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The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory. ... Read more

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Deep but important
The Riemann-Roch theorem has come a long way since its origins in the work of Bernhard Riemann 154 years ago. Riemann was attempting to establish the existence of complex functions on multiply-connected surfaces with no boundary. A surface that is (2p + 1)-connected for a positive integer p can be represented as a 4p-sided polygon after making 2p cuts. Riemann showed using the Dirichlet principle that there are p linearly-independent functions defined inside this polygon that are everywhere holomorphic. The differentials of these functions are also holomorphic integrands. By specifying D points at which a function can have simple poles, and by constraining the functions to change only by a constant amount at the cuts, then Riemann showed that one can obtain functions with only simple poles and constant "jumps" by taking a sum of p linearly independent functions with no poles with functions of the form 1/z at one of the specified points and a constant term. Non-constant meromorphic functions thus exist when p + D + 1 - 2p >= 2, or D > p, which is now called the Riemann inequality. Hence there is a linear space of complex functions of dimension >= D + 1 - p, which contains non-constant functions when D + 1 - p > 1. Roch was Riemann's student and interpreted the quantity D + 1 - p as the dimension of the space of holomorphic integrands. If a function has D simple poles, and if there are Q linearly independent integrands that vanish at these poles, then the function depends on D - p + q + 1 arbitrary constants.

The work of Riemann and Roch is readily seen to be related to the genus of the surface, if viewed in the light of the polygon of 4p sides. The modern view of the Riemann-Roch theorem in fact is naturally viewed as a generalization of a formula for the Euler characteristic, the latter of which involves the genus of a Riemann surface. The "classical" Riemann-Roch theorem is stated in terms of divisors on a Riemann surface X of genus g and reads as r(-D) - i(D) = d(D) - g + 1, where D is a fixed divisor on the surface, r(-D) is the dimension of meromorphic functions of divisors >= -D on X, and i(D) is the dimension of the space of meromorphic 1-forms of divisors >= D on X. Many other statements have been given, one being in terms of holomorphic bundles defined by D over X, where one computes the Euler characteristic of the sheaf of germs of holomorphic sections of the bundle. Another is in the context of holomorphic bundles over nonsingular complex projective varieties, where the Euler characteristic of the sheaf of holomorphic sections of the bundle is given in terms of a formula involving the first Chern class of the variety. The Euler characteristic has of course also been computed in terms of the index of Dirac operators, and so it is not surprising to find that the Riemann-Roch theorem has an analytical formulation also.

This book proves a Riemann-Roch theorem for arithmetic varieties, and the author does so via the formalism of Dirac operators and consequently that of heat kernels. In the first lecture the reader will see the "classical" Riemann-Roch theorem in an even more general context then that mentioned above: that of smooth morphisms of regular schemes. Using familiar constructions involving the K-groups and Chow groups to make the calculations more manageable, the author proves the Riemann-Roch theorem for regular schemes by reducing it to the case for projective bundles. Flag schemes are used in the proof, and the strategy used here repeats itself in the proof of the arithmetic Riemann-Roch theorem. Throughout the book the author outlines the necessary background for the eventual proof of the arithmetic Riemann-Roch theorem. This includes a discussion of how to define Chern classes for arithmetic vector bundles, as well as the background in analysis via a discussion of heat kernels and Laplacians on Riemannian manifolds. The detailed discussion of analysis was done, according to the author, so as to avoid the methods of stochastic integration, which he felt might not be familiar to the average reader with a background in algebraic geometry. ... Read more

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A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms. A valuable source of reference for researchers and graduate students interested in algebraic geometry, Shimura varieties or diophantine geometry. ... Read more

