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The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.

I am aware that there is a related post Suggestions for good notation. The difference is that I would be interested especially in the practical impact of the improved notation, i.e. examples that have actually created a better understanding of a given topic, or have advanced actual research work on a given topic, or communication about results.

I would be interested in two aspects in particular

(1) Visibility and efficiency: Improved and simplified notation that made structures and properties more clearly visible, and that created efficiencies (e.g., using less space and needed less time, dropped unnecessary or redundant details).

(2) Evolving process: Mathematical work is constantly creating new objects, i.e. in constant need of new symbols, or using symbols in new ways. And it needs to create a shared understanding about that. Would you have any practical examples of this evolving process, including dead-ends and breakthroughs?

I would be most interested to learn about practical examples, and would be grateful if you were willing to share.

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    $\begingroup$ I don't completely understand how different this is from the previous question you linked. That already has plenty of answers. $\endgroup$ – Donu Arapura May 21 at 13:58
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    $\begingroup$ @DonuArapura It seems like this one is asking for specific personal examples rather than general suggestions. $\endgroup$ – JoshuaZ May 21 at 14:37
  • $\begingroup$ @DonuArapura thanks a lot for you comment. Indeed, as JoshuaZ says, my key interest is actually in examples that had an actual and practical impact. Maybe let me rephrase my question a bit to make that clearer $\endgroup$ – Claus May 21 at 18:14
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I like to use figures to represent quantities. For example, in this recent preprint, my coauthor and I use simple diagrams to represent certain weighted sums (polynomials). Writing out the sums explicitly would be extremely cumbersome to parse, and any sane reader would just convert it back to a generic figure anyway, in order to understand the sum.

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  • $\begingroup$ @PerAlexaderson thank you for a great example. It is really good $\endgroup$ – Claus May 23 at 21:02
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May I offer the four-vector notation as an example from physics? Quoting Feynman:

The notation for four-vectors is different than it is for three- vectors. [...] We write $p_\mu$ for the four-vector, and $\mu$ stands for the four possible directions $t$, $x$, $y$, or $z$. We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations. The whole idea of a four- vector, in fact, is an improvement in notation so that the transformations can be remembered easily.

The Feynman Lecture on Physics, Volume 1, Chapter 17.

A trigonometric notation that Feynman invented in his youth did not catch on. And then of course Feynman diagrams are perhaps the most celebrated example of an impactful notation in physics.

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    $\begingroup$ thanks a lot for providing a great example and the link. The quote from Feynman is spot on and resonates a lot with me $\endgroup$ – Claus May 21 at 18:16
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Richard Stanley’s symbol for number of ways to make choices with replacement. Looks like a binomial coefficient but with double parentheses. (More here.)

It's a calculation that comes up frequently -- I became more aware of just how frequently it comes up when I started giving it it's own symbol -- and it helps to give it its own notation, even though it reduces to a simple expression in terms of binomial coefficients.

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  • $\begingroup$ Great example. I like this a lot! Thank you $\endgroup$ – Claus May 23 at 20:58
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    $\begingroup$ Please disclose in your posts the fact that you are linking to your own blog. $\endgroup$ – Federico Poloni May 24 at 6:47

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