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Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan-Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

Permutation groups.

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This book presents a complete classification of the transitive permutation representations of rank at most five of the sporadic simple groups and their automorphism groups, together with a comprehensive study of the vertex-transitive graphs associated with these representations. Included is a list of all vertex-transitive, distance-regular graphs on which a sporadic almost simple group acts with rank at most five. In this list are some new, interesting distance-regular graphs of diameter two, which are not distance-transitive. For most of the representations a presentation of the sporadic group is given, with words in the given generators which generate a point stabiliser: this gives readers sufficient information to reconstruct and study the representations and graphs. Practical computational techniques appropriate for analysing finite vertex-transitive graphs are described carefully, making the book an excellent starting point for learning about groups and the graphs on which they act.

Finite simple groups. --- Representations of groups. --- Graph theory.

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Discrete mathematics --- Generating functions. --- Combinatorial analysis. --- Linear algebraic groups. --- Fonctions génératrices --- Analyse combinatoire --- Groupes algébriques linéaires --- Combinatorial analysis --- Generating functions --- Linear algebraic groups --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Functions, Generating --- Combinatorics --- Algebra --- Mathematical analysis --- Fonctions génératrices. --- Analyse combinatoire. --- Groupes algébriques linéaires.

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MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: - Higher Structures in Geometry and Physics - Winter of Disconnectedness - Approximation and Optimisation - Refining C*-Algebraic Invariants for Dynamics using KK-theory - Interactions between Topological Recursion, Modularity, Quantum Invariants and Low- dimensional Topology The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.

Mathematics --- Research. --- Mathematics. --- Category theory (Mathematics). --- Homological algebra. --- Group theory. --- K-theory. --- Mathematical optimization. --- Topology. --- Category Theory, Homological Algebra. --- Group Theory and Generalizations. --- Optimization. --- K-Theory. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Algebraic topology --- Homology theory --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Homological algebra --- Algebra, Abstract --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Math --- Science --- Mathematical research --- Algebra.

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The first World Meeting for Women in Mathematics - (WM)² - was a satellite event of the International Congress of Mathematicians (ICM) 2018 in Rio de Janeiro. With a focus on Latin America, the first (WM)² brought together mathematicians from all over the world to celebrate women mathematicians, and also to reflect on gender issues in mathematics, challenges, initiatives, and perspectives for the future. Its activities were complemented by a panel discussion organized by the Committee for Women in Mathematics (CWM) of the International Mathematical Union (IMU) inside the ICM 2018 entitled "The gender gap in mathematical and natural sciences from a historical perspective”. This historical proceedings book, organized by CWM in coordination with the Association for Women in Mathematics, records the first (WM)² and the CWM panel discussion at ICM 2018. The first part of the volume includes a report of activities with pictures of the first (WM)² and a tribute to Maryam Mirzakhani, the first woman to be awarded the Fields medal. It also comprises survey research papers from invited lecturers, which provide panoramic views of different fields in pure and applied mathematics. The second part of the book contains articles from the panelists of the CWM panel discussion, which consider the historical context of the gender gap in mathematics. It includes an analysis of women lecturers in the ICM since its inception. This book is dedicated to the memory of Maryam Mirzakhani.

Mathematics. --- Social sciences. --- History. --- Mathematics in the Humanities and Social Sciences. --- History of Mathematical Sciences. --- Women in mathematics --- Women mathematicians --- Women as mathematicians --- Mathematicians --- Mathematics --- Math --- Science --- Annals --- Auxiliary sciences of history --- Behavioral sciences --- Human sciences --- Sciences, Social --- Social science --- Social studies --- Civilization