Abstract. In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev's periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane-Mele Z2-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse MayerVietoris exact sequence.

Aquaphotomics is a new discipline that provides a framework for understanding changes in water molecular system presented as a water spectral pattern, to mirror the rest of the solution and to give a holistic description related to system functionality. One of its main purposes is to identify water bands as main coordinates of future absorbance patterns to be used as a system biomarker. This chapter presents the Aquaphotomics methodology and illustrates a way to identify specific water bands using temperature change and addition of solutions of different ionic strength as perturbations. Rapid and precise measurement of low concentration solutes has been given as a strong evidence of the vast information that "the water spectral pattern as molecular mirror" approach provides. Few applications using near infrared spectroscopy and multivariate analysis as main tools of Aquaphotomics have been presented.

In this paper, we study a generalization of twisted (groupoid) equivariant K-theory in the sense of Freed-Moore for Z2-graded C *algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele's K-theory for Z2-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant K-group when the C * -algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

We complete the classification of Bost-Connes systems. We show that two Bost-Connes C*-algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissen-de Smit-Li-Marcolli-Smit, we conclude that two Bost-Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic.

In this paper, the first of two, we introduce an alternative definition of the Chang–Weinberger–Yu relative higher index, which is thought of as a relative analogue of the Mishchenko–Fomenko index pairing. A main result of this paper is that our map coincides with the existing relative higher index maps. We make use of this fact for understanding the relative higher index. First, we relate the relative higher index with the higher index of the corresponding amalgamated free product group. Second, we define the dual relative higher index map and show its rational surjectivity under certain assumptions.