Webster's Ninth Collegiate Dictionary defines an amalgam as a mixture of mercury and another metal, or more generally as a mixture of two different elements. Dental fillings, for example, are made of amalgams.

Mathematically, an amalgam is obtained when we take two structures (two topological spaces, two graphs, two rings), and mix them by identifying some subparts of them; The challenge then is to find a bigger structure where this mixture can live.

For example, if we have two groups G and K, and we have a subgroup H which is common to both, we want to think of G and K as being "glued together along H;" this is an amalgam of the two groups. It is not a group, because there is no way to multiply an element of G which is not in H by an element of K which is not in H. What we want is to find some larger group M, which contains G and K as subgroups in such a way that their intersection is still H. Similar problems exist if we replace "group" and "subgroup" with "topological space" and "topological subspace"; or with "graph" and "subgraph"; or with "manifold" and "submanifold", etc.

This simple question leads very easily to some very powerful mathematics, and of course to even more questions, many of which we cannot yet answer. I will give a tour of amalgams of groups, assuming nothing more than a basic knowledge of them. As opposed to getting to know what a dental amalgam is, it will not hurt at all.