Report on the proof of some conjectures on orbital integrals in Langlands' program (Q1030952)

The paper is a brief survey of the celebrated proof by the author of the so-called fundamental lemma of the Langlands program. In the author's words, he recalls ``the basics about orbital integrals, the natural places in mathematics where we encounter them, the fundamental lemma and the transfer conjectures stated in a precise form only in certain cases''. The author surveys various contributions to the proof of these conjectures, focuses on certain algebraic varieties that play a central role in the understanding of non-Archimedean orbital integrals. The complete proof of the above results exists so far as a preprint of about 200 pages; see \url{http://arxiv.org/abs/0801.0446}. For the case of unitary groups see \textit{G. Laumon} and the author [Ann. Math. (2) 168, No. 2, 477--573 (2008; Zbl 1179.22019)].