Dualities

Duality is a broad concept in mathematics, physics, and related fields, describing a correspondence between structures that often reverses roles or operations. A duality typically assigns to each object a dual object and to each relation a dual relation, so that statements about one side translate into equivalent statements about the other.

In mathematics, examples include linear programming duality, where every optimization problem has a dual and, under suitable conditions, the optimal values coincide; projective duality, relating points to hyperplanes and curves to dual curves in projective space; and matroid duality, which connects bases via complements. Category theory expresses duality by reversing arrows, yielding dual categories.

In physics, dualities connect seemingly different theories. Electric–magnetic duality exchanges electric and magnetic degrees of freedom in certain models. T-duality in string theory relates physics on circles of radius R and 1/R. S-duality links strong and weak coupling. The AdS/CFT correspondence posits a duality between gravity in anti-de Sitter space and a conformal field theory on its boundary.

In logic and computer science, dualities appear as De Morgan dualities in logic and in dual formulations of automata and semantics, where notions are interchanged by taking duals. These perspectives can simplify proofs and constructions.

Significance and limitations: dualities illuminate hidden symmetries, enable problem transformations, and unify disparate theories. They are not universal; a given framework may have many, one, or no duals, depending on definitions and context.

See also: dual object, AdS/CFT, T-duality, S-duality, projective duality, matroid duality, De Morgan duality.