6 shortered the question; cleaned up the language a bit.

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list hell" big-list" questions open at the same time, I did not ask this one. Now its time has come.

Wikipedia has a good page on several forms of "duality" in mathematics, which outlines several notions of duality (geometric, in convex analysis, topology, set theory, etc.) I am very interested in getting help with the following goal:

Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Also welcome are comments / answers that highlight how a particular notion of duality can be extremely useful (in proving theorems, in applications, for computational reasons, etc.)

I got thinking about this question after reading the following amazing paper: The concept of duality in convex analysis, and the characterization of the Legendre transform, by Shiri Artstein-Avidan and Vitali Milman, where the authors talk about duality in more abstract terms (though, largely in the setting of convex analysis). Motivated by their abstract treatment got me thinking whether such abstract treatments of duality have been investigated for other types of duality, which eventually led to this question.

NOTE Ultimately the aim of the question is to seek "analogies between analogies"

Thus, so I will be grateful to the experts out there who could help me figure out the analogies (beyond the obvious of things being dual) amongst the various answers. I hope this is not too ill-defined an aspiration.

** Update **

Can we create an abstract category theoretic formulation of the concept of duality?

And in line with my citation of with the Avidan-Milman results, maybe one can may also immediately ask similar questions about the other forms types of duality (as in, the said i.e., one tries to characterize why and how a chosen notion of duality being is the only natural duality that can arise given "natural" choice under a set of requirements, such as equality under bi-conjugation, order reversal, etc.)axiomatic requirements).

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list hell" questions open at the same time, I did not ask this one. Now its time has come.

Wikipedia has a good page on several forms of "duality" in mathematics, which outlines several notions of duality (geometric, in convex analysis, topology, set theory, etc.) I am very interested in getting help with the following goal:

Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Also welcome are comments / answers that highlight how a particular notion of duality can be extremely useful (in proving theorems, in applications, for computational reasons, etc.)

I got thinking about this question after reading the following amazing paper: The concept of duality in convex analysis, and the characterization of the Legendre transform, by Shiri Artstein-Avidan and Vitali Milman, where the authors talk about duality in more abstract terms (though, largely in the setting of convex analysis). Motivated by their abstract treatment got me thinking whether such abstract treatments of duality have been investigated for other types of duality, which eventually led to this question.

NOTE Ultimately the aim of the question is to seek "analogies between analogies", so I will be grateful to the experts out there who could help me figure out the analogies (beyond the obvious of things being dual) amongst the various answers. I hope this is not too ill-defined an aspiration.

** Update **