Terminology introduced in recent years with more than one meaning

Question

Suppose a term(inology) is recently (in last 20 years) introduced in research mathematics.

It might happen that some one who wish to use it, in the same area of research, for different purposes or see from different point of view realize that, some condition needs to be added or removed for their pov/purpose but still calling by the same name. This creates a slight confusion.

What are some term(inology) introduced recently (in last 20 years) which have more than one possible meaning because of different point of view or different purpose?

Answer 1

I don't think this terminological issue is as recent as you ask for, or arises in exactly the way you describe, but let me give the example anyway.

• The index of an algebraic variety $X$ with canonical (Weil) divisor $K_X$ is the smallest natural number $n$ such that $nK_X$ is a Cartier divisor. An example of this usage is in this paper of Fujino.

But also:

• The index of a nonsingular algebraic variety $X$ with canonical (Cartier) divisor $K_X$ is the largest natural number $n$ such that $\frac{1}{n} K_X$ is a Cartier divisor. An example of this usage is in these notes of Debarre.

Alright, the former sense is only of use for singular varieties, while the latter is used in practice more or less only in the context of smooth (Fano) varieties. Still, it makes me scratch my head that the same word applied in two adjacent contexts in algebraic geometry has two essentially opposite meanings.

Answer 2

One example that I've seen is the use of the word "synthetic," which has multiple uses in differential geometry.

1. There is a field called synthetic differential geometry, which studies differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.

2. There is also a field of synthetic differential geometry, mentioned by Matt F, "in a totally different tradition more closely connected to foundations of math and Finsler geometry." In that tradition Herbert Busemann is the founding figure; here are some sample results.

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight from optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces. One good reference is this paper. Another good reference is Villani's survey paper

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. In Matt F's experience, there aren't too many collisions with the second definition because Busemann's overall approach, despite coming earlier, never attracted many followers.

Answer 3

The word “topological stack” has at least three usages:

1. A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack if there is a a morphism of stacks $p: \underline{M}\rightarrow \mathcal{D}$ for some manifold $M$, such that $p$ is a representable epimorphism. This is Definition 2.22, page number 86 in David Carchedi’s thesis.
2. A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack if there is a morphism of stacks $\underline{M}\rightarrow \mathcal{D}$ for a manifold $M$, such that $p$ is representable and has local sections. This is Definition $2.3$, page number 7 in Jochen Heinloth’s Notes on Differentiable stacks.
3. A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack if there is a a morphism of stacks $p: \underline{M}\rightarrow \mathcal{D}$ for some manifold $M$, such that $p$ is a representable epimorphism and that it is a “local fibration”. This is Definition $13.8$, peg number $42$ in Behrang Noohi’s Foundations of topological stacks, I.

There maybe more. Feel free to add if you know more.

Answer 4

My vote is for the phrase "normal Cayley graph".

Recall that a Cayley graph Cay($G$,$C$) is obtained from a group $G$ and a subset of its elements $C \subseteq G$. The vertex set of Cay($G$,$C$) is $G$ itself, and for each $g \in G$ and $c \in C$ there is an edge from $g$ to $gc$.

Some of my colleagues and co-authors say that Cay($G$,$C$) is a normal Cayley graph if $G$ is a normal subgroup of Aut(Cay($G$,$C$)).

Another set of colleagues and co-authors say that Cay($G$,$C$) is a normal Cayley graph if $C$ is closed under conjugation, (so that $C$ is like a normal subset of $G$).

The first usage involves looking outside $G$ while the second usage involves looking inside $G$.

Answer 5

This is not quite a direct conflict of terminology, but it is a confusing near conflict of terminology, and it happened in the past twenty years:

• The generalized permutohedra are a class of convex polytopes introduced and studied by Postnikov in https://arxiv.org/abs/math/0507163; their defining property is that their normal fans are a coarsening of the normal fan of the permutohedron (i.e., the braid arrangement). (In fact, these polytopes had essentially already been studied for many years under the name polymatroids.) One of the most important examples of a generalized permutohedron, beyond the permutohedron itself, is the associahedron (see the title of Postnikov's paper).
• The generalized associahedra are a class of convex polytopes introduced and studied by Fomin and Zelevinsky in https://arxiv.org/abs/hep-th/0111053. They come from the theory of cluster algebras. Specifically, the cluster complex is a simplicial complex that explains how all the clusters in a cluster algebra fit together. The cluster algebras of finite type (i.e., the ones with finite cluster complexes) are in bijection with root systems. The generalized associahedron of a root system is the polytope which is dual to the cluster complex of this root system. This name comes from the fact that in Type A, the generalized associahedron is the usual associahedron.