Seminar
Higher Structures in Topology and Algebra, Manuel L. Rivera, Purdue University, Fri, 11 October, 2019, 10:00 a.m. NCN II C-204
Abstract
Abstract: It is often useful in different mathematical contexts to relax the notions of “equality” and of “isomorphism” between objects to a weaker notion of equivalence. A classical example is the notion of homotopy between continuous maps of topological spaces and the notion of homotopy equivalence between topological spaces. We also encounter examples of this phenomenon in algebra when equations describing certain identities (like associativity, commutativity, the Jacobi identity in Lie theory, etc.) of operations do not hold strictly (up to equality) but there is a coherent way of describing their failure which allows us to work with them as if the identities were satisfied (i.e. they are true when equality is replaced by a weaker notion). The coherent information encoding the failure of an identity or property often has some interesting and useful combinatorial or algebraic structure, these are called “higher structures”. In this talk I will discuss examples of higher structures in topology and algebra and I will explain why they are important and useful in different fields of mathematics. Time permitting, I will outline an abstract framework, called higher category theory, for handling and relating higher structures coming from different contexts, which is nowadays used all over modern mathematics to describe similar structures and constructions that follow the same pattern.