I have been thinking for sometime about asking this question, but because I did not want to have two "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.)
Some additional context
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.
Thus, in line with the Avidan-Milman results, one may also ask similar questions about other types of duality (i.e., one tries to characterize why and how a chosen notion of duality is the only "natural" choice under a set of axiomatic requirements).
Best Answer
The (1) Fourier transform, (2) mirror symmetry, (3) electric-magnetic duality, and the (4) Pontrjagin and (5) Langlands dualities of Lie groups are all seen to be interrelated by the proposal of Strominger-Yau-Zaslow for mirror symmetry and the work of Kapustin-Witten (foreshadowed by Montonen-Olive) framing the geometric Langlands program in physical terms.