Geometrical and Algebraic Aspects of Condensed Matter Physics

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It has been recognized that some of physically important quantum states are not well characterized by conventional order parameters. In such cases, geometrical phases which are characteristic in quantum physics play an important role. We try to characterize the states by the word "topological orders". Some of such states are polyacetylenes, quantum Hall states, two dimensional Carbon sheets, anisotropic superconductors with time reversal symmetry breaking and so on. In most cases, the topological orders are hidden in the bulk and reveal their characters near boundaries or defects as generalized edge states. Some of low dimensional quantum liquids can be also in the classes where topological invariants based on the Berry's connection may replace the conventional order parameters.