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The Fundamental Lemma is a somewhat obscure combinatorial identity introducedby Robert P. Langlands as an ingredient in the theory of automorphicrepresentations. After many years of deep contributions by mathematiciansworking in representation theory, number theory, algebraic geometry, andalgebraic topology, a proof of the Fundamental Lemma was recently completed byNgo Bao Chau, for which he was awarded a Fields Medal. Our aim here is to touchon some of the beautiful ideas contributing to the Fundamental Lemma and itsproof. We highlight the geometric nature of the problem which allows one toattack a question in p-adic analysis with the tools of algebraic geometry.

The Geometric Nature of the Fundamental Lemma