I matematici italiani e i "misteri riemanniani". La geometria italiana della prima metà del XX secolo tra intuizione e rigore

Research output: Other contribution

Abstract

[automatically translated] The permanence of Riemann in Pisa, its direct relationships with Enrico Betti and, indirectly, with Beltrami, Casorati and Cremona brought profound changes in all fields of mathematics (and the philosophy of mathematics) Italian. In my speech I will focus on some points, especially about the Italian algebraic geometry and its relations with the complex analysis in the first thirty years of the twentieth century, as well as the formation of a way "Italian" to look at the geometry. A careful examination of the historical development of the Italian geometry which will be of crucial information of the contributions from the German and French schools to the development of the geometric interpretation of complex variable function theory Riemann: Clebsch, Nöther and Klein on the one hand, Poincaré , Humbert and Picard on the other hand gave rise dialectically a number of new points of view with which the Italian mathematicians interacted creatively creating an original way to blend a deep geometric vision with great analytical skill. A starting point should be the one formed by the intervention of Corrado Segre at the International Congress of Heidelberg in 1904, through which he made it clear the mutual connections between analysis and geometry, showing a broad overview of some so able to anticipate some trends of mathematics of the twentieth century then just in the bud. In particular Segre will develop its ideas in the sense already outlined some years earlier as a continuous enlargement of the idea of projectivity in the complex and iperccomplessi spaces, according to an address that will find significant developments outside of Italy especially in the work of Cartan (with significant contributions of Fubini); Castelnuovo, Enriques and Severi will develop especially the theory of surfaces and varieties, starting and developing Klein's ideas about the links between group theory no accident Hawkins speaks of Italian surveyors as those who actually developed the Erlangen program before ' decisive intervention of Poincare and Cartan. Given the deep ties between Zariski and Castelnuovo and Enriques (rightly considered his teachers) imposes a survey on the reasons (partly internal to mathematics, but partly also social) type for which the Italian algebraic geometry quickly lost contact with developments - in some great verses - that the discipline was taking through the full assimilation of the ideas generated in the Hilbert school in Göttingen: Van der Waerden, Weil, Artin, Lefschetz, Zariski, ... In conclusion I will examine the role played by the geometric intuition in the development of the ideas of the Italian school. Though of course that role can not be underestimated certainly try to show as if it were an intuition that is not generated spontaneously by the development of purely "visual" faculties of Italian surveyors, but it was hard-won through a careful study of analytical . In the happy phrase of Enriques on "pinned harmonies" close attention should be given to the "answers." The harmonies of algebraic geometry that gave rise to the extraordinary form of intuition for which the Italian surveyors are justly famous were discovered only through a hard apprenticeship began in the 60s of the nineteenth century, a hard apprenticeship that led eventually to unravel the "mysteries Riemannian "mentioned Cremona and at the end to make intuitive and evident to a small group of" initiated "a whole series of" facts "abstract geometric who will need algebraic and sophisticated analytical techniques to be able to really
Original languageItalian
Publication statusPublished - 2009

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