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4 produkter
4 produkter
Häftad, Engelska, 2020
706 kr
Skickas inom 10-15 vardagar
This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers.Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds.This book introduces L2-Betti numbers of groups at an elementary level and thendevelops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school “Random and arithmetic structures in topology” and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.
Del 11 - Surveys and Tutorials in the Applied Mathematical Sciences
Exploring Formalisation
A Primer in Human-Readable Mathematics in Lean 3 with Examples from Simplicial Topology
Häftad, Engelska, 2022
598 kr
Skickas inom 10-15 vardagar
This primer on mathematics formalisation provides a rapid, hands-on introduction to proof verification in Lean.After a quick introduction to Lean, the basic techniques of human-readable formalisation are introduced, illustrated by simple examples on maps, induction and real numbers. Subsequently, typical design options are discussed and brought to life through worked examples in the setting of simplicial complexes (a higher-dimensional generalisation of graph theory). Finally, the book demonstrates how current research in algebraic and geometric topology can be formalised by means of suitable abstraction layers.Informed by the author's recent teaching and research experience, this book allows students and researchers to quickly get started with formalising and checking their proofs. The core material of the book is accessible to mathematics students with basic programming skills. For the final chapter, familiarity with elementarycategory theory and algebraic topology is recommended.
Häftad, Engelska, 2018
921 kr
Skickas inom 10-15 vardagar
Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology.Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability.This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.
Häftad, Tyska, 2019
359 kr
Skickas inom 10-15 vardagar
Wie lassen sich Spiele und Puzzles mathematisch analysieren?Wie kann man unendliche Strukturen zutreffend beschreiben?Wie kann man Nachrichten gut verschlüsseln?Achtzehn ausgewählte mathematische Themen mit Aufgaben und Lösungen laden zum Entdecken und Knobeln ein und bieten Einblicke in die faszinierende Welt der Mathematik – von A wie Aussagenlogik bis Z wie Zahlentheorie. Die Themen wecken so die Neugierde für Mathematik und fördern die Begeisterung von Schülerinnen und Schülern ab Klasse 7. Anleitungen zum mathematischen Problemlösen und Beweisen erleichtern dabei den Einstieg.Das vorliegende Buch enthält das überarbeitete und ergänzte Material des Schülerzirkels Mathematik der Fakultät für Mathematik an der Universität Regensburg aus den Schuljahren 2012/13 bis 2014/15, das für die zweite Auflage um ausgewählte Themen der Schuljahre 2015/16 bis 2017/18 ergänzt wurde. Stimme zur ersten Auflage„Es ist erfreulich, dass die Aufgaben und Lösungen aus dem Schülerzirkel Mathematik der Universität Regensburg einem breiten Leserkreis zur Verfügung gestellt werden. Die Verbindung von pfiffigen Knobelaufgaben als Einstieg in ein Thema mit der Vermittlung des mathematischen Hintergrundwissens wird sicher vielen Schülerinnen und Schülern den Weg in die Welt der Mathematik ebnen."Hanns-Heinrich Langmann, Projektleiter Bundesweite Mathematik-Wettbewerbe bei Bildung & Begabung