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Kapitel: Jürgen Habermas, Christiane Nüsslein-Volhard, Gerd Faltings, Gottfried-Wilhelm-Leibniz-Preis, Berthold Beitz, Heinrich Meier, Gerhard Ertl, Hermann Parzinger, Manfred G. Schmidt, Wolfgang A. Herrmann, Géza Alföldy, Gabriele Brandstetter, Svante Pääbo, Theodor Hänsch, Ulrich Herbert, Jan Veizer, Lothar Gall, Christoph Markschies, Matthias Hentze, Jürgen Mittelstraß, Joachim Milberg, Knut Borchardt, Martin Lohse, Franz-Ulrich Hartl, Herbert W. Roesky, Bert Hölldobler, Stefan Hell, Hubert Wolf, Ferenc Krausz, Peter Schäfer, Detlef Weigel, Petra Schwille, Wolfgang Prinz, Bert Sakmann, Heinrich Detering, Michael Stolleis, Axel Ockenfels, Alois Fürstner, Johannes Buchmann, Claus-Wilhelm Canaris, Axel Haverich, Friedrich Wilhelm Graf, Reinhard Zimmermann, Ute Frevert, Jürgen Kocka, Peter Gruss, Ernst O. Göbel, Armin Falk, Walter Haug, Jürgen Jost, Thomas Jentsch, Hartmut Michel, Antje Boetius, Martin Warnke, Christian Von Bar, Stefan Müller, Günter M. Ziegler, Gyburg Uhlmann, Sigrid Peyerimhoff, Onno Oncken, Dieter Langewiesche, Jürgen Warnatz, Albrecht Koschorke, Gerhard Huisken, Martin Grötschel, Karl Stetter, Barbara Stollberg-Rilinger, Wolfgang Schnick, Jürgen Osterhammel, Angela D. Friederici, Günther Hasinger, Helmut Willke, Matthias Beller, Dominik Perler, Volker Mosbrugger, Ferdi Schüth, Michael Famulok, Konrad Kleinknecht, Glenn W. Most, Irmela Hijiya-Kirschnereit, Stefanie Dimmeler, Bernd Giese, Joachim Küpper, Hans-Joachim Freund, Winfried Schulze, Jürgen Mlynek, Rudolf Thauer, Julius Wess, Herbert Gleiter, Gertrude Lübbe-Wolff, Thomas Carell, Rudolf G. Wagner, Klaus Dieter Pfeffer, Frank Neese, Christian Jung, Günter Hotz, George M. Sheldrick, Holger Fleischer, Randolf Menzel, Thomas Mussweiler, Erwin Neher, Siegfried Bethke, Jochen Feldmann, Karl Joachim Ebeling, Wolfgang Lück, Kurt Mehlhorn, Thomas Hengartner, Ulrich Konrad, Helmut Schwarz, Matthias Kleiner, Christian Haass, Reinhard Genzel, Eduard Arzt, Elmar We...http://booksllc.net/?l=de ... Read more

