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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?

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  • $\begingroup$ This is not just for examples from tagged areas. As number of tags are restricted to 5, I have to choose five, I could only choose 4 :) Examples from other areas are also welcome. $\endgroup$ – Praphulla Koushik Jun 2 at 14:08
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    $\begingroup$ Somewhat related: mathoverflow.net/q/286732/93602 $\endgroup$ – polfosol Jun 3 at 10:24
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    $\begingroup$ I would prefer this without any subject tags (it seems to value at.algebraic-topology above mg.metric-geometry, for no particularly good reason). The tags soft-question and big-list would be appropriate instead. $\endgroup$ – Matt F. Jun 5 at 18:59
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    $\begingroup$ Voting to close as this seems to me like the bad kind of big-list question: limitless potential examples, no clear criteria for the best ones, and no great usefulness in such a big indiscriminate collection of examples. $\endgroup$ – Peter LeFanu Lumsdaine Jun 6 at 5:29
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    $\begingroup$ @PeterLeFanuLumsdaine It is useful to beginning grad students (at least).. Answers to this question are expected to clear some confusion regarding the terminology... I can not defend more.. $\endgroup$ – Praphulla Koushik Jun 6 at 5:33
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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.

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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.

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    $\begingroup$ I thank you for your answer.. I was actually looking for terms which have different meaning but used in same area of research.. As your answer says the word "synthetic" is used in different areas of mathematics... There are many such words. I do not know if they create any confusion as they are in a completely different set up.. Sorry for the unclear question.. I have edited it now... $\endgroup$ – Praphulla Koushik Jun 2 at 16:03
  • $\begingroup$ Fair enough. Just to clarify, both of these usages are in differential geometry, they just have different origins. To be fair, there probably isn't much confusion because there is some distance between them in the literature, as it were. $\endgroup$ – Gabe K Jun 2 at 16:44
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    $\begingroup$ Thanks for taking my comment positively.. :) $\endgroup$ – Praphulla Koushik Jun 2 at 16:47
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    $\begingroup$ @GabeK, there is confusion -- you are talking about Lawvere-style synthetic differential geometry, while there are multiple books by Busemann about synthetic differential geometry in a totally different tradition more closely connected to foundations of math and Finsler geometry. It's not recent, but it's still a problem for the few among us who like both. $\endgroup$ – Matt F. Jun 2 at 17:02
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    $\begingroup$ I was only aware of Lawvere-style synthetic geometry and Villani-style synthetic curvature. I'm somewhat alarmed to find that there is a third definition which I didn't know about. I'll edit the answer to acknowledge your point. $\endgroup$ – Gabe K Jun 2 at 17:13
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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.

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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 $G$.

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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.
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  • $\begingroup$ Thanks for your answer.. I am not sure if this fits in the set up of the question.. These are two different terms, so, I am not able to see the confusion.. I will leave it to the other users to decide if this is ok for the question.. $\endgroup$ – Praphulla Koushik Jun 2 at 16:29
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    $\begingroup$ Probably 'generalised' anywhere in a term is a warning that one should be on the lookout for potential ambiguities, given the number of axes along which one can generalise …. $\endgroup$ – LSpice Jun 2 at 17:19
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    $\begingroup$ @LSpice: yes, I think that is the lesson here. $\endgroup$ – Sam Hopkins Jun 2 at 17:20

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