Associativity in topology

Familiar operations in arithmetic, such as addition and multiplication of numbers, are associative. This means that the answers we obtain don’t depend on the order in which we carry out the operations. For example, (2+3)4 = 2(3+4), and so we do not normally bother writing the brackets.

My work involves the interaction of algebraic conditions like associativity with topology, the study of shapes up to continuous deformations. In topological settings, it turns out that a weaker version of associativity is more natural. This leads to very rich and interesting structures which have become important in many different areas of mathematics, including algebra, geometry and mathematical physics. Similar topological games can be played with other familiar algebraic conditions.

Along the way, I’ll talk about a famous sequence of numbers known as the Catalan numbers. They play a key role, because the Catalan numbers count how many different bracketings there are.

This talk is part of the The Archimedeans (CU Mathematical Society) series.