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Der Erwerb des Buches enthält gleichzeitig die kostenlose Mitgliedschaft im Buchklub des Verlags zum Ausprobieren - dort können Sie von über einer Million Bücher ohne weitere Kosten auswählen. Das Buch besteht aus Wikipedia-Artikeln: Ilya Prigogine, Christiane Nüsslein-Volhard, Kurt Alder, Gerd Faltings, Benedikt XVI., Manfred Trojahn, Gerhard Ertl, Günther Jakobs, Klaus Stern, Heinz Bittel, Reinhart Koselleck, Ernst-Wolfgang Böckenförde, Heinrich Dörrie, Werner Flume, Eugen Ewig, Reinhard Selten, Ernst Otto Fischer, Udo Di Fabio, Alfred Pühler, Paul Mikat, Gottfried Schatz, Helmut Ringsdorf, Karl Dietrich Bracher, Charles Weissmann, Benno Parthier, Martin Lohse, Werner Forßmann, Franz-Ulrich Hartl, Karl Ziegler, Claus Vogel, Klaus Hildebrand, Günther Jachmann, Otmar Schober, Bernhard Kötting, Gerhard Domagk, Herbert Walther, Axel Ockenfels, Arnold Angenendt, Josef Isensee, Peter Schuster, Wolfgang Paul, Theo Kölzer, Rolf Knütel, Hartmut Michel, Heinz Hübner, Wolfgang Kluxen, Enno Poppe, Bruno Schüller, Klaus Tipke, Gerhard Kegel, Theodor Schieffer, Wolfhart Westendorf, Gustav Adolf Lehmann, Ulrich Wannagat, Reinhold Merkelbach, Ekrem Akurgal, Ewald Wicke, Klaus Zechiel-Eckes, Michael Famulok, Günther Wilke, Martin Schermaier, Wolfgang Mathis, Bernhard Großfeld, Konrad Repgen, Rudolf Schieffer, Stefan Hildebrandt, Jörg Becker, Dieter Nörr, Christel Meier-Staubach, Hartmut Krebs, Otto Zwierlein, Herbert Waldmann, Hanns Hatt, Erwin Neher, Werner Schreyer, Hans-Jürgen Engell, Anthony R. Birley, Heinz Mehlhorn, Elmar Weiler, Manfred T. Reetz, Wolfgang Marquardt, Nikolaus Himmelmann, Georg Schöllgen, Hugh Lloyd-Jones, Harald von Petrikovits, Rudolf Kassel, Bettina Heintz, Reinhart Poprawe, Felix Otto, Ilme Schlichting, Klaus Mezger, Wolfgang Dieter Lebek, Gerhard Wegner, Gerrit Walther, Herbert Wiedemann, Manfred J. M. Neumann, Friedrich Hensel, Hélène Esnault, Egon Macher, Fritz Ossenbühl,. Online finden Sie die kostenlose Aktuali...http://booksllc.net/?l=de&id=832524 ... Read more

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Der Erwerb des Buches enthält gleichzeitig die kostenlose Mitgliedschaft im Buchklub des Verlags zum Ausprobieren - dort können Sie von über einer Million Bücher ohne weitere Kosten auswählen. Das Buch besteht aus Wikipedia-Artikeln: Gerd Faltings, Karl-Heinz Petzinka, Johannes Harder, Lew Sinowjewitsch Kopelew, Peter Dienel, Lambert T. Koch, Winfried Matthes, Bazon Brock, Dieter Vieweger, Harald Fritzsch, Gregor Schiemann, Peter Ulrich, Thomas Söding, Harald Mante, Karl Albert, Siegfried Maser, Wolf Erlbruch, Frank R. Werner, Karl-Wilhelm Weeber, Willy Fleckhaus, Friedrich Hofmann, Matthias Kreck, Gerd Aretz, Lutz Becker, Joachim Küpper, Wolfgang Mathis, Bernhard Walz, Uwe Loesch, Michael Badura, Gert Pinkernell, Christel Meier-Staubach, Gerda Breuer, Eckhard Freise, Heinz Rölleke, Michael Tausch, Hans Bode, Erhard Scholz, Udo Margedant, Michael Rapoport, Gesine Weinmiller, Doris Bühler-Niederberger, Claus Bury, Karl-Hans Laermann, Tönis Käo, Manfred Hassebrauck, Eberhard Schmidt, Paul J. J. Welfens, Gerrit Walther, Manfred Brusten, Günter Harder, Wolf F. Fischer-Winkelmann, Joachim Michael Marzinkowski, Dietrich Weber, Michael Scheffel, Uwe Schneidewind, Helmut Goerlich, Elisabeth Stein, Tilman Struve, Ludgera Vogt, Stefan Freund, Matias Martinez, Hans Jochen Boecker, Renate Wald, Winfried Bausback, Wolfgang Orth, Martin Endreß, Mihai Nadin, Martin Topel, Michael Nelles, Günther Wachtler, Carsten Zündorf, Wolfram Boucsein, Ernst Mohr,. Online finden Sie die kostenlose Aktualisierung der Bücher. Nicht dargestellt. Auszug: Karl-Heinz Petzinka (* 7. Januar 1956 in Bocholt) ist ein deutscher Architekt und Professor an der Kunstakademie Düsseldorf, Klasse Baukunst. Karl-Heinz Petzinka studierte ab 1976 an der RWTH Aachen Architektur und schloss das Studium 1982 mit dem akademischen Grad „Diplom-Ingenieur" ab. Nach seinen ersten Berufsjahren 1982-83 als freischaffender Architekt im Büro O. M. Ungers in Köln war er von 1983-1985 al...http://booksllc.net/?l=de&id=280143 ... Read more

