Monday, January 9

CL 0171, 4:30-5:30

A fundamental problem in mathematics consists of determining whether two given mathematical structures are `the same'. For example, knot theorists are interested in knowing when knots are the same, while algebraists like to know when groups are the same. But what exactly do mathematicians mean when they say that two gadgets are the same? Often, they mean "sufficiently the same for our purposes," and that purpose naturally differs from field to field. Higher-dimensional algebra, which enables us to refine our notion of 'sameness', is the study of generalizations of algebraic concepts obtained by developing category-theoretic analogs of set-theoretic concepts. We will see how higher-dimensional algebra can be used to explore mathematical interpretations of being `the same' by carefully examining the concept of equality and comparing it to weaker notions of sameness.