Stefan Müller-Stach - Böcker

The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel-Jacobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson. The immense recent progress in algebraic cycles in the late 1990s, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well. This text offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods.Topics include a discussion of the arithmetic Abel-Jacobi mapping; higher Abel-Jacobi regulator maps; polylogarithms and L-series; candidate Bloch-Beilinson filtrations; applications of Chern-Simons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection; motivic cohomology; Chow groups of singular varieties; and recent progress on the Hodge and Tate conjectures for Abelian varieties.

The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel-Jacobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson. The immense recent progress in algebraic cycles in the late 1990s, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well. This text offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods.Topics include a discussion of the arithmetic Abel-Jacobi mapping; higher Abel-Jacobi regulator maps; polylogarithms and L-series; candidate Bloch-Beilinson filtrations; applications of Chern-Simons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection; motivic cohomology; Chow groups of singular varieties; and recent progress on the Hodge and Tate conjectures for Abelian varieties.

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori’s abelian category of mixed motives.

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori’s abelian category of mixed motives.

Motiviert durch aktuelle Entwicklungen in der abhängigen Typentheorie und bei Unendlichkategorien präsentiert dieses Buch die Ideengeschichte der Begriffe Wahrheit, Beweis, Gleichheit und Äquivalenz. Neben ausgewählten Ideen von Platon, Aristoteles, Leibniz, Kant, Frege und anderen werden Resultate von Gödel und Tarski über Unvollständigkeit, Unentscheidbarkeit und Wahrheit in deduktiven Systemen und ihren semantischen Modellen vorgestellt. Der Hauptgegenstand dieses Textes ist die abhängige Typentheorie und neuere Entwicklungen in der Homotopy Type Theory. Diese Theorien beinhalten Identitätstypen, die neue Möglichkeiten für Gleichheit, Symmetrie, Äquivalenz und Isomorphie auf konzeptuelle Weise eröffnen. Die Interaktion von Typentheorie und Unendlichkategorien ist ein neues Paradigma für eine strukturelle Sichtweise auf die Mathematik. Sie fördert auch den neuen Trend zur Formalisierung von Mathematik in Form von Beweisassistenten.

Die beiden Bücher „Was sind und was sollen die Zahlen?“ (1888) und „Stetigkeit und Irrationale Zahlen“ (1872) sind Dedekinds Beiträge zu den Grundlagen der Mathematik;

Inspired by recent developments in dependent type theory and infinity categories, this book presents a history of ideas around the topics of truth, proof, equality and equivalence. The main focus of this textbook is on dependent type theory and its recent variant homotopy type theory.

The extensive commentary offers a fascinating insight into the life and work of Dedekind's pioneering work and relates the latter to great contemporaries such as Cantor, Dirichlet, Frege, Hilbert, Kronecker, and Riemann.Researchers and students alike will find this work a valuable reference in the history of mathematics.