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Chapters: University of Wuppertal Faculty, Walter Thiel, Gerd Faltings, Karl Albert. Source: Wikipedia. Pages: 18. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The University of Wuppertal (Universität Wuppertal) is a German scientific institution, located in Wuppertal, North Rhine-Westphalia. The university, the full German name of which is Bergische Universität Wuppertal (BUW), was formed in 1972 and is located in the city of Wuppertal, within the state of North Rhine-Westphalia, Germany. In 2007/2008 it had more than 14,000 students in a wide range of subjects with many interdisciplinary linkages in 7 faculties: The main building of the BU Wuppertal is located in the suburb of Elberfeld on Grifflenberg and is a massive, cut honeycomb concrete purpose. The university now has 3 campuses: Since 2004, the University of Wuppertal has the second supercomputer at a German university. ALiCEnext, the supercomputer, is designed as a cluster and consists of 512 so-called Blades. ALiCEnext used in the field of elementary particle physics, applied computer science, astro-particle physics and experimental high energy physics. The university employs 298 professors (as of 2006). Notable people who have taught in Wuppertal are: ...More: http://booksllc.net/?id=12696005 ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 852 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 619 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Der Erwerb des Buches enthält gleichzeitig die kostenlose Mitgliedschaft im Buchklub des Verlags zum Ausprobieren - dort können Sie von über einer Million Bücher ohne weitere Kosten auswählen. Das Buch besteht aus Wikipedia-Artikeln: Gerd Faltings, Alexander Grothendieck, Grigori Jakowlewitsch Perelman, Jean-Pierre Serre, Laurent Schwartz, Stephen Smale, Heisuke Hironaka, Sergei Petrowitsch Nowikow, Lars Valerian Ahlfors, Atle Selberg, Michael Francis Atiyah, John Willard Milnor, Edward Witten, René Thom, Kodaira Kunihiko, David Bryant Mumford, Maxim Lwowitsch Konzewitsch, Shing-Tung Yau, John Griggs Thompson, Terence Tao, Laurent Lafforgue, Simon Donaldson, Enrico Bombieri, Richard Borcherds, Curtis McMullen, Jesse Douglas, Lars Hörmander, Pierre Deligne, Jean Bourgain, Wladimir Wojewodski, Klaus Friedrich Roth, Paul Cohen, Efim Zelmanov, Alain Connes, Jean-Christophe Yoccoz, Michael Freedman, Pierre-Louis Lions, William Thurston, William Timothy Gowers, Charles Fefferman, Vladimir Drinfeld, Grigori Alexandrowitsch Margulis, Shigefumi Mori, Daniel Gray Quillen, Alan Baker, Vaughan F. R. Jones, Andrei Jurjewitsch Okunkow, Wendelin Werner,. Online finden Sie die kostenlose Aktualisierung der Bücher. Nicht dargestellt. Auszug: Alexander Grothendieck (* 28. März 1928 in Berlin) ist ein deutsch-französischer Mathematiker. Er ist Begründer einer eigenen Schule der algebraischen Geometrie, deren Entwicklung er in den 1960er Jahren maßgeblich beeinflusste. 1966 wurde ihm der „Nobelpreis für Mathematik", die Fields-Medaille, verliehen. Nachdem er sich, beeinflusst durch politische Ideen des Pariser Mai 1968, bereits um 1970 weitgehend von seiner zentralen Position im mathematischen Leben von Paris zurückgezogen hatte und z. B. den unter dem Vietnamkrieg leidenden Mathematikern Nordvietnams durch persönlichen Einsatz zu helfen versuchte, verschwand er 1991 völlig aus der Öffentlichkeit. Sein aktueller Aufenthaltsort in den Pyrenäen ist...http://booksllc.net/?l=de&id=451910 ... Read more