Studies on the interaction between analysis and algebra through "Philosophy of Concepts": in the cases of noncommutative geometry and algebraic geometry

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This research puts into practice for modern mathematics "Philosophy of Concepts" developed by such French philosophers, as J. Vuillemin and G.-G. Granger. Using the method of Vuillemin's "Philosophy of Algebra" (1962), which philosophically analyzes the birth of Glois group and its extension to many domains of mathematics, I apply this method for studying the birth and generation of the concepts in noncommutative geometry introduced by A. Connes, as well as of those in algebraic geometry developed by A. Grothendieck. New geometrical objects are born through rich interactions between analytical and algebraic operations in these geometries. There, the concepts such as group, sheaf and (co)homology play important roles for clarifying and relating the structures of objects in many mathematical domains.