliate Professorship at the Royal Institute of Technology (KTH), Stockholm. He also serves as an advisor for the National Institute for Materials Science, Tsukuba, Japan.

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1. Gareth Alexander (University of Warwick), G.P.Alexander@warwick.ac.uk**Soft matter, twisted materials. Assistant Professor in Physics and Complexity Science. His primary interests are in applications of geometry and topology to the understanding of soft material systems, in general and defects in liquid crystals, in particular, and self-propulsion in low Reynolds number hydrodynamics and active matter. Working with Randy Kamien (University of Pennsylvania), he became an authority in this area with a recent publications "Instabilities and Solitons in Minimal Strips", Phys. Rev. Lett. 117, 017801 (2016). with Thomas Machon, Raymond E. Goldstein, and Adriana I. Pesci; "Global Defect Topology in Nematic Liquid Crystals", Proc. R. Soc. A 472, 20160265 (2016) with Thomas Machon; "Umbilic Lines in Orientational Order", Phys. Rev. X 6, 011033 (2016), with Thomas Machon; "Motility of active fluid drops on s urfaces", Phys. Rev. E 92*, 062311 (2015), with Diana Khoromskaia.*

**2. Arun Bansil (Northeastern University), ar.bansil@northeastern.edu **Dirac materials, Weyl semimetals. University Distinguished Professor (Ph.D. Harvard). A world authority on the electronic structure of topological materials, nanosystems and complex materials including topological insulators, Dirac materials, Weyl semimetals, etc. Recently predicted many new materials that have been synthesized. He has multiple high profile publications to his credit. Former DOE program manager, US editor of J. Phys. Chem. Solids. **3. Rossen Dandoloff (Univ. de Cergy-Pontoise, France), rossen.dandoloff@u-cergy.fr, rdandoloff@yahoo.com**Heisenberg magnets and magnetism on curved surfaces. Recently retired professor from the University of Cergy-Pontoise (Paris) and an authority on the geometry and topology of spins systems, geometric phase, topological excitations, knots, parallel transport and quantum mechanics of curved manifolds. He has several important publications to his name in this field and has mentored many students on the topic of geometry and topology. **4. Mark Dennis (Univ. of Bristol), Mark.Dennis@bristol.ac.uk **Geometry and topology of knots: electron vortices and wave dislocations. Professor of Theoretical Physics in the Theoretical Physics Group of the School of Physics. His main research is in Optical Field Theory and Topological Physics. ****These topics involve his primary focus on geometric and topological aspects of wave physics, especially their manifestation in structured light and quantum wave functions:

· Singular and topological optics, including optical and electron vortices, wave dislocations and polarization singularities.
· Statistical geometry and topology of rando m functions, with application to quantum chaos, statistical optics and cosmology.
· Geometry and topology of knots, links, braids and ribbons, especially those in spatially-extended fields.

5. Rumiana Dimova and Roland Knorr (Max-Planck Inst. Of Colloids and Interfaces, Germany), Rumiana.Dimova@mpikg.mpg.de&nbs…