This is a survey on the recent proof of the fundamental lemma. The fundamental lemma and the related transfer conjecture were formulated by R. Langlands in the context of endoscopy theory for automorphic representations in [26]. Important arithmetic applications follow from the endoscopy theory, including the transfer of automorphic representations from classical groups to linear groups and the construction of Galois representations attached to automorphic forms via Shimura varieties. Independent of applications, endoscopy theory is instrumental in building a stable trace formula that seems necessary to any decisive progress toward Langlands’ conjecture on functoriality of automorphic representations.

There are already several expository texts on endoscopy theory and in particular on the fundamental lemma. The original text [26] and articles of Kottwitz [19], [20] are always the best places to learn the theory. The two introductory articles to endoscopy, one by Labesse [24], the other [14] written by Harris for the Book project are highly recommended. So are the reports on the proof of the fundamental lemma in the unitary case written by Dat for Bourbaki [7] and in general written by Dat and Ngô Dac for the Book project [8]. I have also written three expository notes on Hitchin fibration and the fundamental lemma : ["Fibration de Hitchin et structure endoscopique de la formule des traces," International Congress of Mathematicians. Vol. II, 1213–1225, Eur. Math. Soc., Zurich, 2006] reports on endoscopic structure of the cohomology of the Hitchin fibration, [Vietnamese congress of mathematicians (2008)] is a more gentle introduction to the fundamental lemma, and ["Decomposition theorem and abelian fibration"] reports on the support theorem, a key point in the proof of the fundamental lemma written for the Book project. The survey follows the same plan as [Vietnamese congress of mathematicians (2008)] but more details have been added.

This report is written when its author enjoyed the hospitality of the Institute for Advanced Study in Princeton. He acknowledges the support of the Simonyi foundation and the Monell Foundation during his stay in the Institute.

This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper in this volume, The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof. A rigorous account can be found in our preprint The non-existence of elements of Kervaire invariant one on the arXiv and on the third author’s home page. The latter also has numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009.

This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the non-QUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results of Soundararajan completing Lindenstrauss’ QUE result, in the context of matrix elements for Fourier integral operators. Finally, in answer to the potential question “why study matrix elements” it presents an application by the author to the geometry of nodal sets.