This volume contains nine survey articles by the invited speakers of the 30th British Combinatorial Conference, held at Queen Mary University of London in July 2024. Each article provides an overview of recent developments in a current hot research topic in combinatorics. Topics covered include: Latin squares, Erdős covering systems, finite field models, sublinear expanders, cluster expansion, the slice rank polynomial method, and oriented trees and paths in digraphs. The authors are among the world's foremost researchers on their respective topics but their surveys are accessible to nonspecialist readers: they are written clearly with little prior knowledge assumed and with pointers to the wider literature. Taken together these surveys give a snapshot of the research frontier in contemporary combinatorics, helping researchers and graduate students in mathematics and theoretical computer science to keep abreast of the latest developments in the field.

Many areas of current research activity in combinatorics and its applications, including graph theory, designs and probabilistic graphs, are surveyed in lectures presented at the 12th British Combinatorial Conference.

A mixture of survey and research articles by leading experts that will be of interest to specialists in permutation patterns and other researchers in combinatorics and related fields. In addition, the volume provides plenty of material accessible to advanced undergraduates and is a suitable reference for projects and dissertations.

Part two of a two-volume collection exploring recent developments in number theory related to automorphic forms and Galois representations.

A 2010 collection of survey articles by leading experts covering fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. This is a valuable reference for experts and a useful introduction for graduate students entering the field.

A wide-ranging 2010 survey of new and important topics in p-adic analysis for researchers and graduate students.

An extended tour through a selection of the most important trends in modern geometric group theory.

A detailed account of main results in the theory of differential tensor algebras.