53rd Cascade Topology Seminar

Topology is a pervasive subject, with topological ideas arising in almost all branches of mathematics. As a discipline, topology is divided into several branches, including algebraic topology, geometric topology and differential topology. These branches can be very diffuse, with very little in common between different subdisciplines, and indeed each branch can be very widely represented amongst many areas of specialization. The goal of the Cascade Topology Seminar is to establish some common ground between topologists in the Cascade region, including the Pacific Northwest United States and Western Canada. At this particular meeting, there was a focus on algebraic topology. Within this subdiscipline, particular attention was given to the relationship between algebraic topology and K-theory and algebraic geometry, applied topology, and homotopy theory. K-theory is an invariant of categories with symmetric monoidal products, and it is in some sense a universal invariant because it is precisely the closest additive approximation of a symmetric monoidal category. K-theory has several incarnations: algebraic and topological K-theory may be the most frequently mentioned types, but K-theory can be widely applied. Indeed, many early applications of K-theory involved connections between topology and number theory, and in recent years more and diverse applications have been developed, especially to motivic cohomology and algebraic geometry. Applied Topology can refer to a number of emerging relations between topology and statistics and computer science. One of the most thriving areas of applied topology involves the relationship between topology and the study of data. Topology is the study of shape, and in particular topology is well-suited to study the shape of data. Algebraic invariants of topological spaces can be used to distinguish interesting features of data sets, and this has been a particularly fruitful method of data analysis because topology is well-suited to studying highdimensional spaces, and algebraic tools are insensitive to small perturbations of data. Homotopy Theory is the study of topological spaces up to continuous deformations or homotopies. A central question in this field has been the discovery and computation of homotopy invariants which can determine whether two topological spaces are distinct or not.