This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis. Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs. The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than “just” the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account. The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.

This is the first authored book to be dedicated to the new field of directed algebraic topology that arose in the 1990s, in homotopy theory and in the theory of concurrent processes. Its general aim can be stated as 'modelling non-reversible phenomena' and its domain should be distinguished from that of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. Its homotopical tools (corresponding in the classical case to ordinary homotopies, fundamental group and fundamental groupoid) should be similarly 'non-reversible': directed homotopies, fundamental monoid and fundamental category. Homotopy constructions occur here in a directed version, which gives rise to new 'shapes', like directed cones and directed spheres. Applications will deal with domains where privileged directions appear, including rewrite systems, traffic networks and biological systems. The most developed examples can be found in the area of concurrency.

This second volume of Research in Computational Topology is a celebration and promotion of research by women in applied and computational topology, containing the proceedings of the second workshop for Women in Computational Topology (WinCompTop) as well as papers solicited from the broader WinCompTop community. The multidisciplinary and international WinCompTop workshop provided an exciting and unique opportunity for women in diverse locations and research specializations to interact extensively and collectively contribute to new and active research directions in the field. The prestigious senior researchers that signed on to head projects at the workshop are global leaders in the discipline, and two of them were authors on some of the first papers in the field. Some of the featured topics include topological data analysis of power law structure in neural data; a nerve theorem for directional graph covers; topological or homotopical invariants for directed graphs encoding connections among a network of neurons; and the issue of approximation of objects by digital grids, including precise relations between the persistent homology of dual cubical complexes.

This volume highlights the mathematical research presented at the 2019 Association for Women in Mathematics (AWM) Research Symposium held at Rice University, April 6-7, 2019. The symposium showcased research from women across the mathematical sciences working in academia, government, and industry, as well as featured women across the career spectrum: undergraduates, graduate students, postdocs, and professionals. The book is divided into eight parts, opening with a plenary talk and followed by a combination of research paper contributions and survey papers in the different areas of mathematics represented at the symposium: algebraic combinatorics and graph theory algebraic biology commutative algebra analysis, probability, and PDEs topology applied mathematics mathematics education

This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiří Matoušek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition – something Jirka would have approved of. The original research articles, surveys and expository articles, written by leading experts in their respective fields, map Jiří Matoušek’s numerous areas of mathematical interest.

Sir Tony Hoare has had an enormous influence on computer science, from the Quicksort algorithm to the science of software development, concurrency and program verification. His contributions have been widely recognised: He was awarded the ACM’s Turing Award in 1980, the Kyoto Prize from the Inamori Foundation in 2000, and was knighted for “services to education and computer science” by Queen Elizabeth II of England in 2000. This book presents the essence of his various works—the quest for effective abstractions—both in his own words as well as chapters written by leading experts in the field, including many of his research collaborators. In addition, this volume contains biographical material, his Turing award lecture, the transcript of an interview and some of his seminal papers. Hoare’s foundational paper “An Axiomatic Basis for Computer Programming”, presented his approach, commonly known as Hoare Logic, for proving the correctness of programs by using logical assertions. Hoare Logic and subsequent developments have formed the basis of a wide variety of software verification efforts. Hoare was instrumental in proposing the Verified Software Initiative, a cooperative international project directed at the scientific challenges of large-scale software verification, encompassing theories, tools and experiments. Tony Hoare’s contributions to the theory and practice of concurrent software systems are equally impressive. The process algebra called Communicating Sequential Processes (CSP) has been one of the fundamental paradigms, both as a mathematical theory to reason about concurrent computation as well as the basis for the programming language occam. CSP served as a framework for exploring several ideas in denotational semantics such as powerdomains, as well as notions of abstraction and refinement. It is the basis for a series of industrial-strength tools which have been employed in a wide range of applications. This book also presents Hoare’s work in the last few decades. These works include a rigorous approach to specifications in software engineering practice, including procedural and data abstractions, data refinement, and a modular theory of designs. More recently, he has worked with collaborators to develop Unifying Theories of Programming (UTP). Their goal is to identify the common algebraic theories that lie at the core of sequential, concurrent, reactive and cyber-physical computations.