The most overloaded word in math

Last Wednesday, the conversation in my office veered towards the words we hated the most in math. Not surprisingly, the list included the usual suspects like normal, simple, and regular. It's probably the same reason that these words also make it to the top five of this MathOverflow post. These words are overloaded to the point of meaning something different to mathematicians working in different areas of math. On the other hand, we all agreed that some overloading of words was actually fairly useful: for instance, it makes sense to call a normal covering space normal since it actually corresponds to a normal subgroup of the fundamental group. That means calling a cover normal and calling a subgroup normal isn't really a bad thing, since it shows that those two notions are related.

We thought we'd do something similar for all the possible meanings of the word normal: we'd define an equivalence relation between two different meanings if there is some result, deep or otherwise, that links the two notions. Then the number of equivalence classes we get would be a much better metric of the overloaded-ness of the word normal.

Here's a list of the meanings of normal (taken from Wikipedia), along with some additions.

• Normal subgroup: A subgroup which is invariant under conjugation action.
• Normal cover: A covering space whose deck transformation group acts transitively.
• Normal field extension: A field extension such that every irreducible polynomial in the base field splits into linear factor, or is irreducible.

These three uses of normal are really the same, since they all talk about an associated subgroup being a normal subgroup. In the case of the normal cover, the fundamental group of the cover is a normal subgroup of the fundamental group of the base space. When it comes to field extensions, consider the following field extension: \(E \subset F \subset G\), where \(G\) is Galois over \(E\). In this case, \(F\) is a normal extension iff it corresponds to a normal subgroup of \(\mathrm{Gal}(E/G)\).

A few different meanings of the word normal(ize) show up often in algebraic geometry.

• Normal domain: An integral domain which is integrally closed in its field of fractions.
• Normal varieties: A variety \(X\) such that any finite birational map from any variety \(Y\) to \(X\) is an isomorphism.
• Noether normalization: The Noether normalization lemma states that for any finitely generated \(k\)-algebra \(A\), there exist \(\{y_1, \ldots, y_d\} \in A\) such that \(A\) is a finitely generated module over \(k[y_1, \ldots, y_d]\).

These seemingly different notions actually are somewhat equivalent. As it turns out, a variety is normal if the local ring at every point is integrally closed. And while normal varieties are varieties which have maps from "nice" varieties, a geometric interpretation of Noether normalization is that every \(d\)-dimensional affine variety is a ramified cover of \(\mathbb{A}^d\), which is a "nice" variety.

Another meaning of the word normal comes from the geometric notion of being perpendicular. This gives us a lot of different meanings of the word normal which we can collapse to one equivalence class.

• Normal bundle: The normal bundle of an embedded submanifold is the vector bundle such that the fibre over each point consists of vectors perpendicular to the tangent space.
• Normal coordinates: Given a vector bundle with an affine connection, the normal coordinates around a point are coordinates such that the Christoffel symbols of the connection vanish at the point.
• (Ortho)normal basis: A basis of an inner product space such that each vector is of norm \(1\) and every pair is perpendicular.
• Normal operator: An operator which commutes with its Hermitian conjugate.
• Normal modes: (Taken from wikipedia) A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation.

Normal bundle literally comes from the original meaning of the word normal in the sense of being perpendicular. Normal coordinates also come from the same source: in the case of the Levi-Civita connection, one gets a set of normal coordinates by applying the exponential map to an orthonormal basis of the tangent space.

The reason why a normal operator is called a normal operator is that we know from the spectral theorem that its eigenvalues form an orthonormal basis. That is also the source of normal modes. The "normal" in normal modes comes from the fact that the vibrations are the eigenvectors of a certain differential operator, which happens to be self-adjoint, hence normal. That means all the normal modes literally form an orthonormal basis of solutions to the associated PDE.

The above words were some of the different cases of the usage of the word normal that we were able to collapse. And below are the ones we couldn't collapse to anything else, so they sit all by themselves (for now) in their own equivalence class.

• Normal family: A pre-compact family of holomorphic functions.
• Normal space: A topological space which satisfies the \(T_4\) axiom. Here's a bad pun involving this.
• Normal forms: These are a whole class of ways to write matrices of linear operators in a nice form, e.g. Jordan normal form, Smith normal form, etc.
• Normal distribution: The distribution that most sums of random variables converge to, thanks to the Central limit theorem.
• Normal forms part deux: All the normal forms that crop up in formal language theory and computability, e.g. conjunctive normal form, disjunctive normal form, Chomsky normal form, etc.

We started off with 18 different meanings of the word normal, and now, after constructing the equivalence relation, we are left with only 8 different equivalence classes (maybe fewer, if someone discovers some deep result linking normal operators to normal subgroups). That makes one think: maybe it's not so abnormal for mathematicians to overuse normal after all.