GRASP Lecture: The Fundamental Lemma

David Ben-Zvi

I will give a gentle overview of the ideas surrounding the Fundamental Lemma and its solution by Ngo Bao-Chau (recently ranked number 7 in Time Magazine's Top 10 Scientific Discoveries of 2009). The Fundamental Lemma is a key ingredient in the Arthur-Selberg Trace Formula and the entire Langlands program, with  deep implications for number theory. Its conjectural status (to quote Langlands) "rendered progress almost impossible for nearly twenty years". Ngo's solution is a stunning application of the analogy between Riemann surfaces and number fields: it revolves around Hitchin's integrable system, a construction in the geometry of bundles on surfaces motivated by physics.