| Abstract:
The familiar associative law \( (xy)z = x(yz) \) is perhaps one of the
simplest laws (along with its cousin commutative law) everybody encounters
in early mathematics. However, while the proofs or verification of the
commutativity are invariably trivial, the verification of the associativity
will be cumbersome, technical or non-trivial. In this talk we meet some of
the incarnations of the associativity in a variety of situations (with
varying degrees of sophistication) occurring in algebra, analysis and,
notably, geometry. In the process we will go through some of the proofs of
associativity that use sophisticated techniques like the fundamental
functional equation of Cauchy, Birkhoff's theorem on subdirectly
irreducibles, Desargues theorem of projective planes, Pascal's theorem for
conics, the Cayley-Bacharach theorem and the powerful rigidity lemma of
algebraic geometry.
This is an expanded version of a similar talk given in
the Rings and Modules seminar series some 11 years ago as part of a
mini-course on the geometry of non-singular cubic curves
("Associativity Proofs", March 25, 2004
abstract).
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