A festschrift to honour Mel Greenhut, who has long toiled on spatial economics, this book focuses on a single question - if and how economic space matters, discussing theoretical, as well as historical issues of contemporary relevance.;Subjects covered include historical accounts of spatial economics; land use and public goods; and location theory and spatial competition. Hiroshi Ohta is the author of "Theory of Spatial Pricing and Market Areas" (with M.L. Greenhut), "Spatial Price Theory of Imperfect Competition" and "New Frontiers in Regional Science". Jacques-Francois Thisse is the co-author of "Discrete Choice Theory of Product Differentiation".;The contributors include Simon P. Anderson, William J. Baumol, Bruce L. Benson, Andre de Palma, B. Curtis Easton, Nicholas S. Economides, Robert B. Ekelund Jr, Ronald D. Fischer, John G. Greenhut, Timothy J. Gronberg, Robert E. Hebert, Hong Hwang, Amoz Kats, Tatsuhiko Kawashima, Chao-Cheng Mai, Milton H. Marquis, Edwin S. Mills, Gordon M. Myers, Hisao Nishioka, George Norman, Yorgos Y. Papageorgiou, Louis Phlips, Susan Rose-Ackerman, Noboru Sakashita, Paul A. Samuelson, Douglas G. Sauer, Thomas R. Saving, Nicolas Schmitt, Takaaki Takaha

The package of Gromov's pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book's authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, "virtual fundamental class" is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from "geometry" to "homological algebra". Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the "homotopy limit" needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures.

This is a Festschrift to honour Professor Melvin Greenhut who has long toiled on spatial economics. The book accordingly focuses on a single question: in what sense 'economic space' matters in economic theory. Space in economics is an elusive concept, apparently separating and embracing economic agents at the same time. This is why adding it to already overly complicated economic agents at the same time. This is why adding it to already overly complicated economic models may not necessarily help economics to become sufficiently realistic. In this book, leading scholars of international stature try to find ways of introducing space in economic theory which will make it simpler and more realistic, analysing theoretical and historical issues of contemporary relevance, such as land use, congestion and public goods, location theory and spatial competition.

The package of Gromov's pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book's authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, "virtual fundamental class" is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from "geometry" to "homological algebra". Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the "homotopy limit" needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures.