Geometric Methods in Automorphic Forms

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9900324GoreskyIn Langlands' program, one stubborn problem has emerged which, so far, hasdefied all attempts at a general proof: the conjecture of Langlandsand Shelstad which is generally known as "the fundamental lemma". In itsoriginal formulation, the fundamental lemma is an equality betweenorbital integrals on two different algebraic groups. In this project,the fundamental lemma is translated into a statement relating the cohomologyof two different pro-algebraic varieties. Techniques are then developedfor computing these cohomology groups in such a way that they may becompared with each other. This procedure should result in a proof ofthe fundamental lemma in the "unramified case", and may well eventuallylead to a proof in the general case.During the last twenty years, mathematicians have directed tremendousefforts towards proving a collection of conjectures, due to R. Langlands,which unify several apparently very different fields: number theory,group representation theory, and automorphic forms. Each of these fieldswould then benefit from being able to import techniques, problems, and results from the other fields. Progress on these conjectures (which were at one time considered almost hopeless)has been spectacular: many of them are completely solved orverified and all of them have been checked in many special cases. Eventhese partial results are extremely important: for example, A. Wiles'recent and celebrated proof of the Fermat conjecture makesessential use of some of these results from "Langlands' program."Researchers are now optimistic that this program may be successfullycompleted during the next decade.